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22 .1 Charge and Electric Flux 751
GAUSS’S LAW
?This child acquires anelectric charge bytouching the chargedmetal sphere. Thecharged hairs on thechild’s head repel andstand out. If the childstands inside a large,charged metal sphere,will her hair stand onend?
Often, there are both an easy way and a hard way to do a job; the easy waymay involve nothing more than using the right tools. In physics, an impor-tant tool for simplifying problems is the symmetry properties of systems.
Many physical systems have symmetry; for example, a cylindrical body doesn’tlook any different after you’ve rotated it around its axis, and a charged metalsphere looks just the same after you’ve turned it about any axis through its center.
Gauss’s law is part of the key to using symmetry considerations to simplifyelectric-field calculations. For example, the field of a straight-line or plane-sheetcharge distribution, which we derived in Section 21.5 using some fairly strenuousintegrations, can be obtained in a few lines with the help of Gauss’s law. ButGauss’s law is more than just a way to make certain calculations easier. Indeed, itis a fundamental statement about the relationship between electric charges andelectric fields. Among other things, Gauss’s law can help us understand how elec-tric charge distributes itself over conducting bodies.
Here’s what Gauss’s law is all about. Given any general distribution of charge,we surround it with an imaginary surface that encloses the charge. Then we lookat the electric field at various points on this imaginary surface. Gauss’s law is arelationship between the field at all the points on the surface and the total chargeenclosed within the surface. This may sound like a rather indirect way of express-ing things, but it turns out to be a tremendously useful relationship. Above andbeyond its use as a calculational tool, Gauss’s law can help us gain deeperinsights into electric fields. We will make use of these insights repeatedly in thenext several chapters as we pursue our study of electromagnetism.
22.1 Charge and Electric FluxIn Chapter 21 we asked the question, “Given a charge distribution, what is theelectric field produced by that distribution at a point ?” We saw that the answercould be found by representing the distribution as an assembly of point charges,
P
750
22LEARNING GOALSBy studying this chapter, you will learn:
• How you can determine theamount of charge within a closedsurface by examining the electricfield on the surface.
• What is meant by electric flux, andhow to calculate it.
• How Gauss’s law relates the elec-tric flux through a closed surface tothe charge enclosed by the surface.
• How to use Gauss’s law to calcu-late the electric field due to a sym-metrical charge distribution.
• Where the charge is located on acharged conductor.
The discussion of Gauss’s law in this sec-tion is based on and inspired by the inno-vative ideas of Ruth W. Chabay and BruceA. Sherwood in Electric and MagneticInteractions (John Wiley & Sons, 1994).
each of which produces an electric field given by Eq. (21.7). The total field at is then the vector sum of the fields due to all the point charges.
But there is an alternative relationship between charge distributions and elec-tric fields. To discover this relationship, let’s stand the question of Chapter 21 onits head and ask, “If the electric field pattern is known in a given region, what canwe determine about the charge distribution in that region?”
Here’s an example. Consider the box shown in Fig. 22.1a, which may or maynot contain electric charge. We’ll imagine that the box is made of a material thathas no effect on any electric fields; it’s of the same breed as the massless rope andthe frictionless incline. Better still, let the box represent an imaginary surface thatmay or may not enclose some charge. We’ll refer to the box as a closed surfacebecause it completely encloses a volume. How can you determine how much (ifany) electric charge lies within the box?
Knowing that a charge distribution produces an electric field and that an elec-tric field exerts a force on a test charge, you move a test charge around thevicinity of the box. By measuring the force experienced by the test charge atdifferent positions, you make a three-dimensional map of the electric field
outside the box. In the case shown in Fig. 22.1b, the map turns out tobe the same as that of the electric field produced by a positive point charge(Fig. 21.29a). From the details of the map, you can find the exact value of thepoint charge inside the box.
To determine the contents of the box, we actually need to measure only onthe surface of the box. In Fig. 22.2a there is a single positive point chargeinside the box, and in Fig. 22.2b there are two such charges. The field patternson the surfaces of the boxes are different in detail, but in both cases the electricfield points out of the box. Figures 22.2c and 22.2d show cases with one andtwo negative point charges, respectively, inside the box. Again, the details of
on the surface of the box are different, but in both cases the field points intothe box.ES
ES
ES
5 FS
/q0
FS
q0
PES
(a) Positive charge inside box,outward flux
ES
�q
(b) Positive charges inside box,outward flux
ES
�q�q
(d) Negative charges inside box,inward flux
ES
2q
2q
(c) Negative charge inside box,inward flux
ES
�q
22.2 The electric field on the surface of boxes containing (a) a single positive pointcharge, (b) two positive point charges, (c) a single negative point charge, or (d) two nega-tive point charges.
?
q
ES
ES
ES
ES
ES
ES
ES
ES
ES
Test charge q0
�
(b) Using a test charge outside the box to probethe amount of charge inside the box
(a) A box containing an unknown amount ofcharge
22.1 How can you measure the chargeinside a box without opening it?
11.7 Electric Flux
O N L I N E
22 .2 Calculating Electric Flux 753752 C HAPTE R 22 Gauss’s Law
Electric Flux and Enclosed ChargeIn Section 21.4 we mentioned the analogy between electric-field vectors and thevelocity vectors of a fluid in motion. This analogy can be helpful, even though anelectric field does not actually “flow.” Using this analogy, in Figs. 22.2a and22.2b, in which the electric field vectors point out of the surface, we say thatthere is an outward electric flux. (The word “flux” comes from a Latin wordmeaning “flow.”) In Figs. 22.2c and 22.2d the vectors point into the surface,and the electric flux is inward.
Figure 22.2 suggests a simple relationship: Positive charge inside the box goeswith an outward electric flux through the box’s surface, and negative charge insidegoes with an inward electric flux. What happens if there is zero charge inside thebox? In Fig. 22.3a the box is empty and everywhere, so there is no electricflux into or out of the box. In Fig. 22.3b, one positive and one negative pointcharge of equal magnitude are enclosed within the box, so the net charge insidethe box is zero. There is an electric field, but it “flows into” the box on half of itssurface and “flows out of” the box on the other half. Hence there is no net electricflux into or out of the box.
The box is again empty in Fig. 22.3c. However, there is charge present outsidethe box; the box has been placed with one end parallel to a uniformly chargedinfinite sheet, which produces a uniform electric field perpendicular to the sheet(as we learned in Example 21.12 of Section 21.5). On one end of the box, points into the box; on the opposite end, points out of the box; and on the sides,
is parallel to the surface and so points neither into nor out of the box. As inFig. 22.3b, the inward electric flux on one part of the box exactly compensatesfor the outward electric flux on the other part. So in all of the cases shown inFig. 22.3, there is no net electric flux through the surface of the box, and no netcharge is enclosed in the box.
Figures 22.2 and 22.3 demonstrate a connection between the sign (positive,negative, or zero) of the net charge enclosed by a closed surface and the sense(outward, inward, or none) of the net electric flux through the surface. There isalso a connection between the magnitude of the net charge inside the closedsurface and the strength of the net “flow” of over the surface. In bothFigs. 22.4a and 22.4b there is a single point charge inside the box, but inFig. 22.4b the magnitude of the charge is twice as great, and so is every-where twice as great in magnitude as in Fig. 22.4a. If we keep in mind thefluid-flow analogy, this means that the net outward electric flux is also twice asgreat in Fig. 22.4b as in Fig. 22.4a. This suggests that the net electric fluxthrough the surface of the box is directly proportional to the magnitude of thenet charge enclosed by the box.
ES
ES
ES
ES
ES
ES
5 0
ES
(a) No charge inside box,zero flux
E 5 0S
ES
(b) Zero net charge inside box,inward flux cancels outward flux.
(c) No charge inside box,inward flux cancels outward flux.
Uniformlychargedsheet
ES
1q
1s
2q
22.3 Three cases in which there is zero net charge inside a box and no net electric flux through the surface of the box. (a) An emptybox with (b) A box containing one positive and one equal-magnitude negative point charge. (c) An empty box immersed in auniform electric field.
ES
5 0.
Test Your Understanding of Section 22.1 If all of the dimensions of thebox in Fig. 22.2a are increased by a factor of 3, what effect will this change have onthe electric flux through the box? (i) The flux will be times greater; (ii) theflux will be 3 times greater; (iii) the flux will be unchanged; (iv) the flux will be asgreat; (v) the flux will be as great; (vi) not enough information is given to decide.
❚1 1
3 22 5 1
9
1 13 2
32 5 9
This conclusion is independent of the size of the box. In Fig. 22.4c the pointcharge is enclosed by a box with twice the linear dimensions of the box inFig.22.4a. The magnitude of the electric field of a point charge decreases withdistance according to so the average magnitude of on each face of thelarge box in Fig. 22.4c is just of the average magnitude on the correspondingface in Fig. 22.4a. But each face of the large box has exactly four times the areaof the corresponding face of the small box. Hence the outward electric flux is thesame for the two boxes if we define electric flux as follows: For each face of thebox, take the product of the average perpendicular component of and the areaof that face; then add up the results from all faces of the box. With this definitionthe net electric flux due to a single point charge inside the box is independent ofthe size of the box and depends only on the net charge inside the box.
We have seen that there is a relationship between the net amount of chargeinside a closed surface and the electric flux through that surface. For the specialcases of a closed surface in the shape of a rectangular box and charge distribu-tions made up of point charges or infinite charged sheets, we have found:
1. Whether there is a net outward or inward electric flux through a closed sur-face depends on the sign of the enclosed charge.
2. Charges outside the surface do not give a net electric flux through the surface.
3. The net electric flux is directly proportional to the net amount of chargeenclosed within the surface but is otherwise independent of the size of theclosed surface.
These observations are a qualitative statement of Gauss’s law.Do these observations hold true for other kinds of charge distributions and for
closed surfaces of arbitrary shape? The answer to these questions will prove to beyes. But to explain why this is so, we need a precise mathematical statement ofwhat we mean by electric flux. This is developed in the next section.
ES
14
ES
1/r 2,
1q
22.2 Calculating Electric FluxIn the preceding section we introduced the concept of electric flux. Qualitatively,the electric flux through a surface is a description of whether the electric field points into or out of the surface. We used this to give a rough qualitative state-ment of Gauss’s law: The net electric flux through a closed surface is directlyproportional to the net charge inside that surface. To be able to make full use ofthis law, we need to know how to calculate electric flux. To do this, let’s againmake use of the analogy between an electric field and the field of velocity vec-tors in a flowing fluid. (Again, keep in mind that this is only an analogy; anelectric field is not a flow.)
Flux: Fluid-Flow AnalogyFigure 22.5 shows a fluid flowing steadily from left to right. Let’s examine thevolume flow rate (in, say, cubic meters per second) through the wire rec-tangle with area When the area is perpendicular to the flow velocity (Fig. 22.5a) and the flow velocity is the same at all points in the fluid, the volumeflow rate is the area multiplied by the flow speed
dV
dt5 vA
v:AdV/dt
vSA.dV/dt
vSES
ES
1q
(a) A box containing a charge
ES
12q
(b) Doubling the enclosed chargedoubles the flux.
ES
(c) Doubling the box dimensionsdoes not change the flux.
ES
1q
22.4 (a) A box enclosing a positive pointcharge (b) Doubling the charge causesthe magnitude of to double, and it dou-bles the electric flux through the surface.(c) If the charge stays the same but thedimensions of the box are doubled, the fluxstays the same. The magnitude of on thesurface decreases by a factor of but thearea through which “flows” increases bya factor of 4.
ES
14,
ES
ES
1q.
22 .2 Calculating Electric Flux 755754 C HAPTE R 22 Gauss’s Law
When the rectangle is tilted at an angle (Fig. 22.5b) so that its face is not per-pendicular to the area that counts is the silhouette area that we see when welook in the direction of This area, which is outlined in red and labeled inFig. 22.5b, is the projection of the area onto a surface perpendicular to Twosides of the projected rectangle have the same length as the original one, but theother two are foreshortened by a factor of so the projected area is equalto Then the volume flow rate through is
If the wire rectangle is edge-on to the flow, and no fluidpasses through the rectangle.
Also, is the component of the vector perpendicular to the plane of thearea Calling this component we can rewrite the volume flow rate as
We can express the volume flow rate more compactly by using the concept ofvector area a vector quantity with magnitude and a direction perpendicularto the plane of the area we are describing. The vector area describes both thesize of an area and its orientation in space. In terms of we can write the volumeflow rate of fluid through the rectangle in Fig. 22.5b as a scalar (dot) product:
Flux of a Uniform Electric FieldUsing the analogy between electric field and fluid flow, we now define electricflux in the same way as we have just defined the volume flow rate of a fluid; wesimply replace the fluid velocity by the electric field The symbol that we usefor electric flux is (the capital Greek letter phi; the subscript is a reminderthat this is electric flux). Consider first a flat area perpendicular to a uniformelectric field (Fig. 22.6a). We define the electric flux through this area to be theproduct of the field magnitude and the area
Roughly speaking, we can picture in terms of the field lines passing throughIncreasing the area means that more lines of pass through the area, increas-
ing the flux; stronger field means more closely spaced lines of and thereforemore lines per unit area, so again the flux increases.
If the area is flat but not perpendicular to the field then fewer field linespass through it. In this case the area that counts is the silhouette area that we seewhen looking in the direction of This is the area in Fig. 22.6b and is equalto (compare to Fig. 22.5b). We generalize our definition of electric fluxfor a uniform electric field to
(electric flux for uniform flat surface) (22.1)
Since is the component of perpendicular to the area, we can rewrite
Eq. (22.1) as
(22.2)
In terms of the vector area perpendicular to the area, we can write the elec-tric flux as the scalar product of and
(22.3)FE 5 ES
# AS
(electric flux for uniform ES
, flat surface)
AS
:ES
AS
FE 5 E' A (electric flux for uniform ES
, flat surface)
ES
E cos f
ES
,FE 5 EA cos f
A cos fA'E
S
.
ES
,A
ES
ES
A.FE
FE 5 EA
A:EES
AEFE
ES
.vS
dV
dt5 vS # A
S
AS
,AS
AAS
,
dV
dt5 v' A
v' ,A.vSv cos f
dV/dt 5 0;f 5 90°,
dV
dt5 vA cos f
AA cos f.A'cos f,
vS.AA'vS.
vS,f
(a) A wire rectangle in a fluid
AvS
(b) The wire rectangle tilted by an angle f
AA' 5 A cos f
f
AS vS
22.5 The volume flow rate of fluidthrough the wire rectangle (a) is whenthe area of the rectangle is perpendicularto and (b) is when the rectangleis tilted at an angle f.
vA cos fvS
vA
S S S
S
S S
A
E
A
f 5 0
S
S
(a) Surface is face-on to electric field:• E and A are parallel (the angle between E and A is f 5 0).• The flux FE 5 E • A 5 EA.
