PS041
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Spring 2010
Problem Set 4 Solution
Problem 1:
Experiment:
Expt. 2: Faraday Ice Pail
Capacitance of our Experimental SetUp
Part 1
Consider two nested cylindrical conductors of height
h
and radii
a
&
b
respectively.
A charge +
Q
is evenly distributed on the outer surface of the pail (the inner
cylinder),
Q
on the inner surface of the shield (the outer cylinder).
(a)
Calculate the electric field between the two cylinders (
a
<
r
<
b
).
For this we use Gauss’s Law, with a Gaussian cylinder of radius
r
, height
l
00
0
1
2(
)
2
inside
arb
Q
QQ
dr
l
E
lE
r
hr
h
π
ε
επ
<<
⋅=
=
=
⇒
=
∫∫
EA
r
r
±
(b)
Calculate the potential difference between the two cylinders:
The potential difference between the outer shell and the inner cylinder is
0
()
l
n
l
n
22
2
a
a
b
b
QQQ
b
VV
aV
b
d
r
r
rh
h
h a
π ε
πε
⎛⎞
′′
Δ=
−
=
−
=
−
=
⎜⎟
′
⎝⎠
∫
(c)
Calculate the capacitance of this system,
C = Q/
Δ
V
0
2

ln
ln
2
o
h
C
Qb
b
V
ha
a
==
=
Δ
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(d)
Numerically evaluate the capacitance for your experimental setup, given:
h
≅
15 cm,
a
≅
4.75 cm and
b
≅
7.25 cm
9
1
2
11
5
c
m
20 pF
7.25 cm
29 10 m
F
ln
ln
4.75 cm
o
h
C
b
a
πε
==
≅
⋅×
⎛⎞
⎛
⎞
⎜⎟
⎜
⎟
⎝⎠
⎝
⎠
e)
Find the electric field energy density at any point between the conducting
cylinders. How much energy resides in a cylindrical shell between the conductors of
radius
r
(with
arb
<<
), height
h
,
thickness
dr
, and volume
2
rhdr
π
? Integrate your
expression to find the total energy stored in the capacitor and compare your result with
that obtained using
2
(1/ 2) (
)
E
UC
V
Δ
=
.
The total energy stored in the capacitor is
2
2
00
0
22
2
E
Q
uE
rh
εε
Then
2
2
0
1
2
4
E
QQ
d
r
dU
u dV
hr
επ
=
Integrating we find that
ln( / )
44
bb
aa
Qd
r
Q
Ud
U
b
a
h
=
∫∫
.
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 Spring '08
 Hughes
 Capacitance, Charge, Magnetism, Energy, Mass, Magnetic Field, Electric charge, Energy density

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