Phys 2920, Spring 2009
Problem Set #6
1. Find a set of formulae which transforms cylindrical coordinates (ρ, φ, z ) to spherical
coordinates, (r, θ, φ).
2. Express the following (spherical) entities in terms of the Cartesian unit vectors i,jand
k:
a) ˆ
er,ˆ
eθand ˆ
eφfor r= 1, θ=π
2,φ= 0
b) ˆ
er,ˆ
eθand ˆ
eφfor r= 1, θ=π
2,φ=π
2
c) ˆ
erfor θ=π. (Do ˆ
eθand ˆ
eφhave any meaning for this case?)
3. (VA 7.38) Express each of the following loci in spherical coordinates:
a) the sphere x2+y2+z2= 9
b) the cone z2= 3(x2+y2)
c) the paraboloid z=x2+y2
d) the plane z= 0
e) the plane y=x
4. (VA 7.43) Represent the vector a= 2yi−zj+ 3xkin spherical coordinates and determine
ar,aθand aφ.
5. If
A=p0ω2
4π0c2cos θ
rcos[ω(t−r/c)]ˆ
er
find ∇× A.(Note, even though there’s a tin there, which does stand for time, the derivatives
of the curl treat it as any other constant.)
Here, p0,ωare constants. The formula for Ais in fact the vector potential far from an
electric dipole which has magnitude p0and oscillates with angular frequency ω.
6. If
V=α
r+β
r2cos θ
where αand βare constants, show that ∇2V= 0.
7. Prove that for a function Φ given in cylindrical coordinates by
Φ(ρ, φ) = ln ρ
a+Aρn+B
ρn(Csin nφ +Dcos nφ),
where A,B,C,Dare all constants and nis an integer, we have ∇2Φ = 0.
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