AA�
E
A
ff
S
S
(b) Surface is tilted from a face-onorientation by an angle f:• The angle between E and A is f.• The flux FE 5 E • A 5 EA cos f.
S S
S S
A
E
f 5 90°
AS
S
(c) Surface is edge-on to electric field:• E and A are perpendicular (the angle between E and A is f 5 90°).• The flux FE 5 E • A 5 EA cos 90° 5 0.
S S
SS
S S
22.6 A flat surface in a uniform electric field. The electric flux through the surface equals the scalar product of the electric field and the area vector A
S
.ES
FE
Equations (22.1), (22.2), and (22.3) express the electric flux for a flat surface anda uniform electric field in different but equivalent ways. The SI unit for electricflux is Note that if the area is edge-on to the field, and are per-pendicular and the flux is zero (Fig. 22.6c).
We can represent the direction of a vector area by using a unit vector per-pendicular to the area; stands for “normal.” Then
(22.4)
A surface has two sides, so there are two possible directions for and Wemust always specify which direction we choose. In Section 22.1 we related thecharge inside a closed surface to the electric flux through the surface. With aclosed surface we will always choose the direction of to be outward, and wewill speak of the flux out of a closed surface. Thus what we called “outward elec-tric flux” in Section 22.1 corresponds to a positive value of and what wecalled “inward electric flux” corresponds to a negative value of
Flux of a Nonuniform Electric FieldWhat happens if the electric field isn’t uniform but varies from point to pointover the area ? Or what if is part of a curved surface? Then we divide intomany small elements each of which has a unit vector perpendicular to itand a vector area We calculate the electric flux through each elementand integrate the results to obtain the total flux:
(22.5)
We call this integral the surface integral of the component over the area, orthe surface integral of The various forms of the integral all express thesame thing in different terms. In specific problems, one form is sometimes moreconvenient than another. Example 22.3 at the end of this section illustrates theuse of Eq. (22.5).
In Eq. (22.5) the electric flux is equal to the average value of the per-pendicular component of the electric field, multiplied by the area of the surface.This is the same definition of electric flux that we were led to in Section 22.1,now expressed more mathematically. In the next section we will see the connec-tion between the total electric flux through any closed surface, no matter what itsshape, and the amount of charge enclosed within that surface.
∫E' dA
ES
# dAS
.E'
(general definition of electric flux)FE 5 3 E cos f dA 5 3 E' dA 5 3 E
S
# dAS
dAS
5 n dA.ndA,
AAAES
FE .FE ,
n
AS
.n
AS
5 An
nnA
S
AS
ES
1 N # m2/C.
22 .3 Gauss’s Law 757756 C HAPTE R 22 Gauss’s Law
Example 22.1 Electric flux through a disk
A disk with radius 0.10 m is oriented with its normal unit vector at an angle of to a uniform electric field with magnitude
(Fig. 22.7). (Since this isn’t a closed surface, it hasno “inside” or “outside.” That’s why we have to specify the direc-tion of in the figure.) (a) What is the electric flux through thedisk? (b) What is the flux through the disk if it is turned so that itsnormal is perpendicular to (c) What is the flux through the diskif its normal is parallel to
SOLUTION
IDENTIFY: This problem is about a flat surface in a uniform elec-tric field, so we can apply the ideas of this section.
SET UP: The orientation of the disk is like that of the rectangle inFig. 22.6b. We calculate the electric flux using Eq. (22.1).
EXECUTE: (a) The area is and theangle between and is so
(b) The normal to the disk is now perpendicular to so and There is no flux through the disk.FE 5 0.cos f 5 0,90°,
f 5ES
,
5 54 N # m2/C FE 5 EA cos f 5 12.0 3 103 N/C 2 10.0314 m2 2 1 cos 30° 2
f 5 30°,AS
5 AnES
A 5 p 10.10 m 2 2 5 0.0314 m2
ES
?ES
?
n
2.0 3 103 N/CES
30°n (c) The normal to the disk is parallel to so
and the flux has its maximum possible value. From Eq. (22.1),
EVALUATE: As a check on our results, note that the answer topart (a) is smaller than the answer to part (c). Is this as it should be?
5 63 N # m2/C FE 5 EA cos f 5 12.0 3 103 N/C 2 10.0314 m2 2 11 2
cos f 5 1,f 5 0,ES
,
30°
r 5 0.10 m n
ES
22.7 The electric flux through a disk depends on the anglebetween its normal and the electric field E
S
.nFE
Example 22.2 Electric flux through a cube
A cube of side is placed in a region of uniform electric field Find the electric flux through each face of the cube and the totalflux through the cube when (a) it is oriented with two of its facesperpendicular to the field as in Fig. 22.8a; and (b) when thecube is turned by an angle as in Fig. 22.8b.
SOLUTION
IDENTIFY: In this problem we are to find the electric flux througheach face of the cube as well as the total flux (the sum of the fluxesthrough the six faces).
SET UP: Since is uniform and each of the six faces of the cube isa flat surface, we find the flux through each face using Eqs. (22.3)and (22.4). We then calculate the total flux through the cube byadding the six individual fluxes.
EXECUTE: (a) The unit vectors for each face ( through ) areshown in the figure; the direction of each unit vector is outwardfrom the closed surface of the cube. The angle between and is
the angle between and is and the angle between and each of the other four unit vectors is Each face of the cubehas area so the fluxes through each of the faces are
The flux is negative on face 1, where is directed into the cube, andpositive on face 2, where is directed out of the cube. The total fluxthrough the cube is the sum of the fluxes through the six faces:
5 2EL2 1 EL2 1 0 1 0 1 0 1 0 5 0
FE 5 FE1 1 FE2 1 FE3 1 FE4 1 FE5 1 FE6
ES
ES
FE3 5 FE4 5 FE5 5 FE6 5 EL2 cos 90° 5 0
FE2 5 ES # n2 A 5 EL2
cos 0° 5 1EL2
FE1 5 ES # n1 A 5 EL2
cos 180° 5 2EL2
L2,90°.
ES
0°;n2ES
180°;n1E
S
n6n1
ES
u,ES
,
ES
.L
(b) The fluxes through faces 1 and 3 are negative, since isdirected into those faces; the field is directed out of faces 2 and 4,so the fluxes through those faces are positive. We find
The total flux through the surface of the cube is again zero.
EVALUATE: It’s no surprise that the total flux is zero for both ori-entations. We came to this same conclusion in our discussion ofFig. 22.3c in Section 22.1. There we observed that there was zeronet flux of a uniform electric field through a closed surface thatcontains no electric charge.
FE 5 FE1 1 FE2 1 FE3 1 FE4 1 FE5 1 FE6
FE5 5 FE6 5 EL2 cos 90° 5 0
FE4 5 ES # n4
A 5 EL2 cos 190° 2 u 2 5 1EL2 sin u
FE3 5 ES # n3
A 5 EL2
cos 190° 1 u 2 5 2EL2 sin u
FE2 5 ES # n2
A 5 1EL2
cos u
FE1 5 ES # n1 A 5 EL2
cos 1180° 2 u 2 5 2EL2 cos u
ES
(a)
Er
n5n3
n2
n6
n1
n4
(b)
90° 2 uu
Er
n5n3 n2
n6
n1 n4
22.8 Electric flux of a uniform field through a cubical box ofside in two orientations.L
ES
Example 22.3 Electric flux through a sphere
A positive point charge is surrounded by a sphere withradius 0.20 m centered on the charge (Fig. 22.9). Find the electricflux through the sphere due to this charge.
SOLUTION
IDENTIFY: Here the surface is not flat and the electric field is notuniform, so we must use the general definition of electric flux.
SET UP: We use Eq. (22.5) to calculate the electric flux (our targetvariable). Because the sphere is centered on the point charge, at anypoint on the spherical surface, is directed out of the sphere per-pendicular to the surface. The positive direction for both andis outward, so and the flux through a surface element is
This greatly simplifies the integral in Eq. (22.5).
EXECUTE: At any point on the sphere the magnitude of is
Because is the same at every point, it can be taken outside theintegral in Eq. (22.5). What remains is the integral
which is just the total area of the spherical surface.Thus the total flux out of the sphere is
5 3.4 3 105 N # m2/C FE 5 EA 5 16.75 3 105 N/C 2 14p 2 1 0.20 m 2 2
A 5 4pr 2∫dA,FE 5 ∫E dAE
5 6.75 3 105 N/C
E 5q
4pP0 r 2 5 19.0 3 109 N # m2/C2 2
3.0 3 1026 C
10.20 m 2 2
ES
ES
# dAS
5 E dA.dAE' 5 E
E'nES
q 5 3.0 mC
EVALUATE: Notice that we divided by to find then multiplied by to find hence the radius of the sphere cancels out of the result for We would haveobtained the same flux with a sphere of radius 2.0 m or 200 m. Wecame to essentially the same conclusion in our discussion ofFig. 22.4 in Section 22.1, where we considered rectangular closedsurfaces of two different sizes enclosing a point charge. There wefound that the flux of was independent of the size of the surface;the same result holds true for a spherical surface. Indeed, the fluxthrough any surface enclosing a single point charge is independentof the shape or size of the surface, as we’ll soon see.
ES
FE .rFE ;r 2 5 10.20 m 2 2
E,r 2 5 10.20 m 2 2
ES
q r
dAS
22.9 Electric flux through a sphere centered on a point charge.
22.10 Carl Friedrich Gauss helpeddevelop several branches of mathematics,including differential geometry, real analy-sis, and number theory. The “bell curve” ofstatistics is one of his inventions. Gaussalso made state-of-the-art investigations ofthe earth’s magnetism and calculated theorbit of the first asteroid to be discovered.
Test Your Understanding of Section 22.2 Rank the following surfaces inorder from most positive to most negative electric flux. (i) a flat rectangular surfacewith vector area in a uniform electric field (ii) a flatcircular surface with vector area in a uniform electric field
(iii) a flat square surface with vector area in auniform electric field (iv) a flat oval surface with vector area
in a uniform electric field❚
ES
5 14.0 N/C 2 d 2 12.0 N/C 2e.AS
5 13.0 m2 2 d 2 17.0 m2 2eES
5 14.0 N/C 2 d 2 12.0 N/C2e;AS
5 13.0 m2 2 d 1 17.0 m2 2e12.0 N/C2e;ES
5 14.0 N/C2 d 1AS
5 13.0 m2 2eES
5 14.0 N/C2e;AS
5 16.0 m2 2 d
22.3 Gauss’s LawGauss’s law is an alternative to Coulomb’s law. While completely equivalent toCoulomb’s law, Gauss’s law provides a different way to express the relationshipbetween electric charge and electric field. It was formulated by Carl FriedrichGauss (1777–1855), one of the greatest mathematicians of all time. Many areasof mathematics bear the mark of his influence, and he made equally significantcontributions to theoretical physics (Fig. 22.10).
Point Charge Inside a Spherical SurfaceGauss’s law states that the total electric flux through any closed surface (a surfaceenclosing a definite volume) is proportional to the total (net) electric charge insidethe surface. In Section 22.1 we observed this relationship qualitatively for certainspecial cases; now we’ll develop it more rigorously. We’ll start with the field of asingle positive point charge The field lines radiate out equally in all directions.We place this charge at the center of an imaginary spherical surface with radiusThe magnitude of the electric field at every point on the surface is given by
E 51
4pP0
q
R2
ER.
q.
22 .3 Gauss’s Law 759758 C HAPTE R 22 Gauss’s Law
At each point on the surface, is perpendicular to the surface, and its magnitudeis the same at every point, just as in Example 22.3 (Section 22.2). The total elec-tric flux is the product of the field magnitude and the total area ofthe sphere:
(22.6)
The flux is independent of the radius R of the sphere. It depends only on thecharge enclosed by the sphere.
We can also interpret this result in terms of field lines. Figure 22.11 shows twospheres with radii and centered on the point charge Every field line thatpasses through the smaller sphere also passes through the larger sphere, so thetotal flux through each sphere is the same.
What is true of the entire sphere is also true of any portion of its surface. InFig.22.11 an area is outlined on the sphere of radius and then projected ontothe sphere of radius by drawing lines from the center through points on theboundary of The area projected on the larger sphere is clearly 4 But sincethe electric field due to a point charge is inversely proportional to the fieldmagnitude is as great on the sphere of radius as on the sphere of radius Hence the electric flux is the same for both areas and is independent of the radiusof the sphere.
Point Charge Inside a Nonspherical SurfaceThis projection technique shows us how to extend this discussion to nonsphericalsurfaces. Instead of a second sphere, let us surround the sphere of radius by asurface of irregular shape, as in Fig. 22.12a. Consider a small element of areaon the irregular surface; we note that this area is larger than the corresponding ele-ment on a spherical surface at the same distance from If a normal to makesan angle with a radial line from two sides of the area projected onto the spher-ical surface are foreshortened by a factor (Fig. 22.12b). The other two sidesare unchanged. Thus the electric flux through the spherical surface element isequal to the flux through the corresponding irregular surface element.
We can divide the entire irregular surface into elements compute the elec-tric flux for each, and sum the results by integrating, as in Eq. (22.5).Each of the area elements projects onto a corresponding spherical surface ele-ment. Thus the total electric flux through the irregular surface, given by any ofthe forms of Eq. (22.5), must be the same as the total flux through a sphere,which Eq.(22.6) shows is equal to Thus, for the irregular surface,
(22.7)FE 5 C ES # dA
S
5qP0
q/P0 .
E dA cos f
dA,E dA cos f
cos fq,f
dAq.
dAR
R.2R14
r 2,dA.dA.
2RRdA
q.2RR
q
FE 5 EA 51
4pP0
q
R2 1 4pR2 2 5qP0
A 5 4pR2E
ES
2
The same number of field lines and the sameflux pass through both of these area elements.
R
4 dA
dA
q
R
ES
22.11 Projection of an element of area of a sphere of radius onto a concentricsphere of radius The projection multi-plies each linear dimension by 2, so thearea element on the larger sphere is 4 dA.
2R.R
dA
(a) The outward normal to the surface makes an angle f with the direction of E.
dA
E'
dA cos f
(b)
f
f ES
S
qR
r
q
E'
The projection of thearea element dA ontothe spherical surfaceis dA cos f.
dA
f ES
22.12 Calculating the electric fluxthrough a nonspherical surface.
Equation (22.7) holds for a surface of any shape or size, provided only that it is aclosed surface enclosing the charge The circle on the integral sign reminds usthat the integral is always taken over a closed surface.
The area elements and the corresponding unit vectors always point out ofthe volume enclosed by the surface. The electric flux is then positive in areaswhere the electric field points out of the surface and negative where it pointsinward. Also, is positive at points where points out of the surface and nega-tive at points where points into the surface.
If the point charge in Fig. 22.12 is negative, the field is directed radiallyinward; the angle is then greater than its cosine is negative, and the inte-gral in Eq. (22.7) is negative. But since is also negative, Eq. (22.7) still holds.
