Page 198 - Electromagnetics

3.8 Use the expansion

1
−(2n+1)k ρ d
∞
= csch k ρ d = 2 e
sinh k ρ d
n=0
to show that the secondary Green’s function for parallel conducting plates (3.212)may
be written as an infinite sequence of images of the primary point charge. Identify the
geometrical meaning of each image term.

3.9 Find the Green’s functions for a dielectric slab of thickness d placed over a perfectly
conducting ground plane located at z = 0.

3.10 Find the Green’s functions for a dielectric slab of thickness 2d immersed in free
space and centered on the z = 0 plane. Compare to the Green’s function found in
Problem 3.9.

3.11 Referring to the system of Figure 3.9, find the charge density on the surface of
the sphere and integrate to show that the total charge is equal to the image charge.

3.12 Use the method of Green’s functions to find the potential inside a conducting
sphere for ρ inside the sphere.

3.13 Solve for the total potential and electric field of a grounded conducting sphere
centered at the origin within a uniform impressed electric field E = E 0 ˆ z. Find total
charge induced on the sphere.

3.14 Consider a spherical cavity of radius a centered at the origin within a homogeneous
dielectric material of permittivity = 0 r . Solve for total potential and electric field
inside the cavity in the presence of an impressed field E = E 0 ˆ z. Show that the field in
the cavity is stronger than the applied field, and explain this using polarization surface
charge.
3.15 Find the field of a point charge Q located at z = d above a perfectly conducting
ground plane at z = 0. Use the boundary condition to find the charge density on the
plane and integrate to show that the total charge is −Q. Integrate Maxwell’s stress
tensor over the surface of the ground plane and show that the force on the ground plane
is the same as the force on the image charge found from Coulomb’s law.

3.16 Consider in free space a point charge −q at r = r 0 + d, a point charge −q at
r = r 0 − d, and a point charge 2q at r 0 . Find the first three multipole moments and the
resulting potential produced by this charge distribution.

3.17 A spherical charge distribution of radius a in free space has the density
Q
ρ(r) = 3 cos 2θ.
πa
Compute the multipole moments for the charge distribution and find the resulting poten-
tial. Find a suitable arrangement of point charges that will produce the same potential
field for r > a as produced by the spherical charge.

3.18 Compute the magnetic flux density B for the circular wire loop of Figure 3.18 by
(a) using the Biot–Savart law (3.165), and (b) computing the curl of (3.138).

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