Page 200 - Electromagnetics

3.24 Consider two thin conducting parallel plates embedded in a region of permittivity
(Figure 3.27). The bottom plate is connected to ground, and we apply a potential V 0 to
the top plate using a battery. Neglecting fringing, (a)solve Laplace’s equation to show
that
V 0
(z) = z.
d
Use (3.87)to show that
2
V A
0
W = .
2d
(b)Verify W using (3.88). (c) Use F =−ˆ zdW/dz to show that the force on the top plate
is
2
V A
0
F =−ˆ z .
2d 2
(d)Verify F by integrating Maxwell’s stress tensor over a closed surface surrounding the
top plate.
3.25 A group of N perfectly conducting bodies is arranged in free space. Body n is
held at potential V n with respect to ground, and charge Q n is induced upon its surface.
By linearity we may write
N

Q m = c mn V n
n=1
where the c mn are called the capacitance coefficients. Using Green’s reciprocation the-
orem, demonstrate that c mn = c nm . Hint: Use (3.210). Choose one set of voltages so
that V k = 0, k = n, and place V n at some potential, say V n = V 0 , producing the set of
charges {Q k }. For the second set choose V = 0, k = m, and V m = V 0 , producing {Q }.

k k
3.26 For the set of conductors of Problem 3.25, show that we may write

Q m = C mm V m + C mk (V m − V k )
k =m
where
N

C mn =−c mn , m = n, C mm = c mk .
k=1
Here C mm , called the self capacitance, describes the interaction between the mth con-
ductor and ground, while C mn , called the mutual capacitance, describes the interaction
between the mth and nth conductors.

3.27 For the set of conductors of Problem 3.25, show that the stored electric energy is
given by
N N
1
W = c mn V n V m .
2
m=1 n=1
3.28 A group of N wires is arranged in free space as shown in Figure 3.28. Wire n
carries a steady current I n , and a flux n passes through the surface defined by its
contour n . By linearity we may write
N

m = L mn I n
n=1

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