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General Procedure for Calculating Electric Field of Distributed Charges
1. Cut the charge distribution into pieces for which the field is known
2. Write an expression for the electric field due to one piece(i) Choose origin(ii) Write an expression for E and its components
3. Add up the contributions of all the pieces(i) Try to integrate symbolically(ii) If impossible – integrate numerically
4. Check the results:(i) Direction(ii) Units(iii) Special cases
Distance dependence:
Far from the ring (z>>R):
Close to the ring (z<<R): Ez~z
Ez~1/z2
A Uniformly Charged Thin Ring
A total charge Q is uniformly distributed over a half ring with radius R. The total charge inside a small element dθ is given by:
1. Choice One
2. Choice Two
3. Choice Three
4. Choice Four
5. Choice Five
6. Choice Six
θdθ
Q
R
dRQ
2
A.
dRQ
B.
RQ
C.
dQD.
dQ2
E.
Clicker Question
A total charge Q is uniformly distributed over a half ring with radius R. The y component of electric field at the center created by a short element dθ is given by:
1. Choice One
2. Choice Two
3. Choice Three
4. Choice Four
θdθ
Q
R
sin2R
kQdA.
cos2R
kQdB.
sin3R
kQdC.
cos3R
kQdD.
+yClicker Question
A Uniformly Charged Disk
Close to the disk (0 < z < R)
Along z axis
Approximations:
If z/R is extremely small
Very close to disk (0 < z << R)
Field Far From the Disk
Exact
For z>>R
Point Charge
Uniformly Charged Disk Edge On
Two disks of opposite charges, s<<R: charges distribute uniformly:
+Q-Q
s
A single metal disk cannot be uniformly charged: charges repel and concentrate at the edges
We will calculate E both inside and outside of the disk close to the center
Two uniformly charged metal disks of radius R placed very near each other
Almost all the charge is nearly uniformly distributed on the inner surfaces of the disks; very little charge on the outer surfaces.
Capacitor
+Q-Q
s
We know the field for a single disk There are only 2 “pieces”
E-
E+Enet
Step 1: Cut Charge Distribution into Pieces
Step 2: Contribution of one Piece
Origin: left disk, center
E-
E+Enet
sz
0
Location of disks: z=0, z=sDistance from disk to “2”
z, (s-z)
Left:
Right:
Step 3: Add up Contributions
Location: “2” (inside a capacitor)
Does not depend on z
Step 3: Add up Contributions
E-
E+Enet
sz
0
Location: “3” (fringe field)
For s<<R: E1=E30
Far from the capacitor (z>>R>>s): E1=E3~1/z3 (like dipole)
Fringe field is very small compared to the field inside the capacitor.
E-
E+Enet
sz
0Units:
Inside:
Fringe:
Step 4: check the results:
Electric Field of a Capacitor
Which arrow best represents the field at the “X”?
A)
B)
C) E=0
D)
E)
Clicker Question
Field inside:
Field outside: (like point charge)
Electric Field of a Spherical Shell of Charge
E1+E4
E2E3
E6 E5
Divide into 6 areas:
Direction: radial - due to the symmetry
E of a Sphere Outside
Magnitude: E=0
Note: E is not always 0 inside – other charges in the Universemay make a nonzero electric field inside.
E of a Sphere Inside
E=0: Implications
Fill charged sphere with plastic.Will plastic be polarized? No!
Solid metal sphere: since it is a conductor, any excess charges on the sphere arranges itself uniformly on the outer surface.There will be no field nor excess charges inside!
In general: there is no electric field inside metals
E of a Sphere Inside
Divide shell into rings of charge, each delimited by the angle and the angle +From ring to point: d=(r-Rcos)Surface area of ring:
Integrating Spherical Shell
R
R
Rcos
Rsin
r
d
Q
A mess of math
A solid metal ball bearing a charge –17 nC is located near a solid plastic ball bearing a uniformly distributed charge +7 nC (on surface). Show approximate charge distribution in each ball.
What is electric field field inside the metal ball?
Metal-17 nC
Plastic+7 nC
Exercise
Two uniformly charged thin plastic shells.
Find electric field at 3, 7 and 10 cm from the center
3 cm: E=0
7 cm:
10 cm:
Exercise
What if charges are distributed throughout an object?
Step 1: Cut up the charge into shells
r E
RFor each spherical shell:
outside:
inside: dE = 0
Outside a solid sphere of charge:
for r>R
A Solid Sphere of Charge
Inside a solid sphere of charge:
E
R
r
for r<R
Why is E~r?
On surface:
A Solid Sphere of Charge
What is in the box?
no charges? vertical charged plate?
Patterns of Fields in Space
Box versus open surface
Seem to be able to tellif there are charges inside
…no clue…
Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.
Patterns of Fields in Space