Page 199 - Electromagnetics

Figure 3.27: Parallel plate capacitor.

3.19 Two circular current-carrying wires are arranged coaxially along the z-axis. Loop
1 has radius a 1 , carries current I 1 , and is centered in the z = 0 plane. Loop 2 has radius
a 2 , carries current I 2 , and is centered in the z = d plane. Find the force between the
loops.
1

3.20 Choose Q =∇ × c in (3.162)and derive the following expression for B:
R
1
µ

B(r) = J(r ) ×∇ dV −
4π V R
1
1
1

− [ˆ n × B(r )] ×∇ + [ˆ n · B(r )]∇ dS ,
4π S R R
where ˆ n is the normal vector outward from V . Compare to the Stratton–Chu formula
(6.8).
3.21 Compute the curl of (3.163)to obtain the integral expression for B given in Prob-
lem 3.20. Compare to the Stratton–Chu formula (6.8).
3.22 Obtain (3.170)by integration of Maxwell’s stress tensor over the xz-plane.
3.23 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 an excess charge
+Q to the top plate (and thus −Q is drawn onto the bottom plate.)Neglecting fringing,
(a)solve Laplace’s equation to show that
Q
(z) = z.
A
Use (3.87)to show that
2
Q d
W = .
2A
(b)Verify W using (3.88). (c) Use F =−ˆ zdW/dz to show that the force on the top plate
is

Q 2
F =−ˆ z .
2A
(d)Verify F by integrating Maxwell’s stress tensor over a closed surface surrounding the
top plate.

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