Description
Problems
1. Spherical multipoles, Jackson, 4.1: Calculate the multipole moments qlm of the
charge distributions shown as parts (a) (15 pts) and (b) (15 pts). Try to obtain results
for the nonvanishing moments valid for all l, but in each case find the first two sets of
nonvanishing moments at the very least.
(c) (10 pts) For the charge distribution of the second set (b) write down the multipole
expansion for the potential. Keeping only the lowest-order term in the expansion,
plot the potential in the x − y plane as a function of distance from the origin for
distances greater than a.
(d) (10 pts) Calculate directly from Coulomb’s law the exact potential for (b) in the
x − y plane. Plot it as a function of distance and compare with the result found
in part (c).
Divide out the asymptotic form in parts (c) and (d) to see the behavior at large
distances more clearly.
2. The potential outside a charged disk, Zangwill 4.22: The z-axis is the symmetry
axis of a disk of radius R which lies in the x − y plane and carries a uniform charge
per unit area σ. Let Q be the total charge on the disk.
(a) (15 pts) Evaluate the exterior multipole moments and show that
φ(r, θ) = Q
4π0r
X∞
l=0
R
r
l
2
l + 2
Pl(0)Pl(cos θ) r > R. (1)
(b) (10 pts) Compute the potential at any point on the z-axis by elementary means
and confirm that your answer agrees with the part (a) when z > R. Note:
Pl(1) = 1.
3. Potential in a box: Consider a rectangular empty box with lengths (a, b, c) in (x, y, z)
direction. All surfaces of the box have zero potential, except for the side at z = c, where
the potential is V (x, y) = d0 x y with a constant d0.
(a) (20 pts) Solve the Laplace equation in Cartesian coordinates using a product
ansatz and separation of variables. Derive a general solution for the potential
with generic boundary constants.
(b) (5 pts) Determine the boundary constants by requiring the potential to take the
prescribed values on the surfaces of the box.