For a closed surface enclosing no charge,
This is the mathematical statement that when a region contains no charge, anyfield lines caused by charges outside the region that enter on one side must leaveagain on the other side. (In Section 22.1 we came to the same conclusion by con-sidering the special case of a rectangular box in a uniform field.) Figure 22.13illustrates this point. Electric field lines can begin or end inside a region of spaceonly when there is charge in that region.
General Form of Gauss’s LawNow comes the final step in obtaining the general form of Gauss’s law. Supposethe surface encloses not just one point charge but several charges
The total (resultant) electric field at any point is the vector sum ofthe fields of the individual charges. Let be the total charge enclosed bythe surface: Also let be the total field at the posi-tion of the surface area element and let be its component perpendicular tothe plane of that element (that is, parallel to ). Then we can write an equationlike Eq. (22.7) for each charge and its corresponding field and add the results.When we do, we obtain the general statement of Gauss’s law:
(Gauss’s law) (22.8)
The total electric flux through a closed surface is equal to the total (net) electriccharge inside the surface, divided by
CAUTION Gaussian surfaces are imaginary Remember that the closed surface inGauss’s law is imaginary; there need not be any material object at the position of the sur-face. We often refer to a closed surface used in Gauss’s law as a Gaussian surface. ❚
Using the definition of and the various ways to express electric flux givenin Eq. (22.5), we can express Gauss’s law in the following equivalent forms:
(22.9)
As in Eq. (22.5), the various forms of the integral all express the same thing, thetotal electric flux through the Gaussian surface, in different terms. One form issometimes more convenient than another.
As an example, Fig. 22.14a shows a spherical Gaussian surface of radiusaround a positive point charge The electric field points out of the Gaussian sur-face, so at every point on the surface is in the same direction as andf 5 0,dA
S
,ES
1q.r
(various forms of Gauss’s law)
FE 5 CE cos f dA 5 CE' dA 5 CES # dA
S
5Qencl
P0
Qencl
`0 .
FE 5 C ES # dA
S
5Qencl
P0
dAS
E'dAS
,ES
Qencl 5 q1 1 q2 1 q3 1 c .QenclE
S
ES
q3 , c.q2 ,q1 ,q
FE 5 C ES # dA
S
5 0
q90°,f
ES
ES
ES
E'
ndAS
q.
ES
Field lineentering surface Same field line
leaving surface
22.13 A point charge outside a closed sur-face that encloses no charge. If an electricfield line from the external charge entersthe surface at one point, it must leave atanother.
22 .4 Applications of Gauss’s Law 761760 C HAPTE R 22 Gauss’s Law
is equal to the field magnitude Since is the same at all pointson the surface, we can take it outside the integral in Eq. (22.9). Then the remainingintegral is the area of the sphere. Hence Eq. (22.9) becomes
The enclosed charge is just the charge so this agrees with Gauss’s law.If the Gaussian surface encloses a negative point charge as in Fig. 22.14b, then
points into the surface at each point in the direction opposite Thenand is equal to the negative of the field magnitude:
Equation (22.9) then becomes
This again agrees with Gauss’s law because the enclosed charge in Fig. 22.14b is
In Eqs. (22.8) and (22.9), is always the algebraic sum of all the positiveand negative charges enclosed by the Gaussian surface, and is the total field ateach point on the surface. Also note that in general, this field is caused partly bycharges inside the surface and partly by charges outside. But as Fig. 22.13 shows,the outside charges do not contribute to the total (net) flux through the surface. SoEqs. (22.8) and (22.9) are correct even when there are charges outside the surfacethat contribute to the electric field at the surface. When the total fluxthrough the Gaussian surface must be zero, even though some areas may havepositive flux and others may have negative flux (see Fig. 22.3b).
Gauss’s law is the definitive answer to the question we posed at the beginningof Section 22.1: “If the electric field pattern is known in a given region, what canwe determine about the charge distribution in that region?” It provides a relation-ship between the electric field on a closed surface and the charge distributionwithin that surface. But in some cases we can use Gauss’s law to answer thereverse question: “If the charge distribution is known, what can we determineabout the electric field that the charge distribution produces?” Gauss’s law mayseem like an unappealing way to address this question, since it may look asthough evaluating the integral in Eq. (22.8) is a hopeless task. Sometimes it is,but other times it is surprisingly easy. Here’s an example in which no integrationis involved at all; we’ll work out several more examples in the next section.
Qencl 5 0,
ES
Qencl
Qencl 5 2q.
FE 5 CE' dA 5 C 1 2q
4pP0 r 2 2 dA 52q
4pP0 r 2 CdA 52q
4pP0 r 2 4pr 2 52qP0
2 02q 0 /4pP0 r 2 5 2q/4pP0 r 2.E' 5 2E 5E'180°
f 5dAS
.ES
1q,Qencl
FE 5 CE' dA 5 C
q
4pP0 r 2 dA 5q
4pP0 r 2 CdA 5q
4pP0 r 2 4pr 2 5qP0
∫dA 5 A 5 4pr 2,
EE 5 q/4pP0 r 2.E'
(a) Gaussian surface around positive charge:positive (outward) flux
(b) Gaussian surface around negative charge:negative (inward) flux
1qr
dA
ES
S
2qr
dA
ES
S
22.14 Spherical Gaussian surfaces around(a) a positive point charge and (b) a nega-tive point charge.
22.4 Applications of Gauss’s LawGauss’s law is valid for any distribution of charges and for any closed surface.Gauss’s law can be used in two ways. If we know the charge distribution, and if ithas enough symmetry to let us evaluate the integral in Gauss’s law, we can findthe field. Or if we know the field, we can use Gauss’s law to find the charge dis-tribution, such as charges on conducting surfaces.
In this section we present examples of both kinds of applications. As youstudy them, watch for the role played by the symmetry properties of each system.We will use Gauss’s law to calculate the electric fields caused by several simplecharge distributions; the results are collected in a table in the chapter summary.
In practical problems we often encounter situations in which we want to knowthe electric field caused by a charge distribution on a conductor. These calcula-tions are aided by the following remarkable fact: When excess charge is placedon a solid conductor and is at rest, it resides entirely on the surface, not in theinterior of the material. (By excess we mean charges other than the ions and freeelectrons that make up the neutral conductor.) Here’s the proof. We know fromSection 21.4 that in an electrostatic situation (with all charges at rest) the electricfield at every point in the interior of a conducting material is zero. If werenot zero, the excess charges would move. Suppose we construct a Gaussian sur-face inside the conductor, such as surface in Fig. 22.17. Because every-where on this surface, Gauss’s law requires that the net charge inside the surfaceis zero. Now imagine shrinking the surface like a collapsing balloon until itencloses a region so small that we may consider it as a point then the charge atthat point must be zero. We can do this anywhere inside the conductor, so therecan be no excess charge at any point within a solid conductor; any excess chargemust reside on the conductor’s surface. (This result is for a solid conductor. Inthe next section we’ll discuss what can happen if the conductor has cavities in itsinterior.) We will make use of this fact frequently in the examples that follow.
P;
ES
5 0A
ES
ES
Conceptual Example 22.4 Electric flux and enclosed charge
Figure 22.15 shows the field produced by two point charges and of equal magnitude but opposite sign (an electric dipole).Find the electric flux through each of the closed surfaces and
SOLUTION
The definition of electric flux given in Eq. (22.5) involves a sur-face integral, and so it might seem that integration is called for. ButGauss’s law says that the total electric flux through a closed sur-face is equal to the total enclosed charge divided by By inspec-tion of Fig. 22.15, surface (shown in red) encloses the positivecharge, so surface (shown in blue) encloses the neg-ative charge, so surface (shown in yellow), whichencloses both charges, has and surface
(shown in purple), which has no charges enclosed within it, alsohas Hence without having to do any integration, we canQencl 5 0.D
Qencl 5 1q 1 12q 2 5 0;CQencl 5 2q;
BQencl 5 1q;A
P0 .
D.C,B,A,
2q1q
conclude that the total fluxes for the various surfaces are for surface for surface and for
both surface and surface These results depend only on the charges enclosed within each
Gaussian surface, not on the precise shapes of the surfaces. Forexample, compare surface to the rectangular surface shown inFig. 22.3b, which also encloses both charges of an electric dipole.In that case as well, we concluded that the net flux of was zero;the inward flux on one part of the surface exactly compensates forthe outward flux on the remainder of the surface.
We can draw similar conclusions by examining the electric fieldlines. Surface encloses only the positive charge; in Fig. 22.15,A
ES
C
D.CFE 5 0B,FE 5 2q/P0A,1q/P0
FE 5 18 lines are depicted crossing in an outward direction. Surfaceencloses only the negative charge; it is crossed by these same18 lines, but in an inward direction. Surface encloses bothcharges. It is intersected by lines at 16 points; at 8 intersections thelines are outward, and at 8 they are inward. The net number of linescrossing in an outward direction is zero, and the net charge insidethe surface is also zero. Surface is intersected at 6 points; at3 points the lines are outward, and at the other 3 they are inward.The net number of lines crossing in an outward direction and thetotal charge enclosed are both zero. There are points on the sur-faces where is not perpendicular to the surface, but this doesn’taffect the counting of the field lines.
ES
D
C
BA
C
DB A
2q
ES
1q
22.15 The net number of field lines leaving a closed surface isproportional to the total charge enclosed by that surface.
Test Your Understanding of Section 22.3 Figure 22.16 shows six pointcharges that all lie in the same plane. Five Gaussian surfaces— and
—each enclose part of this plane, and Fig. 22.16 shows the intersection of each surfacewith the plane. Rank these five surfaces in order of the electric flux through them, frommost positive to most negative.
❚
S5
S1 , S2 , S3 , S4 ,
11.0 mC
19.0 mC
S1S2
S4
S3S5
210.0 mC
27.0 mC
15.0 mC
18.0 mC
22.16 Five Gaussian surfaces and sixpoint charges.
Conductor(shown in
cross section)
Charge on surfaceof conductor
Gaussian surface Ainside conductor
(shown incross section)
22.17 Under electrostatic conditions(charges not in motion), any excess chargeon a solid conductor resides entirely on theconductor’s surface.
22 .4 Applications of Gauss’s Law 763762 C HAPTE R 22 Gauss’s Law
Problem-Solving Strategy 22.1 Gauss’s Law
IDENTIFY the relevant concepts: Gauss’s law is most useful in sit-uations where the charge distribution has spherical or cylindricalsymmetry or is distributed uniformly over a plane. In these situa-tions we determine the direction of from the symmetry of thecharge distribution. If we are given the charge distribution, we canuse Gauss’s law to find the magnitude of Alternatively, if we aregiven the field, we can use Gauss’s law to determine the details ofthe charge distribution. In either case, begin your analysis by ask-ing the question: What is the symmetry?
SET UP the problem using the following steps:1. Select the surface that you will use with Gauss’s law. We often
call it a Gaussian surface. If you are trying to find the field ata particular point, then that point must lie on your Gaussiansurface.
2. The Gaussian surface does not have to be a real physical sur-face, such as a surface of a solid body. Often the appropriatesurface is an imaginary geometric surface; it may be in emptyspace, embedded in a solid body, or both.
3. Usually you can evaluate the integral in Gauss’s law (withoutusing a computer) only if the Gaussian surface and the chargedistribution have some symmetry property. If the charge distri-bution has cylindrical or spherical symmetry, choose the Gauss-ian surface to be a coaxial cylinder or a concentric sphere,respectively.
EXECUTE the solution as follows:1. Carry out the integral in Eq. (22.9). This may look like a daunt-
ing task, but the symmetry of the charge distribution and yourcareful choice of a Gaussian surface make it straightforward.
ES
.
ES
2. Often you can think of the closed Gaussian surface as beingmade up of several separate surfaces, such as the sides and endsof a cylinder. The integral over the entire closed surfaceis always equal to the sum of the integrals over all the separatesurfaces. Some of these integrals may be zero, as in points 4and 5 below.
3. If is perpendicular (normal) at every point to a surface witharea if it points outward from the interior of the surface, andif it also has the same magnitude at every point on the surface,then and over that surface isequal to If instead is perpendicular and inward, then
and 4. If is tangent to a surface at every point, then and the
integral over that surface is zero.5. If at every point on a surface, the integral is zero.6. In the integral is always the perpendicular compo-
nent of the total electric field at each point on the closed Gauss-ian surface. In general, this field may be caused partly bycharges within the surface and partly by charges outside it. Evenwhen there is no charge within the surface, the field at points onthe Gaussian surface is not necessarily zero. In that case, how-ever, the integral over the Gaussian surface—that is, the totalelectric flux through the Gaussian surface—is always zero.
7. Once you have evaluated the integral, use Eq. (22.9) to solvefor your target variable.
EVALUATE your answer: Often your result will be a function thatdescribes how the magnitude of the electric field varies with posi-tion. Examine this function with a critical eye to see whether itmakes sense.
E'rE' dA,ES
5 0
E' 5 0ES
∫E' dA 5 2EA.E' 5 2EES
EA.∫ E' dAE' 5 E 5 constant,
A,ES
rE' dA
Example 22.5 Field of a charged conducting sphere
We place positive charge on a solid conducting sphere withradius (Fig. 22.18). Find at any point inside or outside thesphere.
SOLUTION
IDENTIFY: As we discussed earlier in this section, all the chargemust be on the surface of the sphere. The system has sphericalsymmetry.
SET UP: To take advantage of the symmetry, we take as ourGaussian surface an imaginary sphere of radius centered on theconductor. To calculate the field outside the conductor, we take tobe greater than the conductor’s radius to calculate the fieldinside, we take to be less than In either case, the point wherewe want to calculate lies on the Gaussian surface.
EXECUTE: The role of symmetry deserves careful discussionbefore we do any calculations. When we say that the system isspherically symmetric, we mean that if we rotate it through anyangle about any axis through the center, the system after rotation isindistinguishable from the original unrotated system. The charge isfree to move on the conductor, and there is nothing about the con-
ES
R.rR;
rr
ES
Rq
ductor that would make it tend to concentrate more in someregions than others. So we conclude that the charge is distributeduniformly over the surface.
Symmetry also shows that the direction of the electric fieldmust be radial, as shown in Fig. 22.18. If we again rotate the sys-tem, the field pattern of the rotated system must be identical to thatof the original system. If the field had a component at some pointthat was perpendicular to the radial direction, that componentwould have to be different after at least some rotations. Thus therecan’t be such a component, and the field must be radial. For thesame reason the magnitude of the field can depend only on thedistance from the center and must have the same value at allpoints on a spherical surface concentric with the conductor.
