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Week of February 17 05
Electric Field 1
Lecture 04
The Electric Field
Chapter 22 - HRW
Electric Field 2Week of February 17 05
Physics 2049 News
WebAssign was due today Another one is posted for
Friday You should be reading
chapter 22; The Electric Field.This is a very important concept. It is a little “mathy”
There will be a QUIZ on Friday.Material from chapters 21-22.Studying Works!
Electric Field 3Week of February 17 05
This is WAR
You are fighting the enemy on the planet Mongo.
The evil emperor Ming’s forces are behind a strange green haze.
You aim your blaster and fire … but ……
Ming themerciless
this guy is
MEAN!
Electric Field 4Week of February 17 05
Nothing Happens! The Green thing is a Force Field!
The Force may not be with you ….
Electric Field 5Week of February 17 05
Side View
TheFORCE FIELD
Force
Positiono
|Force| Big!
Electric Field 6Week of February 17 05
Properties of a FORCE FIELD
It is a property of the position in space.
There is a cause but that cause may not be known.
The force on an object is usually proportional to some property of an object which is placed into the field.
Electric Field 7Week of February 17 05
EXAMPLE: The Gravitational Field That We Live In.
m M
mgMg
Electric Field 8Week of February 17 05
The gravitational field: g
The gravitational field strength is defined as the Force per unit mass that the field creates on an object
This becomes g=(F/m)=(mg/m)=g
The field strength is a VECTOR. For this case, the gravitational
field is constant.magnitude=g (9.8 m/s) direction= down
Electric Field 9Week of February 17 05
Final Comment on Gravitational Field:
ggF
g m
m
massunit
orce
_
Even though we know what is causing the force, we really don’t usually think
about it.
Electric Field 10Week of February 17 05
Newton’s Law of Gravitation
R
MEarth
m
Electric Field 11Week of February 17 05
The Calculation
2
26
2411
2
2
/77.9
)104.6(
1061067.6
smg
x
xxg
R
M
m
R
mM
Earth
Earth
GF
g
GF
Electric Field 12Week of February 17 05
Not quite correct ….
Earth and the Moon (in background), seen from space)
Electric Field 13Week of February 17 05
More better …
FEarth
MEarth
m
Moon
Fmoon
mg
Electric Field 14Week of February 17 05
To be more precise …
g is caused byEarth (MAJOR)moon (small)Sun (smaller yet)Mongo (extremely teeny tiny)
g is therefore a function of position on the Earth and even on the time of the year or day.
Electric Field 15Week of February 17 05
The Electric Field E
In a SIMILAR WAYWe DEFINE the ELECTRIC FIELD
STRENGTH AS BEING THE FORCE PER UNIT CHARGE.
Place a charge q at a point in space.Measure (or sense) the force on the
charge – F Calculate the Electric Field by dividing
the Force by the charge,
Coulomb
Newtons
q
q
FE
EF
Electric Field 16Week of February 17 05
Electric Field 17Week of February 17 05
Electric Field Near a Charge
Electric Field 18Week of February 17 05
Two (+) Charges
Electric Field 19Week of February 17 05
Two Opposite Charges
Electric Field 20Week of February 17 05
A First Calculation
Q
r
q
A Charge
The spot where we wantto know the Electric Field
Place a “test chargeat the point and
measure the Force on it.
Electric Field 21Week of February 17 05
Doing itQ
r
q
A Charge
The spot where we wantto know the Electric Field
unit
unit
r
Qk
q
r
qQk
rF
E
rF
2
2
F
Electric Field 22Week of February 17 05
General-
unitjj
jjj
unit
unit
r
Qk
q
General
r
Qk
q
r
qQk
,2
2
2
rF
EE
rF
E
rF
Electric Field 23Week of February 17 05
Continuous Charge Distribution
Electric Field 24Week of February 17 05
ymmetry
Electric Field 25Week of February 17 05
Let’s Do it Real Time
Concept – Charge perunit length
dq= ds
Electric Field 26Week of February 17 05
The math
)sin(2
)cos(2
)cos()2(
)cos()2(
0
0
0
02
02
0
0
0
r
kd
r
kE
r
rdkE
r
dqkE
E
rdds
x
x
x
y
Why?
Electric Field 27Week of February 17 05
A Harder Problem
A line of charge=charge/length
setupsetup
dx
L
r
x
dE dEy
Electric Field 28Week of February 17 05
2/
02/322
2/
02/322
22
2
2
22
)(2
)(2
)()cos(
)(
)cos(
L
x
L
x
L
Lx
xr
dxkrE
xr
dxrkE
xr
r
xr
dxkE
(standard integral)
Electric Field 29Week of February 17 05
Completing the Math
r
kL
r
klE
Lr
L
Lrr
kLE
x
x
2
2
4
:line long VERY a oflimit In the
4
:nintegratio theDoing
22
22
1/r dependence
Electric Field 30Week of February 17 05
Dare we project this??
Point Charge goes as 1/r2
Infinite line of charge goes as 1/r1
Could it be possible that the field of an infinite plane of charge could go as 1/r0? A constant??
Electric Field 31Week of February 17 05
The Geometry
Define surface charge density=charge/unit-area
dq=dA
dA=2rdr
(z2+r2)1/2
dq= x dA = 2rdr
Electric Field 32Week of February 17 05
(z2+r2)1/2
R
z
z
rz
rdrzkE
rz
z
rz
drrk
rz
dqkdE
02/322
2/1222222
2
2)cos(
Electric Field 33Week of February 17 05
(z2+r2)1/2
Final Result
0z
220
2E
,R
12
When
Rz
zEz
Electric Field 34Week of February 17 05
Look at the “Field Lines”
Electric Field 35Week of February 17 05
What did we learn in this chapter??
