~ From Hollow Sphere
E
~ From Solid Sphere
E
E~ From Hollow and Solid Spheres
PHYS 272 - David Blasing
Wednesday June 19th
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~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A solid sphere charged throughout its volume
Note: What we are doing today is only an approximation when you
get very close to the surface of an object.
2 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A solid sphere charged throughout its volume
Note: What we are doing today is only an approximation when you
get very close to the surface of an object.
Effects from precisely how the individual charges have arranged
themselves (each electron’s current position) become important
when you are less than about 1000 atomic diameters (< 10−7 m)
away from the surface.
2 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A solid sphere charged throughout its volume
Note: What we are doing today is only an approximation when you
get very close to the surface of an object.
Effects from precisely how the individual charges have arranged
themselves (each electron’s current position) become important
when you are less than about 1000 atomic diameters (< 10−7 m)
away from the surface.
Outside this range, the charges can be treated as uniform sheets of
charge rather than individual charges. This is how we are treating
them from here on out.
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~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
Note on Gauss’s Law
1
Gauss’s law can get E~net from certain highly symmetric charge
distributions very easily.
3 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
Note on Gauss’s Law
1
2
Gauss’s law can get E~net from certain highly symmetric charge
distributions very easily.
Gauss’s law is a topic covered later, and we will be able to
prove the following then.
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~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
Note on Gauss’s Law
1
Gauss’s law can get E~net from certain highly symmetric charge
distributions very easily.
2
Gauss’s law is a topic covered later, and we will be able to
prove the following then.
3
It can also be done through direct integration, but its messy.
For the sake of time I won’t present the derivation, but you
should go through it on your own.
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~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
E~ From Hollow Sphere
Q is the total charge on the
sphere’s surface, R is the radius
of the sphere
For r > R, E~sphere =
1 Q
4π0 r 2 rˆ
For r < R, E~sphere = 0
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~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A spherical shell of charge
Note: a uniformly charged spherical shell (as long as its stays
uniformly charged) interacts with external fields like a point
charge
The shell must be spherically symmetric and thin enough as
to neglect any effects arising from this thickness
5 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A spherical shell of charge
Note: a uniformly charged spherical shell (as long as its stays
uniformly charged) interacts with external fields like a point
charge
The shell must be spherically symmetric and thin enough as
to neglect any effects arising from this thickness
It does not matter what the spherical shell is made of
It only matters that the charge is distributed in a thin,
spherically symmetric shell
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~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A spherical shell of charge
A qualitative drawing showing that E~net sums to that of a point
charge outside the spherical shell of charge:
6 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
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Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A spherical shell of charge
A qualitative drawing showing that E~net sums to ~0 inside a uniform
spherical shell of charge:
7 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
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Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A spherical shell of charge
Group question: you have a spherical shell that is uniformly
charged on only its surface and filled with a plastic. What is
the polarization of the molecules in the plastic inside of the
charged shell? Why?
~ different very close to the charged surface of the sphere
Is P
as compared to the center? Why?
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~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A spherical shell of charge
Answer:
~ = αE~ . So no E~ , no P.
~ It doesn’t matter at all
Recall that P
if there are charges around, it only matters what the local E~net
is at the location of the plastic/dielectric material.
Since the E~net = ~0 inside a uniformly charged spherical shell,
~ = ~0 every in the plastic
P
Even if the molecules are very close to the charged surface,
E~net is still zero. There will be no polarization of those
molecules.
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~ From Hollow Sphere
E
~ From Solid Sphere
E
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
A solid sphere charged throughout its volume
Summary: E~ from a thin uniformly charged shell:

for r < R, E~ = ~0
for r > R, E~ = 1 Q rˆ
4π0 r 2
Outside, E~ is exactly that of a point charge with the total charge
(Q) located at the center
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Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
~ From Hollow Sphere
E
~ From Solid Sphere
E
A spherical shell of charge
~ of Hollow Shell vs Distance
| E|
1
0.9
0.7
0.6
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
0.8
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
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Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
~ From Hollow Sphere
E
~ From Solid Sphere
E
Clicker Question 1
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Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
~ From Hollow Sphere
E
~ From Solid Sphere
E
Clicker Question 2
13 / 32
Validity
Note on Gauss’s Law
~ From Hollow Sphere
E
Clicker Questions
~ From Hollow Sphere
E
~ From Solid Sphere
E
Clicker Question 3
14 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A Vague Group Question
Discuss in small group the following question, why can’t
1 Q
E~ = 4π
ˆ be the whole story for the E~ field from a point charge?
2r
0 r
15 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
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Hollow Vs Solid
A Vague Group Question
Discuss in small group the following question, why can’t
1 Q
E~ = 4π
ˆ be the whole story for the E~ field from a point charge?
2r
0 r
It is undefined at r=0. E~ can be made arbitrarily big by making r
arbitrarily small. Fields take energy to set up, so that formula says
the field from point charges carry an unlimited amount of energy.
15 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A Vague Group Question
Discuss in small group the following question, why can’t
1 Q
E~ = 4π
ˆ be the whole story for the E~ field from a point charge?
2r
0 r
It is undefined at r=0. E~ can be made arbitrarily big by making r
arbitrarily small. Fields take energy to set up, so that formula says
the field from point charges carry an unlimited amount of energy.
When specifically do you think this formula breaks down?
15 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A Vague Group Question
Discuss in small group the following question, why can’t
1 Q
E~ = 4π
ˆ be the whole story for the E~ field from a point charge?
2r
0 r
It is undefined at r=0. E~ can be made arbitrarily big by making r
arbitrarily small. Fields take energy to set up, so that formula says
the field from point charges carry an unlimited amount of energy.
When specifically do you think this formula breaks down?
This formula is not accurate when the distance becomes smaller
than the radius of the point charge. Next, we are going to write
down a better approximation for the electric field “inside” a point
charge.
15 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
E~ from of a solid sphere (radius R, total charge Q) charged
throughout its volume:
Model the sphere as a series of concentric spherical shells
At a location outside the sphere, we are also outside of all
spherical shells
Each spherical shell creates an electric field like a point charge
would at the center of the shell
Question: what is the electric field outside a charged solid sphere?
16 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
Electric field of a solid sphere charged throughout its volume, for
r > R:
E=
1 Q
4π0 r 2
17 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
Electric field of a solid sphere charged throughout its volume:
Now r < R, same ”model” as above
At a location inside the sphere we are inside some of the
spherical shells - they contribute nothing to E~net
At a location inside the sphere we are outside some of the
spherical shells - they contribute to E~net
18 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
For r < R
19 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
For r < R
20 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
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~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
For r < R
21 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
For r < R
22 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
For r < R
23 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
For r < R
24 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
For r < R
25 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
For r < R
26 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
Summary: E~ from a solid sphere charged throughout its volume:

1 Q


r rˆ
for r < R, E~ =
4π0 R 3
1 Q


for r > R, E~ =
rˆ
4π0 r 2
Inside E~ grows linearly; outside it falls ∝
E~ of a point charge.
1
r2
and is exactly like the
27 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
~ of Solid Sphere vs Distance
| E|
1
0.9
0.7
0.6
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
0.8
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
28 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
~ of Solid Sphere vs Distance
| E|
1
Sanity Check:
0.9
E~ = ~0 at r=0 X
0.7
0.6
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
0.8
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
29 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
~ of Solid Sphere vs Distance
| E|
1
Sanity Check:
0.9
E~ = ~0 at r=0 X
0.7
E~ grows as we
enclose more charge
for r < R X
0.6
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
0.8
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
29 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
~ of Solid Sphere vs Distance
| E|
1
Sanity Check:
0.9
E~ = ~0 at r=0 X
0.7
E~ grows as we
enclose more charge
for r < R X
0.6
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
0.8
0.5
0.4
0.3
E~ = E~Point Charge
for r > R X
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
29 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
~ of Solid Sphere vs Distance
| E|
0.9
0.8
0.8
Q
1
4 π ǫ 0 R2 )
0.9
0.7
0.7
0.6
~ (Units of
| E|
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0
~ of Hollow Shell vs Distance
| E|
1
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
1
0.1
0.5
1
1.5
2
2.5
3
Radial Distance (Units of R)
3.5
4
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
Inside they are different, but outside both are exactly the same as
a point charge
30 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
~ of Solid Sphere vs Distance
| E|
0.9
0.8
0.8
Q
1
4 π ǫ 0 R2 )
0.9
0.7
0.7
0.6
~ (Units of
| E|
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0
~ of Hollow Shell vs Distance
| E|
1
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
1
0.1
0.5
1
1.5
2
2.5
3
Radial Distance (Units of R)
3.5
4
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
Inside they are different, but outside both are exactly the same as
a point charge
So the sanity check of reducing to a point charge far away doesen’t
help in this case
30 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
Note: E~ always jumps by σ0 across a uniformly charged thin
surface (can be proved with Gauss’s law).
31 / 32
~ From Hollow Sphere
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~ From Solid Sphere
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~ From Solid Sphere
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Hollow Vs Solid
A solid sphere charged throughout its volume
E~ jumps by
σ
0
across a uniformly thing charged surface
~ of Hollow Shell vs Distance
| E|
1
0.9
0.7
0.6
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
0.8
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
Group discussion question: use this to verify the jump in the above
graph.
32 / 32
~ From Hollow Sphere
E
~ From Solid Sphere
E
~ From Solid Sphere
E
Hollow Vs Solid
A solid sphere charged throughout its volume
E~ jumps by
σ
0
across a uniformly thing charged surface
~ of Hollow Shell vs Distance
| E|
1
0.9
0.7
0.6
~ (Units of
| E|
Q
1
4 π ǫ 0 R2 )
0.8
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Radial Distance (Units of R)
Group discussion question: use this to verify the jump in the above
graph.
For the hollow shell, σ, the charge per area ( Q
A ), is
Q
σ
jumps from 0 to 0 = 4π0 R 2
Q
.
4πR 2
So E~
32 / 32