Our choice of a sphere as a Gaussian surface takes advantage ofthese symmetry properties. We first consider the field outside theconductor, so we choose The entire conductor is within theGaussian surface, so the enclosed charge is The area of theGaussian surface is is uniform over the surface and per-pendicular to it at each point. The flux integral in Gauss’slaw is therefore just and Eq. (22.8) gives
This expression for the field at any point outside the sphereis the same as for a point charge; the field due to the
charged sphere is the same as though the entire charge were con-centrated at its center. Just outside the surface of the sphere, where
(at the surface of a charged conducting sphere)
E 51
4pP0
q
R2
r 5 R,
1 r . R 2
E 51
4pP0 q
r 2 (outside a charged conducting sphere)
E 14pr 2 2 5q
P0 and
E 14pr 2 2 ,rE' dA
ES
4pr 2;q.
r . R.
rE
CAUTION Flux can be positive or negative Remember thatwe have chosen the charge to be positive. If the charge is nega-tive, the electric field is radially inward instead of radially out-ward, and the electric flux through the Gaussian surface isnegative. The electric field magnitudes outside and at the surfaceof the sphere are given by the same expressions as above, exceptthat denotes the magnitude (absolute value) of the charge. ❚
To find inside the conductor, we use a spherical Gaussiansurface with radius The spherical symmetry again tells usthat But because all of the charge is on thesurface of the conductor, our Gaussian surface (which lies entirelywithin the conductor) encloses no charge. So and, there-fore, the electric field inside the conductor is zero.
EVALUATE: We already knew that inside the conductor, asit must be inside any solid conductor when the charges are at rest.Figure 22.18 shows as a function of the distance from the cen-ter of the sphere. Note that in the limit as the spherebecomes a point charge; there is then only an “outside,” and thefield is everywhere given by Thus we havededuced Coulomb’s law from Gauss’s law. (In Section 22.3 wededuced Gauss’s law from Coulomb’s law, so this completes thedemonstration of their logical equivalence.)
We can also use this method for a conducting spherical shell (aspherical conductor with a concentric spherical hole in the center)if there is no charge inside the hole. We use a spherical Gaussiansurface with radius less than the radius of the hole. If there were afield inside the hole, it would have to be radial and sphericallysymmetric as before, so But now there is noenclosed charge, so and inside the hole.
Can you use this same technique to find the electric field in theinterspace between a charged sphere and a concentric hollow con-ducting sphere that surrounds it?
E 5 0Qencl 5 0E 5 Qencl/4pP0 r 2.
r
E 5 q/4pP0 r 2.
R S 0,rE
ES
5 0
Qencl 5 0
E 14pr 2 2 5 Qencl/P0 .r , R.
ES
q
q
++ ++
++
++ ++
+
Outside the sphere, the magnitudeof the electric field decreases withthe square of the radial distancefrom the center of the sphere:
Inside the sphere, theelectric field is zero:E 5 0.
E
Or
R 2R 3R
E 1R 2/4
Gaussian surfacesat r 5 2R and r 5 3R
E 1R 2/9
E 1R 2 5
R
E 5 qr2
qR2
14pP0
14pP0
22.18 Calculating the electric field of a conducting sphere withpositive charge q. Outside the sphere, the field is the same as if allof the charge were concentrated at the center of the sphere.
Example 22.6 Field of a line charge
Electric charge is distributed uniformly along an infinitely long,thin wire. The charge per unit length is (assumed positive). Findthe electric field. (This is an approximate representation of the fieldof a uniformly charged finite wire, provided that the distance fromthe field point to the wire is much less than the length of the wire.)
SOLUTION
IDENTIFY: The system has cylindrical symmetry. The field mustpoint away from the positive charges. To determine the direction of
more precisely, as well as how its magnitude can depend onposition, we use symmetry as in Example 22.5.
SET UP: Cylindrical symmetry means that we can rotate the sys-tem through any angle about its axis, and we can shift it by anyamount along the axis; in each case the resulting system is indistin-guishable from the original. Hence at each point can’t changewhen either of these operations is carried out. The field can’t haveany component parallel to the wire; if it did, we would have toexplain why the field lines that begin on the wire pointed in onedirection parallel to the wire and not the other. Also, the field can’thave any component tangent to a circle in a plane perpendicular tothe wire with its center on the wire. If it did, we would have toexplain why the component pointed in one direction around the
ES
ES
lwire rather than the other. All that’s left is a component radiallyoutward from the wire at each point. So the field lines outside auniformly charged, infinite wire are radial and lie in planes per-pendicular to the wire. The field magnitude can depend only on theradial distance from the wire.
These symmetry properties suggest that we use as a Gaussiansurface a cylinder with arbitrary radius and arbitrary length with its ends perpendicular to the wire (Fig. 22.19).
l,r
GaussianE' 5 E
dAS
surface
l
E' 5 0
r
22.19 A coaxial cylindrical Gaussian surface is used to find theelectric field outside an infinitely long, charged wire.
Continued
22 .4 Applications of Gauss’s Law 765764 C HAPTE R 22 Gauss’s Law
expression for the field magnitude we must interpret as themagnitude (absolute value) of the charge per unit length.
EVALUATE: Note that although the entire charge on the wire con-tributes to the field, only the part of the total charge that is withinthe Gaussian surface is considered when we apply Gauss’s law.This may seem strange; it looks as though we have somehowobtained the right answer by ignoring part of the charge and thefield of a short wire of length would be the same as that of a verylong wire. But we do include the entire charge on the wire whenwe make use of the symmetry of the problem. If the wire is short,the symmetry with respect to shifts along the axis is not present,and the field is not uniform in magnitude over our Gaussian sur-face. Gauss’s law is then no longer useful and cannot be used tofind the field; the problem is best handled by the integration tech-nique used in Example 21.11.
We can use a Gaussian surface like that in Fig. 22.19 to showthat the field at points outside a long, uniformly charged cylinder isthe same as though all the charge were concentrated on a linealong its axis. We can also calculate the electric field in the spacebetween a charged cylinder and a coaxial hollow conducting cylin-der surrounding it. We leave these calculations to you (see Prob-lems 22.37 and 22.40).
l
lEEXECUTE: We break the surface integral for the flux into anintegral over each flat end and one over the curved side walls.There is no flux through the ends because lies in the plane of thesurface and To find the flux through the side walls, notethat is perpendicular to the surface at each point, so bysymmetry, has the same value everywhere on the walls. The areaof the side walls is (To make a paper cylinder with radius and height you need a paper rectangle with width height and area ) Hence the total flux through the entire cylinderis the sum of the flux through the side walls, which is and the zero flux through the two ends. Finally, we need the totalenclosed charge, which is the charge per unit length multiplied bythe length of wire inside the Gaussian surface, or FromGauss’s law, Eq. (22.8),
This is the same result that we found in Example 21.11 (Sec-tion 21.5) by much more laborious means.
We have assumed that is positive. If it is negative, isdirected radially inward toward the line of charge, and in the above
ES
l
E 51
2pP0 l
r (field of an infinite line of charge)
FE 5 1E 2 1 2prl 2 5llP0 and
Qencl 5 ll.
1E 2 12prl 2 ,FE2prl.
l,2pr,l,r2prl.
EE 5 E' ;E
S
E' 5 0.ES
FE
Example 22.7 Field of an infinite plane sheet of charge
Find the electric field caused by a thin, flat, infinite sheet on whichthere is a uniform positive charge per unit area
SOLUTION
IDENTIFY: The field must point away from the positively chargedsheet. As in Examples 22.5 and 22.6, before doing calculations weuse the symmetry (in this case, planar symmetry) to learn moreabout the direction and position dependence of
SET UP: Planar symmetry means that the charge distribution does-n’t change if we slide it in any direction parallel to the sheet. Fromthis we conclude that at each point, is perpendicular to the sheet.The symmetry also tells us that the field must have the same mag-nitude at any given distance on either side of the sheet. To takeE
ES
ES
.
s.advantage of these symmetry properties, we use as our Gaussiansurface a cylinder with its axis perpendicular to the sheet ofcharge, with ends of area (Fig. 22.20).
EXECUTE: The charged sheet passes through the middle of thecylinder’s length, so the cylinder ends are equidistant from thesheet. At each end of the cylinder, is perpendicular to the surfaceand is equal to hence the flux through each end is
Because is perpendicular to the charged sheet, it is parallel tothe curved side walls of the cylinder, so at these walls is zeroand there is no flux through these walls. The total flux integral inGauss’s law is then ( from each end and zero from the sidewalls). The net charge within the Gaussian surface is the chargeper unit area multiplied by the sheet area enclosed by the surface,or Hence Gauss’s law, Eq. (22.8), gives
This is the same result that we found in Example 21.12 (Sec-tion 21.5) using a much more complex calculation. The field isuniform and directed perpendicular to the plane of the sheet. Itsmagnitude is independent of the distance from the sheet. The fieldlines are therefore straight, parallel to each other, and perpendicu-lar to the sheet.
If the charge density is negative, is directed toward the sheet,the flux through the Gaussian surface in Fig. 22.20 is negative, and
in the expression denotes the magnitude (absolutevalue) of the charge density.
E 5 s/2P0s
ES
E 5s
2P0 (field of an infinite sheet of charge)
2EA 5sAP0
and
Qencl 5 sA.
EA2EA
E'
ES
1EA.E;E'
ES
A
++
++
+
++
++ +
++
++
+
++
++
+
++
++
+
++
++
+Gaussiansurface
E' 5 E
E
A
22.20 A cylindrical Gaussian surface is used to find the field ofan infinite plane sheet of charge.
Example 22.8 Field between oppositely charged parallel conducting plates
Two large plane parallel conducting plates are given charges ofequal magnitude and opposite sign; the charge per unit area is for one and for the other. Find the electric field in the regionbetween the plates.
SOLUTION
IDENTIFY: The field between and around the plates is approxi-mately as shown in Fig. 22.21a. Because opposite charges attract,most of the charge accumulates at the opposing faces of the plates.A small amount of charge resides on the outer surfaces of theplates, and there is some spreading or “fringing” of the field at theedges. But if the plates are very large in comparison to the distancebetween them, the amount of charge on the outer surfaces is negligi-bly small, and the fringing can be neglected except near the edges.In this case we can assume that the field is uniform in the interiorregion between plates, as in Fig. 22.21b, and that the charges aredistributed uniformly over the opposing surfaces.
SET UP: To exploit this symmetry, we can use the shaded Gauss-ian surfaces and These surfaces are cylinders withends of area like the one shown in perspective in Fig. 22.20; theyare shown in a side view in Fig. 22.21b. One end of each surfacelies within one of the conducting plates.
EXECUTE: For the surface labeled the left-hand end is withinplate 1 (the positive plate). Since the field is zero within the vol-ume of any solid conductor under electrostatic conditions, there isno electric flux through this end. The electric field between theplates is perpendicular to the right-hand end, so on that end, isE'
S1 ,
AS4 .S2 , S3 ,S1 ,
2s1s
equal to and the flux is this is positive, since is directedout of the Gaussian surface. There is no flux through the side wallsof the cylinder, since these walls are parallel to So the total fluxintegral in Gauss’s law is The net charge enclosed by thecylinder is so Eq. (22.8) yields
(field between oppositely charged conducting plates)
The field is uniform and perpendicular to the plates, and its magni-tude is independent of the distance from either plate. This sameresult can be obtained by using the Gaussian surface further-more, the surfaces and can be used to show that to theleft of plate 1 and to the right of plate 2. We leave these calcula-tions to you (see Exercise 22.27).
EVALUATE: We obtained the same results in Example 21.13(Section 21.5) by using the principle of superposition of electricfields. The fields due to the two sheets of charge (one on eachplate) are and from Example 22.7, both of these have mag-nitude The total (resultant) electric field at any point is thevector sum At points and in Fig. 22.21b, and
have opposite directions, and their resultant is zero. This is alsotrue at every point within the material of each plate, consistentwith the requirement that with charges at rest there can be no fieldwithin a solid conductor. At any point between the plates, and
have the same direction; their resultant has magnitudejust as we found above using Gauss’s law.E 5 s/P0 ,
ES
2
ES
1b
ES
2
ES
1caES
5 ES
1 1 ES
2 .s/2P0 .
ES
2 ;ES
1
E 5 0S3S2
S4 ;
EA 5sAP0
and E 5s
P0
sA,EA.
ES
.
ES
EA;E
+
–
+
+
+
+
+
+
+
+
–
–
–
–
–
–
–
–
–
–
+
+
+
+
–
–
+++
+ ++
+ ++
+ ++
+ ++
+ +++
+
–––
–––––––––
–
–
–
––
–
– –
In the idealized casewe ignore “fringing”at the plate edges andtreat the field betweenthe plates as uniform.
Cylindrical Gaussiansurfaces (seen fromthe side)
ES
Between the twoplates the electric fieldis nearly uniform,pointing from thepositive plate towardthe negative one.
ab
c
1 2
S1
S4S2
S3
E1S
E1r
E2S
E2r
E2S
ES
E1S
(a) Realistic drawing (b) Idealized model
22.21 Electric field between oppositely charged parallel plates.
EVALUATE: The assumption that the sheet is infinitely large is anidealization; nothing in nature is really infinitely large. But theresult is a good approximation for points that are closeE 5 s/2P0
to the sheet (compared to the sheet’s dimensions) and not too nearits edges. At such points, the field is very nearly uniform and per-pendicular to the plane.
22 .5 Charges on Conductors 767766 C HAPTE R 22 Gauss’s Law
++ ++
++
++ ++
+++ +
++
+ +++
E
rR
Rr
O
Gaussiansurface
Spherical insulatorr
E(R) 5
Q
r 2E 5 1
4pP0
Q
R21
4pP0
E 5 Qr
R31
4pP0
22.22 The magnitude of the electric field of a uniformly chargedinsulating sphere. Compare this with the field for a conductingsphere (Fig. 22.18).
Example 22.10 Field of a hollow charged sphere
A thin-walled, hollow sphere of radius 0.250 m has an unknownamount of charge distributed uniformly over its surface. At a dis-tance of 0.300 m from the center of the sphere, the electric fieldpoints directly toward the center of the sphere and has magnitude
How much charge is on the sphere?1.80 3 102 N/C.
SOLUTION
IDENTIFY: The charge distribution is spherically symmetric. Asin Examples 22.5 and 22.9, it follows that the electric field is radialeverywhere and its magnitude is a function only of the radial dis-tance from the center of the sphere.r
SET UP: We again use a spherical Gaussian surface that is concen-tric with the charge distribution and that passes through the pointof interest at
EXECUTE: The charge distribution is the same as if the chargewere on the surface of a 0.250-m-radius conducting sphere. Hencewe can borrow the results of Example 22.5. A key difference fromthat example is that because the electric field here is directedtoward the sphere, the charge must be negative. Furthermore,because the electric field is directed into the Gaussian surface,
and the flux is By Gauss’s law, the flux is equal to the charge on the sphere
(all of which is enclosed by the Gaussian surface) divided by Solving for we findq,
P0 .q
rE' dA 5 2E 14pr 2 2 .E' 5 2E
r 5 0.300 m.
EVALUATE: To determine the charge, we had to know the electricfield at all points on the Gaussian surface so that we could calcu-late the flux integral. This was possible here because the chargedistribution is highly symmetric. If the charge distribution is irreg-ular or lacks symmetry, however, Gauss’s law is not very useful forcalculating the charge distribution from the field, or vice versa.
