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C:\User\Teaching\505506\F10\HW02sol.doc
PHY 505
Classical Electrodynamics I
Fall 2010
Homework 2 with solutions
Problem 2.1
(to be graded of 10 points). Use the Gauss law to calculate the
mutual capacitance of the following 2electrode systems, with the same cross
section (see Fig. on the right):
(i) a conducting sphere inside a concentric spherical cavity in another
conductor, and
(ii) a conducting cylinder inside a coaxial cavity in another conductor.
(In this case, we speak about capacitance per unit length.)
Compare the results with those obtained in class using the Laplace
equation solution.
Solutions:
(i) Applying the Gauss Law to a sphere of radius
r
in the range
a
<
r
<
b
, for the electric field
E
= E(r)
n
r
, we get
,
4
)
(
2
0
r
Q
r
E
where
Q
is the charge of the inner conductor. Integrating this result, we find that voltage
V
(
b
) 
(
a
)
between the electrodes is
,
1
1
4
4
)
(
0
2
0
b
a
Q
r
dr
Q
dr
r
E
V
b
a
b
a
so that the mutual capacitance
a
b
ab
b
a
V
Q
C
m
0
1
0
4
1
1
4
,
in agreement with Eq. (2.56) of the lecture notes.
(ii) An absolutely similar calculation, but applied to a cylinder, with charge
Q
/
L
per unit
length, gives
)
/
ln(
2
,
2
)
(
0
0
a
b
L
C
r
r
E
m
,
in agreement with Eq. (2.49).
Problem 2.2
(10 points). Following the class discussion of two weakly coupled conducting spheres, find
an approximate expression for the mutual capacitance (per unit length) between two thin, parallel wires,
each with a round crosssection, but its own diameter. Both diameters,
D
1
and
D
2
, are much smaller than
the distance
d
between the wires. Compare the result with that for two spheres, and interpret the
difference.
a
b
a
0
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C:\User\Teaching\505506\F10\HW02sol.doc
Solution
: This problem is close to that solved in class for two small spheres, but we should be more
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 Fall '08
 Stephens,P
 Capacitance, Work

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