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Electrodynamics Solutions
Solution 1
The ADaxis of the cube has threefold symmetry, i.e. the cube is invariant under
rotations by
about that axis.
0
120
±
Hence, the corners
,
and
are equivalent and have the same potential.
1
B
2
B
3
B
Also, the corners
,
and
are equivalent and have the same potential.
1
C
2
C
3
3
C
If a current I enters A, then:

A current
3
I
circulates through each of the resistors
.
i
AB

A current
6
I
circulates through each of the six resistors
.
j
i
C
B

A current
3
I
passes through each of the three resistors
.
D
C
j

The potential drop between A and
is
i
B
3
rI
.

The potential drop between
and
is
i
B
j
C
6
rI
.

The potential drop between
and D is
j
C
6
rI
.
The potential difference between A and D is:
rI
rI
6
5
3
1
6
1
3
1
=
⎟
⎠
⎞
⎜
⎝
⎛
+
+
Hence
r
R
6
5
=
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View Full DocumentSolution 2
a) Calculate the capacitance of the capacitor.
To find the capacitance we would like to find the voltage between the plates for a given
charge +Q and
−
Q on the two plates. We can use Gauss’s law (both with and without a
dielectric) to find the electric field due to the charge on the plates, using the
approximation that there are no end effects. Gauss’s law reads, in general
∫
=
⋅
ε
enclosed
Q
S
d
E
r
r
where
o
κε
=
is the general form of the electric permittivity in the presence of a
dielectric (note that in free space
κ
= 1).
If we use a pillbox shaped Gaussian surface with end area A
′
and with one end below the
bottom plate, which is charged to
−
Q, and the other end inside the dielectric, then Gauss’s
law gives us
,
'
'
'
A
A
Q
A
Q
A
E
S
d
E
enc
d
d
εε
σ
=
=
=
=
⋅
∫
r
r
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