5 28.01 3 10210 C 5 20.801 nC
3 18.854 3 10212 C2/N # m2 2 10.300 m 2 2
q 5 2E 14pP0 r 2 2 5 2 11.80 3 102 N/C 2 14p 2
Test Your Understanding of Section 22.4 You place a known amount of chargeon the irregularly shaped conductor shown in Fig. 22.17. If you know the size and
shape of the conductor, can you use Gauss’s law to calculate the electric field at an arbi-trary position outside the conductor?
❚
Q
22.5 Charges on ConductorsWe have learned that in an electrostatic situation (in which there is no net motionof charge) the electric field at every point within a conductor is zero and that anyexcess charge on a solid conductor is located entirely on its surface (Fig. 22.23a).But what if there is a cavity inside the conductor (Fig. 22.23b)? If there is nocharge within the cavity, we can use a Gaussian surface such as (which liescompletely within the material of the conductor) to show that the net charge on thesurface of the cavity must be zero, because everywhere on the Gaussiansurface. In fact, we can prove in this situation that there can’t be any chargeanywhere on the cavity surface. We will postpone detailed proof of this statementuntil Chapter 23.
Suppose we place a small body with a charge inside a cavity within a con-ductor (Fig. 22.23c). The conductor is uncharged and is insulated from thecharge Again everywhere on surface so according to Gauss’s law thetotal charge inside this surface must be zero. Therefore there must be a charge
distributed on the surface of the cavity, drawn there by the charge insidethe cavity. The total charge on the conductor must remain zero, so a chargemust appear either on its outer surface or inside the material. But we showed inSection 22.4 that in an electrostatic situation there can’t be any excess chargewithin the material of a conductor. So we conclude that the charge mustappear on the outer surface. By the same reasoning, if the conductor originallyhad a charge then the total charge on the outer surface must be afterthe charge is inserted into the cavity.q
qC 1 qqC ,
1q
1qq2q
A,ES
5 0q.
q
ES
5 0
A
(a) Solid conductor with charge qC
qC
The charge qC resides entirely on the surface ofthe conductor. The situation is electrostatic, soE 5 0 within the conductor.
E 5 0 withinconductor
S
S
(b) The same conductor with an internal cavity
qC ArbitraryGaussiansurface A
Cavity
Because E 5 0 at all points within the conductor,the electric field at all points on the Gaussiansurface must be zero.
S
qC 1 q
(c) An isolated charge q placed in the cavity
–
–––
––– –
q
–––––––
–
For E to be zero at all points on the Gaussiansurface, the surface of the cavity must have atotal charge 2q.
S
22.23 Finding the electric field within a charged conductor.
Example 22.9 Field of a uniformly charged sphere
Positive electric charge is distributed uniformly throughout thevolume of an insulating sphere with radius Find the magnitudeof the electric field at a point a distance from the center of thesphere.
SOLUTION
IDENTIFY: As in Example 22.5, the system is spherically sym-metric. Hence we can use the conclusions of that example aboutthe direction and magnitude of
SET UP: To make use of the symmetry, we choose as our Gauss-ian surface a sphere with radius concentric with the chargedistribution.
EXECUTE: From symmetry the magnitude of the electric fieldhas the same value at every point on the Gaussian surface, and thedirection of is radial at every point on the surface, so Hence the total electric flux through the Gaussian surface is theproduct of and the total area of the surface that is,
The amount of charge enclosed within the Gaussian surfacedepends on the radius Let’s first find the field magnitude insidethe charged sphere of radius the magnitude is evaluated at theradius of the Gaussian surface, so we choose The volumecharge density is the charge divided by the volume of theentire charged sphere of radius
r 5Q
4pR3/3
R:Qr
r , R.ER;
r.
FE 5 4pr 2E.A 5 4pr 2,E
E' 5 E.ES
E
r,
ES
.
rPR.
Q The volume enclosed by the Gaussian surface is so thetotal charge enclosed by that surface is
Then Gauss’s law, Eq. (22.8), becomes
The field magnitude is proportional to the distance of the fieldpoint from the center of the sphere. At the center
To find the field magnitude outside the charged sphere, we usea spherical Gaussian surface of radius This surface enclosesthe entire charged sphere, so and Gauss’s law gives
(field outside a uniformly charged sphere)
For any spherically symmetric charged body the electric fieldoutside the body is the same as though the entire charge wereconcentrated at the center. (We made this same observation inExample 22.5.)
Figure 22.22 shows a graph of as a function of for thisproblem. For is directly proportional to and for
varies as If the charge is negative instead of positive, isradially inward and in the expressions for is interpreted as themagnitude (absolute value) of the charge.
EVALUATE: Notice that if we set in either of the two expres-sions for (inside or outside the sphere), we get the same result
for the magnitude of the field at the surface of thesphere. This is because the magnitude is a continuous function of
By contrast, for the charged conducting sphere of Example 22.5the electric-field magnitude is discontinuous at (it jumpsfrom just inside the sphere to just outsidethe sphere). In general, the electric field is discontinuous in mag-nitude, direction, or both wherever there is a sheet of charge, suchas at the surface of a charged conducting sphere (Example 22.5), atthe surface of an infinite charged sheet (Example 22.7), or at thesurface of a charged conducting plate (Example 22.8).
The general technique used in this example can be applied toany spherically symmetric distribution of charge, whether it is uni-form or not. Such charge distributions occur within many atomsand atomic nuclei, which is why Gauss’s law is a useful tool inatomic and nuclear physics.
ES
E 5 Q/4pP0 R2E 5 0r 5 R
r.E
E 5 Q/4pP0 R2E
r 5 R
EQES
1/r 2.Er . R,r,Er , R,
rE
E 51
4pP0 Q
r 2
4pr 2E 5Q
P0 or
Qencl 5 Q,r . R.
E 5 0.1 r 5 0 2 ,r
E 51
4pP0 Qr
R3 (field inside a uniformly charged sphere)
4pr 2E 5Q
P0
r 3
R3 or
Qencl 5 rVencl 5 1 Q
4pR3/3 2 14
3 pr 3 2 5 Q
r 3
R3
Qencl
43 pr 3,Vencl
22 .5 Charges on Conductors 769768 C HAPTE R 22 Gauss’s Law
+ ++ ++ ++ +
Chargedconductingball
Insulatingthread
+
(a)
Insulatingstand
Metalcontainer
––––––
–
––––
––
–
– – – –
– –– –
+ ++ ++ ++ +
++++++
++
++++
++
++
+ + + +
+ +
+
Charged ball induces charges on theinterior and exterior of the container.
(b) Metal lid
++++++
++++
++
+++ + + + +
+ + +++
+ + +
Once the ball touches the container, itis part of the interior surface; all thecharge moves to the container’s exterior.
(c) Metal lid
22.25 (a) A charged conducting ball suspended by an insulating thread outside a conducting container on an insulating stand. (b) Theball is lowered into the container, and the lid is put on. (c) The ball is touched to the inner surface of the container.
Conceptual Example 22.11 A conductor with a cavity
A solid conductor with a cavity carries a total charge of Within the cavity, insulated from the conductor, is a point chargeof How much charge is on each surface (inner and outer)of the conductor?
SOLUTION
Figure 22.24 shows the situation. If the charge in the cavity isthe charge on the inner cavity surface must be
The conductor carries a total charge ofnone of which is in the interior of the material. If is
on the inner surface of the cavity, then there must be on the outer surface of the conductor.115 nC 2 5 12 nC
117 nC 2 215 nC17 nC,
2 125 nC 2 5 15 nC.2q 5q 5 25 nC,
25 nC.
17 nC. 22.24 Our sketch for this problem. There is zero electric fieldinside the bulk conductor and hence zero flux through the Gauss-ian surface shown, so the charge on the cavity wall must be theopposite of the point charge.
Testing Gauss’s Law ExperimentallyWe can now consider a historic experiment, shown in Fig. 22.25. We mount aconducting container, such as a metal pail with a lid, on an insulating stand. Thecontainer is initially uncharged. Then we hang a charged metal ball from an insu-lating thread (Fig. 22.25a), lower it into the pail, and put the lid on (Fig. 22.25b).Charges are induced on the walls of the container, as shown. But now we let theball touch the inner wall (Fig. 22.25c). The surface of the ball becomes, in effect,part of the cavity surface. The situation is now the same as Fig. 22.23b; if Gauss’slaw is correct, the net charge on the cavity surface must be zero. Thus the ballmust lose all its charge. Finally, we pull the ball out; we find that it has indeedlost all its charge.
This experiment was performed in the 19th century by the English scientistMichael Faraday, using a metal icepail with a lid, and it is called Faraday’s ice-pail experiment. (Similar experiments were carried out in the 18th century byBenjamin Franklin in America and Joseph Priestley in England, although withmuch less precision.) The result confirms the validity of Gauss’s law and there-fore of Coulomb’s law. Faraday’s result was significant because Coulomb’sexperimental method, using a torsion balance and dividing of charges, was notvery precise; it is very difficult to confirm the dependence of the electrostaticforce with great precision by direct force measurements. By contrast, experi-ments like Faraday’s test the validity of Gauss’s law, and therefore of Coulomb’slaw, with much greater precision.
1/r 2
A modern version of Faraday’s experiment is shown in Fig. 22.26. The detailsof the box labeled “Power supply” aren’t important; its job is to place charge onthe outer sphere and remove it, on demand. The inner box with a dial is a sensi-tive electrometer, an instrument that can detect motion of extremely smallamounts of charge between the outer and inner spheres. If Gauss’s law is correct,there can never be any charge on the inner surface of the outer sphere. If so, thereshould be no flow of charge between spheres while the outer sphere is beingcharged and discharged. The fact that no flow is actually observed is a very sensi-tive confirmation of Gauss’s law and therefore of Coulomb’s law. The precisionof the experiment is limited mainly by the electrometer, which can be astonish-ingly sensitive. Experiments have shown that the exponent 2 in the ofCoulomb’s law does not differ from precisely 2 by more than So there isno reason to suspect that it is anything other than exactly 2.
The same principle behind Faraday’s icepail experiment is used in a Van deGraaff electrostatic generator (Fig. 22.27). The charged conducting sphere ofFig. 22.26 is replaced by a charged belt that continuously carries charge to theinside of a conducting shell, only to have it carried away to the outside surface ofthe shell. As a result, the charge on the shell and the electric field around it canbecome very large very rapidly. The Van de Graaff generator is used as an accel-erator of charged particles and for physics demonstrations.
This principle also forms the basis for electrostatic shielding. Suppose wehave a very sensitive electronic instrument that we want to protect from strayelectric fields that might cause erroneous measurements. We surround the instru-ment with a conducting box, or we line the walls, floor, and ceiling of the room witha conducting material such as sheet copper. The external electric field redistributesthe free electrons in the conductor, leaving a net positive charge on the outer
10216.1/r 2
Conductingshell
Motorfor belt
Electronsink
Insulatingsupport
Insulatingbelt
–
–
–
–
–
–
–
–––––
–––
+
+ +
+
+
++
+
++
+ ++ +
+ +
+
+
+
+
+
+
+
22.27 Cutaway view of the essential parts of a Van de Graaff electrostatic generator. Theelectron sink at the bottom draws electrons from the belt, giving it a positive charge; atthe top the belt attracts electrons away from the conducting shell, giving the shell a posi-tive charge.
E
Powersupply
0 +–
22.26 The outer spherical shell can bealternately charged and discharged by thepower supply. If there were any flow ofcharge between the inner and outer shells,it would be detected by the electrometerinside the inner shell.
?
22 .5 Charges on Conductors 771770 C HAPTE R 22 Gauss’s Law
surface in some regions and a net negative charge in others (Fig. 22.28). Thischarge distribution causes an additional electric field such that the total field atevery point inside the box is zero, as Gauss’s law says it must be. The charge dis-tribution on the box also alters the shapes of the field lines near the box, as the fig-ure shows. Such a setup is often called a Faraday cage. The same physics tells youthat one of the safest places to be in a lightning storm is inside an automobile; if thecar is struck by lightning, the charge tends to remain on the metal skin of the vehi-cle, and little or no electric field is produced inside the passenger compartment.
Field at the Surface of a ConductorFinally, we note that there is a direct relationship between the field at a pointjust outside any conductor and the surface charge density at that point. In gen-eral, varies from point to point on the surface. We will show in Chapter 23 thatat any such point, the direction of is always perpendicular to the surface (seeFig. 22.28a).
To find a relationship between at any point on the surface and the perpendi-cular component of the electric field at that point, we construct a Gaussian sur-face in the form of a small cylinder (Fig. 22.29). One end face, with area lieswithin the conductor and the other lies just outside. The electric field is zero at allpoints within the conductor. Outside the conductor the component of perpendi-cular to the side walls of the cylinder is zero, and over the end face the perpendi-cular component is equal to (If is positive, the electric field points out ofthe conductor and is positive; if is negative, the field points inward and is negative.) Hence the total flux through the surface is The charge enclosedwithin the Gaussian surface is so from Gauss’s law,
(22.10)
We can check this with the results we have obtained for spherical, cylindrical,and plane surfaces.
We showed in Example 22.8 that the field magnitude between two infinite flatoppositely charged conducting plates also equals In this case the field mag-nitude is the same at all distances from the plates, but in all other cases itdecreases with increasing distance from the surface.
s/P0 .
(field at the surfaceof a conductor)E' A 5
sAP0
and E' 5s
P0
sA,E' A.
E'sE'
sE' .
ES
A,
s
ES
ss
ES
++++ +++++
Outersurface ofchargedconductor
Gaussiansurface
E� � E
E� � 0
E � 0A
A
22.29 The field just outside a charged con-ductor is perpendicular to the surface, andits perpendicular component is equal tos/P0 .
E'
Conceptual Example 22.12 Field at the surface of a conducting sphere
Verify Eq. (22.10) for a conducting sphere with radius and totalcharge
SOLUTION
In Example 22.5 (Section 22.4) we showed that the electric fieldjust outside the surface is
E 51
4pP0
q
R2
q.R The surface charge density is uniform and equal to divided by
the surface area of the sphere:
Comparing these two expressions, we see that asEq. (22.10) states.
E 5 s/P0 ,
s 5q
4pR2
q
Example 22.13 Electric field of the earth
The earth (a conductor) has a net electric charge. The resulting elec-tric field near the surface can be measured with sensitive electronicinstruments; its average value is about directed towardthe center of the earth. (a) What is the corresponding surface chargedensity? (b) What is the total surface charge of the earth?
SOLUTION
IDENTIFY: We are given the electric field magnitude at the sur-face of the conducting earth, and we are asked to calculate the sur-face charge density and the total charge on the entire surface of theearth.
SET UP: Given the perpendicular electric field, we determine thesurface charge density using Eq. (22.10). The total surface chargeon the earth is then the product of and the earth’s surface area.