We introduced the concept of the Electric FIELDFIELD.We may not know what causes
the field. (The evil Emperor Ming)
If we know where all the charges are we can CALCULATE E.
E is a VECTOR.The equation for E is the same
as for the force on a charge from Coulomb’s Law but divided by the “q of the test charge”.
Electric Field 36Week of February 17 05
What else did we learn in this chapter?
We introduced continuous distributions of charge rather than individual discrete charges.
Instead of adding the individual charges we must INTEGRATE the (dq)s.
There are three kinds of continuously distributed charges.
Electric Field 37Week of February 17 05
Kinds of continuously distributed charges Line of charge
or sometimes = the charge per unit length.
dq=ds (ds= differential of length along the line)
Area = charge per unit area dq=dA dA = dxdy (rectangular coordinates) dA= 2rdr for elemental ring of charge
Volume =charge per unit volume dq=dV dV=dxdydz or 4r2dr or some other
expressions we will look at later.
Electric Field 38Week of February 17 05
The Sphere
dqr
thk=dr
dq=dV= x surface area x thickness= x 4r2 x dr
Electric Field 39Week of February 17 05
Summary
222
,2
2
2
)()()(
r
rdsk
r
rdAk
r
rdVk
r
Qk
q
General
r
Qk
q
r
qQk
unitjj
jjj
unit
unit
E
rF
EE
rF
E
rF
(Note: I left off the unit vectors in the lastequation set, but be aware that they should
be there.)
Electric Field 40Week of February 17 05
To be remembered …
If the ELECTRIC FIELD at a point is E, then
E=F/q (This is the definition!)
Using some advanced mathematics we can derive from this equation, the fact that:
EF q
Electric Field 41Week of February 17 05
Example:
(2,8)? coordinate at the
placed isit if experience charge
coulomb 0.5 a wouldforceWhat
3xE
:expression by thegiven is
space ofregion ain field electric The
2
Electric Field 42Week of February 17 05
Solution
Newtons 6F
or
(N/C) 12 C 0.5 qE force The
matter.t doesn' coordinatey The
)/(12433xE
one!easy an is This2
CN
Electric Field 43Week of February 17 05
In the Figure, particle 1 of charge q1 = -9.00q and
particle 2 of charge q2 = +2.00q are fixed to an x
axis.
(a) As a multiple of distance L, at what coordinate on the axis is the net electric field of the particles zero?[1.89] L(b) Plot the strength of the electric field as a function of position (z).
q1 = -9q q2=+2q
Electric Field 44Week of February 17 05
Let’s do it backwards…
for this. EXCEL use sLet' brackets.
inquantity plot thejust and of
function a as at thislook can We
9
)1(
2)(
9
)(
2)(
Let
9
)(
2)(
...Then L.an greater th isthat
coordinate a hasit ofheck for the
whichposition xarbitrary an Take
222
222
22
L
qkxE
LLLqkxE
αLx
x
q
Lx
qkxE
Electric Field 45Week of February 17 05
EXCEL
a First Term Second Term Sum
-3 0.13 -1.00 -0.88
-2.9 0.13 -1.07 -0.94
-2.8 0.14 -1.15 -1.01
-2.7 0.15 -1.23 -1.09
-2.6 0.15 -1.33 -1.18
-2.5 0.16 -1.44 -1.28
-2.4 0.17 -1.56 -1.39
-2.3 0.18 -1.70 -1.52
-2.2 0.20 -1.86 -1.66
-2.1 0.21 -2.04 -1.83
-2 0.22 -2.25 -2.03
-1.9 0.24 -2.49 -2.26
-1.8 0.26 -2.78 -2.52
ETC ….
Electric Field 46Week of February 17 05
Bracket
-100.00
-50.00
0.00
50.00
100.00
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
alphaalpha=1.89
??
Electric Field 47Week of February 17 05
The mystery solved!!!
.
9
)(
2)(
0For x
9
)(
2)(
LFor x
9
)(
2)(
LFor x
22
22
22
x
q
Lx
qkxE
x
q
Lx
qkxE
x
q
Lx
qkxE
BE CAREFULL!BE CAREFULL!
Electric Field 48Week of February 17 05
In the Figure, the four particles are fixed in place and have charges q1 = q2 = +5e, q3 = +3e,
and q4 = -12e. Distance d = 9.0 mm. What is
the magnitude of the net electric field at point P due to the particles?
Electric Field 49Week of February 17 05
Electric Field 50Week of February 17 05
Figure 22-34 shows two charged particles on an x axis, q = -3.20 10-19 C at x = -4.20 m and q = +3.20 10-19 C at x = +4.20 m.
(a) What is the magnitude of the net electric field produced at point P at y = -5.60 m?[7.05e-11] N/C(b) What is its direction?[180]° (counterclockwise from the positive x axis)
Electric Field 51Week of February 17 05
Figure 22-40 shows two parallel nonconducting rings arranged with their central axes along a common line. Ring 1 has uniform charge q1 and radius R; ring 2 has
uniform charge q2 and the same radius R. The rings
are separated by a distance d = 3.00R. The net electric field at point P on the common line, at distance R from ring 1, is zero. What is the ratio q1/q2?
[0.506]
Electric Field 52Week of February 17 05
In the Figure, eight charged particles form a square array; charge q = +e and distance d = 1.8 cm. What are the magnitude and direction of the net electric field at the center?