EXECUTE: (a) We know from the direction of the field that isnegative (corresponding to being directed into the surface, so is negative). From Eq. (22.10),
5 21.33 3 1029 C/m2 5 21.33 nC/m2
s 5 P0 E' 5 18.85 3 10212 C2/N # m2 2 12150 N/C 2
E'ES
s
ss
150 N/C,
(b) The earth’s surface area is whereis the radius of the earth (see Appendix F). The total charge is theproduct or
EVALUATE: You can check our result in part (b) using the result ofExample 22.5. Solving for we find
One electron has a charge of Hence this muchexcess negative electric charge corresponds to there being
excess elec-trons on the earth, or about 7 moles of excess electrons. This iscompensated by an equal deficiency of electrons in the earth’supper atmosphere, so the combination of the earth and its atmos-phere is electrically neutral.
126.8 3 105 C 2 / 121.60 3 10219 C 2 5 4.2 3 1024
21.60 3 10219 C.
5 26.8 3 105 C
51
9.0 3 109 N # m2/C2 16.38 3 106 m 2 2 12150 N/C 2
Q 5 4pP0 R2E'
Q,
5 26.8 3 105 C 5 2680 kC
Q 5 4p 16.38 3 106 m 2 2 121.33 3 1029 C/m2 2
4pRE
2s,Q
RE 5 6.38 3 106 m4pRE
2,
Test Your Understanding of Section 22.5 A hollow conducting sphere has nonet charge. There is a positive point charge at the center of the spherical cavity withinthe sphere. You connect a conducting wire from the outside of the sphere to ground. Willyou measure an electric field outside the sphere?
❚
q
(b)
E 5 0
(a)
Field perpendicular to conductor surface
Field pushes electronstoward left side.
Net positive chargeremains on right side.
ES
ES
S
––––––––
+++++++
+
22.28 (a) A conducting box (a Faradaycage) immersed in a uniform electric field.The field of the induced charges on the boxcombines with the uniform field to givezero total field inside the box. (b) Electro-static shielding can protect you from adangerous electric discharge.
772
Discussion Questions 773CHAPTER 22 SUMMARY
Electric flux: Electric flux is a measure of the “flow” ofelectric field through a surface. It is equal to the productof an area element and the perpendicular component of
integrated over a surface. (See Examples 22.1–22.3.)ES
,(22.5) 5 3E' dA 5 3E
S # dAS
FE 5 3E cos f dA
Gauss’s law: Gauss’s law states that the total electricflux through a closed surface, which can be written asthe surface integral of the component of normal to thesurface, equals a constant times the total charge enclosed by the surface. Gauss’s law is logically equiva-lent to Coulomb’s law, but its use greatly simplifiesproblems with a high degree of symmetry. (See Exam-ples 22.4–22.10.)
When excess charge is placed on a conductor and isat rest, it resides entirely on the surface, and everywhere in the material of the conductor. (See Exam-ples 22.11–22.13.)
ES
5 0
Qencl
ES
(22.8), (22.9) 5Qencl
P0
5 CE' dA 5 CES # dA
S
FE 5 CE cos f dA
AA�
ES
AS
ff
R
r
q
dA
E�
Outward normalto surface
f Er
Electric field of various symmetric charge distributions: The following table lists electric fields caused by several symmetric chargedistributions. In the table, and refer to the magnitudes of the quantities.
Point in Electric FieldCharge Distribution Electric Field Magnitude
Single point charge Distance from
Charge on surface of conducting sphere with radius Outside sphere,
Inside sphere,
Infinite wire, charge per unit length Distance from wire
Infinite conducting cylinder with radius charge per Outside cylinder,unit length
Inside cylinder,
Solid insulating sphere with radius charge distributed Outside sphere,uniformly throughout volume
Inside sphere,
Infinite sheet of charge with uniform charge per unit area Any point
Two oppositely charged conducting plates with surface Any point between platescharge densities and 2s1s
E 5s
P0
E 5s
2P0s
E 51
4pP0 Qr
R3r , R
E 51
4pP0 Q
r 2r . RQR,
E 5 0r , R
lE 5
1
2pP0 l
rr . RR,
E 51
2pP0 l
rrl
E 5 0r , R
E 51
4pP0 q
r 2r . RRq
E 51
4pP0 q
r 2qrq
sq, Q, l,
Key Termsclosed surface, 751electric flux, 752
surface integral, 755Gauss’s law, 757
Gaussian surface, 759Faraday’s icepail experiment, 768
Answer to Chapter Opening Question ?No. The electric field inside a cavity within a conductor is zero, sothere is no electric effect on the child. (See Section 22.5.)
Answers to Test Your Understanding Questions22.1 Answer: (iii) Each part of the surface of the box will bethree times farther from the charge so the electric field will be
as strong. But the area of the box will increase by a factorof Hence the electric flux will be multiplied by a factor of
In other words, the flux will be unchanged.22.2 Answer: (iv), (ii), (i), (iii) In each case the electric field inuniform, so the flux is We use the relationships for thescalar products of unit vectors: In case (i)we have (the electric field andvector area are perpendicular, so there is zero flux). In case (ii) wehave
Similarly, in case (iii) we have
andin case (iv) we have
22.3 Answer: and (tie) Gauss’s law tells us thatthe flux through a closed surface is proportional to the amountof charge enclosed within that surface. So an ordering of these
S3S2 , S5 , S4 , S1
17.0 m2 2 5 26 N # m2/C.12.0 N/C 2 #14.0 N/C 2 1 3.0 m2 2 13 13.0 m2 2 d 2 17.0 m2 2e 4 5
FE 5 3 14.0 N/C 2 d 2 12.0 N/C 2e 4 #14.0 N/C 2 1 3.0 m2 2 2 12.0 N/C 2 1 7.0 m2 2 5 22 N # m2/C,
3 14.0 N/C 2 d 2 12.0 N/C 2e 4 # 3 13.0 m2 2 d 1 17.0 m2 2e 4 5FE 513.0 m2 2 5 6.0 N # m2/C.
FE 3 14.0 N/C 2 d 1 12.0 N/C 2e 4 # 13.0 m2 2e 5 12.0 N/C 2 #
FE 5 14.0 N/C 2 16.0 m2 2 d # e 5 0d # e 5 0.d # d 5 e # e 5 1,
FE 5 ES # A
S
.
1 19 2 19 2 5 1.
32 5 9.1 1
3 22 5 1
9
1q,
surfaces by their fluxes is the same as an ordering by theamount of enclosed charge. Surface encloses no charge, sur-face encloses sur-face encloses surface
encloses and surfaceencloses
22.4 Answer: no You might be tempted to draw a Gaussian sur-face that is an enlarged version of the conductor, with the sameshape and placed so that it completely encloses the conductor.While you know the flux through this Gaussian surface (byGauss’s law, it’s ), the direction of the electric fieldneed not be perpendicular to the surface and the magnitude of thefield need not be the same at all points on the surface. It’s not pos-sible to do the flux integral and we can’t calculate theelectric field. Gauss’s law is useful for calculating the electric fieldonly when the charge distribution is highly symmetric.22.5 Answer: no Before you connect the wire to the sphere, thepresence of the point charge will induce a charge on the innersurface of the hollow sphere and a charge on the outer surface(the net charge on the sphere is zero). There will be an electric fieldoutside the sphere due to the charge on the outer surface. Once youtouch the conducting wire to the sphere, however, electrons willflow from ground to the outer surface of the sphere to neutralizethe charge there (see Fig. 21.7c). As a result the sphere will haveno charge on its outer surface and no electric field outside.
q2q
rE' dA,
FE 5 Q/P0
19.0 mC 2 1 15.0 mC 2 5 6.0 mC.8.0 mC 1 127.0 mC 2 1 1210.0 mC 2 1 11.0 mC 2 1
S58.0 mC 1 127.0 mC 2 5 1.0 mC,S4
9.0 mC 1 1.0 mC 1 1210.0 mC 2 5 0,S3
9.0 mC 1 5.0 mC 1 127.0 mC 2 5 7.0 mC,S2
S1
PROBLEMS For instructor-assigned homework, go to www.masteringphysics.com
Discussion QuestionsQ22.1. A rubber balloon has a single point charge in its interior.Does the electric flux through the balloon depend on whether ornot it is fully inflated? Explain your reasoning.Q22.2. Suppose that in Fig. 22.15 both charges were positive.What would be the fluxes through each of the four surfaces in theexample?Q22.3. In Fig. 22.15, suppose a third point charge were placed out-side the purple Gaussian surface Would this affect the electricflux through any of the surfaces or in the figure? Why orwhy not?Q22.4. A certain region of space bounded by an imaginary closedsurface contains no charge. Is the electric field always zero every-where on the surface? If not, under what circumstances is it zeroon the surface?Q22.5. A spherical Gaussian surface encloses a point charge Ifthe point charge is moved from the center of the sphere to a pointaway from the center, does the electric field at a point on the sur-face change? Does the total flux through the Gaussian surfacechange? Explain.Q22.6. You find a sealed box on your doorstep. You suspect thatthe box contains several charged metal spheres packed in insulat-
q.
DA, B, C,C.
ing material. How can you determine the total net charge inside thebox without opening the box? Or isn’t this possible?Q22.7. During the flow of electric current in a conducting wire,one or more electrons from each atom are free to move along thewire, somewhat like water flowing through a pipe. Would youexpect to find an electric field outside a wire carrying such a steadyflow of electrons? Explain.Q22.8. If the electric field of a point charge were proportional to
instead of would Gauss’s law still be valid? Explain yourreasoning. (Hint: Consider a spherical Gaussian surface centeredon a single point charge.)Q22.9. Suppose the disk in Example 22.1 (Section 22.2), insteadof having its normal vector oriented at just two or three particularangles to the electric field, began to rotate continuously, so that itsnormal vector was first parallel to the field, then perpendicular toit, then opposite to it, and so on. Sketch a graph of the resultingelectric flux versus time, for an entire rotation of Q22.10. In a conductor, one or more electrons from each atom arefree to roam throughout the volume of the conductor. Does thiscontradict the statement that any excess charge on a solid conduc-tor must reside on its surface? Why or why not?
360°.
1/r 2,1/r 3
Exercises 775774 C HAPTE R 22 Gauss’s Law
Q22.11. You charge up the van de Graaff generator shown inFig. 22.27, and then bring an identical but uncharged hollow con-ducting sphere near it, without letting the two spheres touch.Sketch the distribution of charges on the second sphere. What isthe net flux through the second sphere? What is the electric fieldinside the second sphere?Q22.12. The magnitude of at the surfaceof an irregularly shaped solid conductormust be greatest in regions where the sur-face curves most sharply, such as point in Fig. 22.30, and must be least in flatregions such as point in Fig. 22.30.Explain why this must be so by consider-ing how electric field lines must bearranged near a conducting surface. Howdoes the surface charge density compare atpoints and ? Explain.Q22.13. A lightning rod is a rounded copper rod mounted on top ofa building and welded to a heavy copper cable running down intothe ground. Lightning rods are used to protect houses and barnsfrom lightning; the lightning current runs through the copper ratherthan through the building. Why? Why should the end of the rod berounded? (Hint: The answer to Discussion Question Q22.12 maybe helpful.)Q22.14. A solid conductor has a cavity in its interior. Would thepresence of a point charge inside the cavity affect the electric fieldoutside the conductor? Why or why not? Would the presence of apoint charge outside the conductor affect the electric field insidethe cavity? Again, why or why not?Q22.15. Explain this statement: “In a static situation, the electricfield at the surface of a conductor can have no component parallelto the surface because this would violate the condition that thecharges on the surface are at rest.” Would this same statement bevalid for the electric field at the surface of an insulator? Explainyour answer and the reason for any differences between the casesof a conductor and an insulator.Q22.16. A solid copper sphere has a net positive charge. Thecharge is distributed uniformly over the surface of the sphere, andthe electric field inside the sphere is zero. Then a negative pointcharge outside the sphere is brought close to the surface of thesphere. Is all the net charge on the sphere still on its surface? If so,is this charge still distributed uniformly over the surface? If it isnot uniform, how is it distributed? Is the electric field inside thesphere still zero? In each case justify your answers.Q22.17. Some modern aircraft are made primarily of compositematerials that do not conduct electricity. The U.S. Federal AviationAdministration requires that such aircraft have conducting wiresembedded in their surfaces to provide protection when flying nearthunderstorms. Explain the physics behind this requirement.
ExercisesSection 22.2 Calculating Electric Flux22.1. A flat sheet of paper of area is oriented so that thenormal to the sheet is at an angle of to a uniform electric fieldof magnitude (a) Find the magnitude of the electric fluxthrough the sheet. (b) Does the answer to part (a) depend on theshape of the sheet? Why or why not? (c) For what angle betweenthe normal to the sheet and the electric field is the magnitude of theflux through the sheet (i) largest and (ii) smallest? Explain youranswers.
f
14 N/C.60°
0.250 m2
BA
B
A
ES
Figure 22.30Question Q22.12.
A
B
Section 22.3 Gauss’s Law22.8. The three small spheres shown in Fig. 22.33 carry charges
and Find the netelectric flux through each of the following closed surfaces shownin cross section in the figure: (a) (b) (c) (d) (e) (f ) Do your answers to parts (a)– (e) depend on how the charge isdistributed over each small sphere? Why or why not?
S5 .S4 ;S3 ;S2 ;S1 ;
q3 5 2.40 nC.q1 5 4.00 nC, q2 5 27.80 nC,
Section 22.4 Applications of Gauss’s Law andSection 22.5 Charges on Conductors22.16. A solid metal sphere with radius 0.450 m carries a netcharge of 0.250 nC. Find the magnitude of the electric field (a) at apoint 0.100 m outside the surface of the sphere and (b) at a pointinside the sphere, 0.100 m below the surface.22.17. On a humid day, an electric field of isenough to produce sparks about an inch long. Suppose that in yourphysics class, a van de Graaff generator (see Fig. 22.27) with asphere radius of 15.0 cm is producing sparks 6 inches long. (a) UseGauss’s law to calculate the amount of charge stored on the surfaceof the sphere before you bravely discharge it with your hand.(b) Assume all the charge is concentrated at the center of thesphere, and use Coulomb’s law to calculate the electric field at thesurface of the sphere.22.18. Some planetary scientists have suggested that the planetMars has an electric field somewhat similar to that of the earth, pro-ducing a net electric flux of at the planet’ssurface. Calculate: (a) the total electric charge on the planet; (b) theelectric field at the planet’s surface (refer to the astronomical datainside the back cover); (c) the charge density on Mars, assuming allthe charge is uniformly distributed over the planet’s surface.22.19. How many excess electrons must be added to an isolatedspherical conductor 32.0 cm in diameter to produce an electricfield of just outside the surface?22.20. The electric field 0.400 m from a very long uniform line ofcharge is How much charge is contained in a 2.00-cmsection of the line?22.21. A very long uniform line of charge has charge per unitlength and lies along the x-axis. A second long uniformline of charge has charge per unit length and is paral-lel to the -axis at What is the net electric field (mag-nitude and direction) at the following points on the y-axis:(a) and (b) 22.22. (a) At a distance of 0.200 cm from the center of a chargedconducting sphere with radius 0.100 cm, the electric field is
What is the electric field 0.600 cm from the center of thesphere? (b) At a distance of 0.200 cm from the axis of a very longcharged conducting cylinder with radius 0.100 cm, the electricfield is What is the electric field 0.600 cm from the axisof the cylinder? (c) At a distance of 0.200 cm from a large uniformsheet of charge, the electric field is What is the electricfield 1.20 cm from the sheet?22.23. A hollow, conducting sphere with an outer radius of0.250 m and an inner radius of 0.200 m has a uniform surfacecharge density of A charge of isnow introduced into the cavity inside the sphere. (a) What is thenew charge density on the outside of the sphere? (b) Calculate thestrength of the electric field just outside the sphere. (c) What isthe electric flux through a spherical surface just inside the innersurface of the sphere?22.24. A point charge of is located in the center of aspherical cavity of radius 6.50 cm inside an insulating chargedsolid. The charge density in the solid is Calculate the electric field inside the solid at a distance of 9.50 cmfrom the center of the cavity.22.25. The electric field at a distance of 0.145 m from the surfaceof a solid insulating sphere with radius 0.355 m is (a) Assuming the sphere’s charge is uniformly distributed, what isthe charge density inside it? (b) Calculate the electric field insidethe sphere at a distance of 0.200 m from the center.
1750 N/C.
r 5 7.35 3 1024 C/m3.
22.00 mC
20.500 mC16.37 3 1026 C/m2.
480 N/C.
480 N/C.
480 N/C.
y 5 0.600 m?y 5 0.200 m
y 5 0.400 m.x22.40 mC/m
4.80 mC/m
840 N/C.
1150 N/C
3.63 3 1016 N # m2/C
2.00 3 104 N/C
22.9. A charged paint is spread in a very thin uniform layer overthe surface of a plastic sphere of diameter 12.0 cm, giving it acharge of Find the electric field (a) just inside the paintlayer; (b) just outside the paint layer; (c) 5.00 cm outside the sur-face of the paint layer.22.10. A point charge is located on the x-axis at
and a second point charge is on they-axis at What is the total electric flux due to thesetwo point charges through a spherical surface centered at the originand with radius (a) 0.500 m, (b) 1.50 m, (c) 2.50 m?22.11. In a certain region of space, the electric field is uniform.(a) Use Gauss’s law to prove that this region of space must be elec-trically neutral; that is, the volume charge density must be zero.(b) Is the converse true? That is, in a region of space where there isno charge, must be uniform? Explain.22.12. (a) In a certain region of space, the volume charge density has a uniform positive value. Can be uniform in this region?Explain. (b) Suppose that in this region of uniform positive thereis a “bubble” within which Can be uniform within thisbubble? Explain.22.13. A point charge is at the center of a cube with sidesof length 0.500 m. (a) What is the electric flux through one of thesix faces of the cube? (b) How would your answer to part (a)change if the sides were 0.250 m long? Explain.22.14. Electric Fields in an Atom. The nuclei of large atoms,such as uranium, with 92 protons, can be modeled as sphericallysymmetric spheres of charge. The radius of the uranium nucleus isapproximately (a) What is the electric field thisnucleus produces just outside its surface? (b) What magnitude ofelectric field does it produce at the distance of the electrons, whichis about (c) The electrons can be modeled as form-ing a uniform shell of negative charge. What net electric field dothey produce at the location of the nucleus?22.15. A point charge of is located on the x-axis at
next to a spherical surface of radius 3.00 m centeredat the origin. (a) Calculate the magnitude of the electric field at
(b) Calculate the magnitude of the electric field at(c) According to Gauss’s law, the net flux through
the sphere is zero because it contains no charge. Yet the field due tothe external charge is much stronger on the near side of the sphere(i.e., at ) than on the far side (at ). How,then, can the flux into the sphere (on the near side) equal the fluxout of it (on the far side)? Explain. A sketch will help.
x 5 23.00 mx 5 3.00 m
x 5 23.00 m.x 5 3.00 m.
x 5 4.00 m,15.00 mC
1.0 3 10210 m?
7.4 3 10215 m.
9.60-mC
ES
r 5 0.r
ES
rES
r
ES
y 5 1.00 m.q2 5 26.00 nCx 5 2.00 m,
q1 5 4.00 nC
215.0 mC.
20°0.400 m
0.600 m
ES
Figure 22.31 Exercise 22.2.
S3 (right side)
S5 (front)
LL
L
S4 (bottom)
y
z
x
S6 (back)
S1 (left side)
S2 (top)
Figure 22.32 Exercises 22.4 and 22.6; Problem 22.32.
22.5. A hemispherical surface with radius in a region of uni-form electric field has its axis aligned parallel to the directionof the field. Calculate the flux through the surface.22.6. The cube in Fig. 22.32 has sides of length Theelectric field is uniform, has magnitude andis parallel to the -plane at an angle of measured from the
-axis toward the -axis. (a) What is the electric flux througheach of the six cube faces and (b) What is thetotal electric flux through all faces of the cube?22.7. It was shown in Example 21.11 (Section 21.5) that the elec-tric field due to an infinite line of charge is perpendicular to theline and has magnitude Consider an imaginarycylinder with radius and length that hasan infinite line of positive charge running along its axis. Thecharge per unit length on the line is (a) What isthe electric flux through the cylinder due to this infinite line ofcharge? (b) What is the flux through the cylinder if its radius isincreased to (c) What is the flux through the cylinderif its length is increased to l 5 0.800 m?
r 5 0.500 m?
l 5 6.00 mC/m.
l 5 0.400 mr 5 0.250 mE 5 l/2pP0 r.
S6?S1 , S2 , S3 , S4 , S5 ,1y1x
36.9°xyE 5 4.00 3 103 N/C,
L 5 10.0 cm.
ES
r
22.3. You measure an electric field of at a dis-tance of 0.150 m from a point charge. (a) What is the electric fluxthrough a sphere at that distance from the charge? (b) What is themagnitude of the charge?22.4. A cube has sides of length It is placed withone corner at the origin as shown in Fig. 22.32. The electricfield is not uniform but is given by
(a) Find the electric flux through each of the sixcube faces and (b) Find the total electriccharge inside the cube.
S6 .S1 , S2 , S3 , S4 , S5 ,13.00 N/C # m 2 zk.
ES
5 125.00 N/C # m 2 x d 1
L 5 0.300 m.
1.25 3 106 N/C
22.2. A flat sheet is in the shape of a rectangle with sides oflengths 0.400 m and 0.600 m. The sheet is immersed in a uniformelectric field of magnitude that is directed at fromthe plane of the sheet (Fig. 22.31). Find the magnitude of the elec-tric flux through the sheet.
20°75.0 N/C
q2
q3
q1
S4
S3
S1 S2
S5
SurfaceS1S2S3S4S5
What it enclosesq1q2q1 and q2q1 and q3q1 and q2 and q3
Figure 22.33 Exercise 22.8.
Problems 777776 C HAPTE R 22 Gauss’s Law
22.26. A conductor with an inner cavity, like that shown inFig. 22.23c, carries a total charge of The charge withinthe cavity, insulated from the conductor, is How muchcharge is on (a) the inner surface of the conductor and (b) the outersurface of the conductor?22.27. Apply Gauss’s law to the Gaussian surfaces and inFig. 22.21b to calculate the electric field between and outside theplates.22.28. A square insulating sheet 80.0 cm on a side is held horizon-tally. The sheet has 7.50 nC of charge spread uniformly over itsarea. (a) Calculate the electric field at a point 0.100 mm above thecenter of the sheet. (b) Estimate the electric field at a point 100 mabove the center of the sheet. (c) Would the answers to parts (a)and (b) be different if the sheet were made of a conducting mate-rial? Why or why not?22.29. An infinitely long cylindrical conductor has radius anduniform surface charge density (a) In terms of and what isthe charge per unit length for the cylinder? (b) In terms of what is the magnitude of the electric field produced by the chargedcylinder at a distance from its axis? (c) Express the result ofpart (b) in terms of and show that the electric field outside thecylinder is the same as if all the charge were on the axis. Compareyour result to the result for a line of charge in Example 22.6 (Sec-tion 22.4).22.30. Two very large, nonconductingplastic sheets, each 10.0 cm thick, carryuniform charge densities and on their surfaces, as shown inFig. 22.34. These surface chargedensities have the values
andUse Gauss’s law to find
the magnitude and direction of the elec-tric field at the following points, farfrom the edges of these sheets: (a) point
5.00 cm from the left face of the left-hand sheet; (b) point 1.25 cm from the inner surface of the right-hand sheet; (c) point in the middle of the right-hand sheet.22.31. A negative charge is placed inside the cavity of a hol-low metal solid. The outside of the solid is grounded by connectinga conducting wire between it and the earth. (a) Is there any excesscharge induced on the inner surface of the piece of metal? If so,find its sign and magnitude. (b) Is there any excess charge on theoutside of the piece of metal? Why or why not? (c) Is there an elec-tric field in the cavity? Explain. (d) Is there an electric field withinthe metal? Why or why not? Is there an electric field outside thepiece of metal? Explain why or why not. (e) Would someone out-side the solid measure an electric field due to the charge Is itreasonable to say that the grounded conductor has shielded theregion from the effects of the charge In principle, could thesame thing be done for gravity? Why or why not?
Problems22.32. A cube has sides of length It is placed with one corner atthe origin as shown in Fig. 22.32. The electric field is uniform andgiven by where and are positiveconstants. (a) Find the electric flux through each of the six cubefaces and (b) Find the electric flux through theentire cube.
S6 .S1 , S2 , S3 , S4 , S5 ,
DC,B,ES
5 2Bd 1 Ce 2 Dk,
L.
2Q?
2Q?
2Q
C,B,A,
14.00 mC/m2.s4 5s3 5 12.00 mC/m2,
s2 5 15.00 mC/m2,26.00 mC/m2,s1 5
s4
s1 , s2 , s3 ,
lr . R
s,lR,ss.
R
S4S2 , S3 ,
26.00 nC.15.00 nC.
22.33. The electric field in Fig. 22.35is everywhere parallel to the -axis, sothe components and are zero. The-component of the field depends onbut not on and At points in the-plane (where ),
(a) What is the electric flux throughsurface I in Fig. 22.35? (b) What is theelectric flux through surface II? (c) Thevolume shown in the figure is a smallsection of a very large insulating slab1.0 m thick. If there is a total charge of
within the volume shown,what are the magnitude and direction of at the face opposite sur-face I? (d) Is the electric field produced only by charges within theslab, or is the field also due to charges outside the slab? How canyou tell?22.34. A flat, square surface with sides of length is described bythe equations
(a) Draw this square and show the -, -, and -axes. (b) Find theelectric flux through the square due to a positive point charge located at the origin (Hint: Think of thesquare as part of a cube centered on the origin.)22.35. The electric field at oneface of a parallelepiped is uniformover the entire face and is directedout of the face. At the opposite face,the electric field is also uniformover the entire face and is directedinto that face (Fig. 22.36). The twofaces in question are inclined at
from the horizontal, while and are both horizontal; has amagnitude of and
has a magnitude of (a) Assuming that no otherelectric field lines cross the surfaces of the parallelepiped, deter-mine the net charge contained within. (b) Is the electric field pro-duced only by the charges within the parallelepiped, or is the fieldalso due to charges outside the parallelepiped? How can you tell?22.36. A long line carrying a uniform linear charge density
runs parallel to and 10.0 cm from the surface of alarge, flat plastic sheet that has a uniform surface charge density of
on one side. Find the location of all points where anparticle would feel no force due to this arrangement of charged
objects.22.37. The Coaxial Cable. A long coaxial cable consists of aninner cylindrical conductor with radius and an outer coaxialcylinder with inner radius and outer radius The outer cylinderis mounted on insulating supports and has no net charge. The innercylinder has a uniform positive charge per unit length Calculatethe electric field (a) at any point between the cylinders a distancefrom the axis and (b) at any point outside the outer cylinder.(c) Graph the magnitude of the electric field as a function of thedistance from the axis of the cable, from to(d) Find the charge per unit length on the inner surface and on theouter surface of the outer cylinder.22.38. A very long conducting tube (hollow cylinder) has innerradius and outer radius It carries charge per unit length where is a positive constant with units of A line of chargeC/m.a
1a,b.a
r 5 2c.r 5 0r
rl.
c.ba
a2100 mC/m2
150.0 mC/m
7.00 3 104 N/C.ES
2
2.50 3 104 N/C,ES
1ES
2
ES
130.0°
ES
2
ES
1
1 x 5 0, y 5 0, z 5 0 2 .q
zyx
x 5 L 10 # y # L, 0 # z # L 2
L
ES
224.0 nC
Ex 5 125 N/C.x 5 0yzz.yx
ExxEzEy
xES
10 cm 10 cm12 cm
s1 s2 s4s3
CBA
Figure 22.34Exercise 22.30.
z
3.0 m
I
II
x
y
1.0 m O
2.0 m ES
Figure 22.35Problem 22.33.
lies along the axis of the tube. The line of charge has charge perunit length (a) Calculate the electric field in terms of and thedistance from the axis of the tube for (i) (ii) (iii) Show your results in a graph of as a function of (b) What is the charge per unit length on (i) the inner surface of thetube and (ii) the outer surface of the tube?22.39. Repeat Problem 22.38, but now let the conducting tubehave charge per unit length As in Problem 22.38, the line ofcharge has charge per unit length 22.40. A very long, solid cylinder with radius has positivecharge uniformly distributed throughout it, with charge per unitvolume (a) Derive the expression for the electric field inside thevolume at a distance from the axis of the cylinder in terms of thecharge density (b) What is the electric field at a point outsidethe volume in terms of the charge per unit length in the cylinder?(c) Compare the answers to parts (a) and (b) for (d) Graphthe electric-field magnitude as a function of from to
22.41. A small sphere with a massof 0.002 g and carrying a chargeof hangs from athread near a very large, chargedconducting sheet, as shown inFig. 22.37. The charge density onthe sheet is Find the angle of the thread.22.42. A Sphere in a Sphere.A solid conducting sphere carry-ing charge has radius It isinside a concentric hollow con-ducting sphere with inner radius and outer radius The hollowsphere has no net charge.(a) Derive expressions for the electric-field magnitude in terms ofthe distance from the center for the regions
and (b) Graph the magnitude of the electric fieldas a function of from to (c) What is the charge onthe inner surface of the hollow sphere? (d) On the outer surface?(e) Represent the charge of the small sphere by four plus signs.Sketch the field lines of the system within a spherical volume ofradius 2 22.43. A solid conducting sphere with radius that carries posi-tive charge is concentric with a very thin insulating shell ofradius that also carries charge The charge is distributeduniformly over the insulating shell. (a) Find the electric field(magnitude and direction) in each of the regions
and (b) Graph the electric-field magnitude asa function of 22.44. A conducting spherical shell with innerradius and outer radius has a positive pointcharge located at its center. The total chargeon the shell is and it is insulated from itssurroundings (Fig. 22.38). (a) Derive expres-sions for the electric-field magnitude in terms ofthe distance from the center for the regions
and (b) What is the surface charge den-sity on the inner surface of the conducting shell? (c) What is thesurface charge density on the outer surface of the conducting shell?(d) Sketch the electric field lines and the location of all charges.(e) Graph the electric-field magnitude as a function of 22.45. Concentric Spherical Shells. A small conducting spheri-cal shell with inner radius and outer radius is concentric with aba
r.
r . b.a , r , b,r , a,r
23Q,Q
ba
r.r . 2R.R , r , 2R,
0 , r , R,
QQ.2RQ
Rc.
r 5 2c.r 5 0rr . c.b , r , c,
a , r , b,r , a,r
c.b
a.q
2.50 3 1029 C/m2.
5.00 3 1028 C
r 5 3R.r 5 0r
r 5 R.l
r.r
r.
R1a.
2a.
r.Er . b.a , r , b;r , a;r
a1a.larger conducting spherical shell with innerradius and outer radius (Fig. 22.39). Theinner shell has total charge and theouter shell has charge (a) Calculatethe electric field (magnitude and direction)in terms of and the distance from thecommon center of the two shells for(i) (ii) (iii)(iv) (v) Show your resultsin a graph of the radial component of as afunction of (b) What is the total charge on the (i) inner surface ofthe small shell; (ii) outer surface of the small shell; (iii) inner sur-face of the large shell; (iv) outer surface of the large shell?22.46. Repeat Problem 22.45, but now let the outer shell havecharge As in Problem 22.45, the inner shell has charge 22.47. Repeat Problem 22.45, but now let the outer shell havecharge As in Problem 22.45, the inner shell has charge 22.48. A solid conducting sphere with radius carries a positivetotal charge The sphere is surrounded by an insulating shellwith inner radius and outer radius The insulating shell has auniform charge density (a) Find the value of so that the netcharge of the entire system is zero. (b) If has the value found inpart (a), find the electric field (magnitude and direction) in each ofthe regions and Show yourresults in a graph of the radial component of as a function of (c) As a general rule, the electric field is discontinuous only atlocations where there is a thin sheet of charge. Explain how yourresults in part (b) agree with this rule.22.49. Negative charge is distributed uniformly over the sur-face of a thin spherical insulating shell with radius Calculate theforce (magnitude and direction) that the shell exerts on a positivepoint charge located (a) a distance from the center of theshell (outside the shell) and (b) a distance from the center ofthe shell (inside the shell).22.50. (a) How many excess electrons must be distributed uni-formly within the volume of an isolated plastic sphere 30.0 cm indiameter to produce an electric field of just outside thesurface of the sphere? (b) What is the electric field at a point 10.0 cmoutside the surface of the sphere?22.51. A single isolated, largeconducting plate (Fig. 22.40)has a charge per unit area onits surface. Because the plate isa conductor, the electric field atits surface is perpendicular tothe surface and has magnitude
(a) In Example 22.7(Section 22.4) it was shownthat the field caused by a large,uniformly charged sheet withcharge per unit area has magnitude exactly half asmuch as for a charged conducting plate. Why is there a difference?(b) Regarding the charge distribution on the conducting plate asbeing two sheets of charge (one on each surface), each with chargeper unit area use the result of Example 22.7 and the principle ofsuperposition to show that inside the plate and outside the plate.22.52. Thomson’s Model of the Atom. In the early years of the20th century, a leading model of the structure of the atom was thatof the English physicist J. J. Thomson (the discoverer of the elec-tron). In Thomson’s model, an atom consisted of a sphere of posi-tively charged material in which were embedded negatively
E 5 s/P0E 5 0s,
E 5 s/2P0 ,s
E 5 s/P0 .
s
1150 N/C
r , Rr . Rq
R.2Q
r.ES
r . 2R.R , r , 2R,0 , r , R,
rrr.
2R.RQ.
R12q.24q.
12q.22q.
r.ES
r . d.c , r , d;b , r , c;a , r , b;r , a;
rq
14q.12q,
dc
u
Figure 22.37Problem 22.41.
ab
c
d
Figure 22.39Problem 22.45.
Figure 22.38Problem 22.44.
Q
ab
23Q
30°5.00 cm
6.00cm
E2S
E1S
Figure 22.36Problem 22.35.
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
s s
E 5 s/P0 E 5 s/P0
Figure 22.40 Problem 22.51.
Challenge Problems 779778 C HAPTE R 22 Gauss’s Law
charged electrons, like chocolate chips in a ball of cookie dough.Consider such an atom consisting of one electron with mass andcharge which may be regarded as a point charge, and a uni-formly charged sphere of charge and radius (a) Explain whythe equilibrium position of the electron is at the center of thenucleus. (b) In Thomson’s model, it was assumed that the positivematerial provided little or no resistance to the motion of the elec-tron. If the electron is displaced from equilibrium by a distanceless than show that the resulting motion of the electron will besimple harmonic, and calculate the frequency of oscillation. (Hint:Review the definition of simple harmonic motion in Section 13.2.If it can be shown that the net force on the electron is of this form,then it follows that the motion is simple harmonic. Conversely, ifthe net force on the electron does not follow this form, the motionis not simple harmonic.) (c) By Thomson’s time, it was known thatexcited atoms emit light waves of only certain frequencies. In hismodel, the frequency of emitted light is the same as the oscillationfrequency of the electron or electrons in the atom. What would theradius of a Thomson-model atom have to be for it to produce redlight of frequency Compare your answer to theradii of real atoms, which are of the order of (see Appen-dix F for data about the electron). (d) If the electron were displacedfrom equilibrium by a distance greater than would the electronoscillate? Would its motion be simple harmonic? Explain your rea-soning. (Historical note: In 1910, the atomic nucleus was discov-ered, proving the Thomson model to be incorrect. An atom’spositive charge is not spread over its volume as Thomson sup-posed, but is concentrated in the tiny nucleus of radius to
)22.53. Thomson’s Model of the Atom,Continued. Using Thomson’s (outdated)model of the atom described in Prob-lem 22.52, consider an atom consisting oftwo electrons, each of charge embed-ded in a sphere of charge and radius In equilibrium, each electron is a distance from the center of the atom (Fig. 22.41).Find the distance in terms of the otherproperties of the atom.22.54. A Uniformly Charged Slab. Aslab of insulating material has thickness and is oriented sothat its faces are parallel to the -plane and given by the planes
and The - and -dimensions of the slab are verylarge compared to and may be treated as essentially infinite.The slab has a uniform positive charge density (a) Explainwhy the electric field due to the slab is zero at the center of theslab (b) Using Gauss’s law, find the electric field due tothe slab (magnitude and direction) at all points in space.22.55. A Nonuniformly Charged Slab. Repeat Problem 22.54,but now let the charge density of the slab be given by
where is a positive constant.22.56. Can Electric Forces Alone Give Stable Equilibrium?In Chapter 21, several examples were given of calculating theforce exerted on a point charge by other point charges in its sur-roundings. (a) Consider a positive point charge Give an exam-ple of how you would place two other point charges of yourchoosing so that the net force on charge will be zero. (b) If thenet force on charge is zero, then that charge is in equilibrium.The equilibrium will be stable if, when the charge is displacedslightly in any direction from its position of equilibrium, the netforce on the charge pushes it back toward the equilibrium position.For this to be the case, what must the direction of the electric field
1q1q
1q
1q.
r0r0 1 x/d 2 2,r 1 x 2 5
1 x 5 0 2 .
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d
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2e,
10215 m.10214
R,
10210 m4.57 3 1014 Hz?
R,
R.1e2e,
mbe due to the other charges at points surrounding the equilibrium
position of (c) Imagine that the charge is moved very faraway, and imagine a small Gaussian surface centered on the posi-tion where was in equilibrium. By applying Gauss’s law to thissurface, show that it is impossible to satisfy the condition for sta-bility described in part (b). In other words, a charge cannot beheld in stable equilibrium by electrostatic forces alone. This resultis known as Earnshaw’s theorem. (d) Parts (a)– (c) referred to theequilibrium of a positive point charge Prove that Earnshaw’stheorem also applies to a negative point charge 22.57. A nonuniform, but spherically symmetric, distribution ofcharge has a charge density given as follows:
where is a positive constant. (a) Show that the totalcharge contained in the charge distribution is (b) Show that theelectric field in the region is identical to that produced by apoint charge at (c) Obtain an expression for the electricfield in the region (d) Graph the electric-field magnitude as a function of (e) Find the value of at which the electric fieldis maximum, and find the value of that maximum field.22.58. A nonuniform, but spherically symmetric, distribution ofcharge has a charge density given as follows:
where is a positive constant. (a) Find the total charge containedin the charge distribution. (b) Obtain an expression for the electricfield in the region (c) Obtain an expression for the electricfield in the region (d) Graph the electric-field magnitude as a function of (e) Find the value of at which the electric fieldis maximum, and find the value of that maximum field.22.59. Gauss’s Law for Gravitation. The gravitational forcebetween two point masses separated by a distance is proportionalto just like the electric force between two point charges.Because of this similarity between gravitational and electric inter-actions, there is also a Gauss’s law for gravitation. (a) Let be theacceleration due to gravity caused by a point mass at the origin,so that Consider a spherical Gaussian surfacewith radius centered on this point mass, and show that the flux of
through this surface is given by
(b) By following the same logical steps used in Section 22.3 toobtain Gauss’s law for the electric field, show that the flux of through any closed surface is given by
where is the total mass enclosed within the closed surface.22.60. Applying Gauss’s Law for Gravitation. Using Gauss’slaw for gravitation (derived in part (b) of Problem 22.59), showthat the following statements are true: (a) For any spherically sym-metric mass distribution with total mass the acceleration due togravity outside the distribution is the same as though all the masswere concentrated at the center. (Hint: See Example 22.5 in Sec-tion 22.4.) (b) At any point inside a spherically symmetric shell ofmass, the acceleration due to gravity is zero. (Hint: See Exam-ple 22.5.) (c) If we could drill a hole through a spherically sym-
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would find that the magnitude of is directly proportional to thedistance from the center. (Hint: See Example 22.9 in Sec-tion 22.4.) We proved these results in Section 12.6 using somefairly strenuous analysis; the proofs using Gauss’s law for gravita-tion are much easier.22.61. (a) An insulating sphere with radiushas a uniform charge density The sphereis not centered at the origin but at Show that the electric field inside the sphereis given by (b) An insu-lating sphere of radius has a spherical holeof radius located within its volume and cen-tered a distance from the center of thesphere, where (a cross section ofthe sphere is shown in Fig. 22.42). The solid part of the sphere hasa uniform volume charge density Find the magnitude and direc-tion of the electric field inside the hole, and show that is uni-form over the entire hole. [Hint: Use the principle of superpositionand the result of part (a).]22.62. A very long, solid insulatingcylinder with radius has a cylindricalhole with radius bored along its entirelength. The axis of the hole is a distance from the axis of the cylinder, where
(Fig. 22.43). The solid mate-rial of the cylinder has a uniform volumecharge density Find the magnitude anddirection of the electric field inside thehole, and show that is uniform over theentire hole. (Hint: See Problem 22.61.)22.63. Positive charge is dis-tributed uniformly over each oftwo spherical volumes withradius One sphere of chargeis centered at the origin and theother at (Fig. 22.44).Find the magnitude and direc-tion of the net electric field dueto these two distributions of charge at the following points on the -axis: (a) (b) (c) (d)
22.64. Repeat Problem 22.63, but now let the left-hand spherehave positive charge and let the right-hand sphere have negativecharge 22.65. Electric Field Inside a Hydrogen Atom. A hydrogenatom is made up of a proton of charge andan electron of charge The proton may beregarded as a point charge at the center of the atom. Themotion of the electron causes its charge to be “smeared out” into aspherical distribution around the proton, so that the electron isequivalent to a charge per unit volume of
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a
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where is called the Bohr radius. (a) Find thetotal amount of the hydrogen atom’s charge that is enclosed withina sphere with radius centered on the proton. Show that as the enclosed charge goes to zero. Explain this result. (b) Find theelectric field (magnitude and direction) caused by the charge of thehydrogen atom as a function of (c) Graph the electric-field mag-nitude as a function of
Challenge Problems22.66. A region in space contains a total positive charge that isdistributed spherically such that the volume charge density isgiven by
Here is a positive constant having units of (a) Determinein terms of and (b) Using Gauss’s law, derive an expression
for the magnitude of as a function of Do this separately for allthree regions. Express your answers in terms of the total charge Be sure to check that your results agree on the boundaries of theregions. (c) What fraction of the total charge is contained withinthe region (d) If an electron with charge isoscillating back and forth about (the center of the distribu-tion) with an amplitude less than show that the motion is sim-ple harmonic. (Hint: Review the discussion of simple harmonicmotion in Section 13.2. If, and only if, the net force on the electronis proportional to its displacement from equilibrium, then themotion is simple harmonic.) (e) What is the period of the motion inpart (d)? (f) If the amplitude of the motion described in part (e) isgreater than is the motion still simple harmonic? Why or whynot?22.67. A region in space contains a total positive charge that isdistributed spherically such that the volume charge density isgiven by
Here is a positive constant having units of (a) Determinein terms of and (b) Using Gauss’s law, derive an expression
for the magnitude of the electric field as a function of Do thisseparately for all three regions. Express your answers in terms ofthe total charge (c) What fraction of the total charge is con-tained within the region (d) What is the magnitudeof at (e) If an electron with charge is releasedfrom rest at any point in any of the three regions, the resultingmotion will be oscillatory but not simple harmonic. Why? (SeeChallenge Problem 22.66.)
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R/2 # r # R?Q.
r.R.Qa
C/m3.a
r 1 r 2 5 0 for r $ R
r 1 r 2 5 a 31 2 1 r/R 2 2 4 for R/2 # r # R
r 1 r 2 5 3ar/ 12R 2 for r # R/2
r 1 r 2Q
R/2,
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S
R.QaC/m3.a
r 1 r 2 5 0 for r $ R
r 1 r 2 5 2a 11 2 r/R 2 for R/2 # r # R
r 1 r 2 5 a for r # R/2
r 1 r 2Q
r.Er.
r S `,r
a0 5 5.29 3 10211 m
R
d d
2e
12e
2e
Figure 22.41Problem 22.53.
ab
R
Charge density r
Figure 22.42Problem 22.61.
xR RRO
Q Q
y
Figure 22.44 Problem 22.63.
abR
Chargedensity r
Figure 22.43Problem 22.62.
