PHZ 3113 Fall 2010
Homework #5, Due Wednesday, October 13
1. Work out in detail the value of ∇ · (∇φ × ∇ψ).
2. Let v(x, y) be the vector field
v=
(x − y)
(x + y)
x̂ +
ŷ,
3
r
r3
p
where r = x2 + y 2 . Let C be the circumference of the circle of radius r = 1 in the x-y
plane centered at the origin.
I
(a) Compute
v · n̂ ds two different ways.
C
I
(b) Compute
v · dr two different ways.
C
(c) What happens when the denominator in v is r 2 instead of r 3 ?
3. (a) Apply the divergence theorem to compute the integral
Z
(φ1 ∇2 φ2 − φ2 ∇2 φ1 ) d3 v
V
as
involving ∂φ1 /∂n = n̂ · ∇φ1 , and ∂φ2 /∂n = n̂ · ∇φ2 . [Hint: start with
R a surface integral
3 v; do it again exchanging φ and φ , and subtract.]
∇
·
(φ
∇φ
)
d
1
2
1
2
V
(b) Suppose φ1 is the electrostatic potential obtained from charge density ρ1 in the interior
of V with surface charge density σ1 on the surface S of V , such that
ρ
∇2 φ 1 = − 1 ,
0
∂φ1 σ1
= ;
∂n S
0
and φ2 is found similarly from ρ2 and σ2 . Prove the reciprocity theorem,
Z
Z
Z
Z
3
2
3
ρ1 φ2 d v +
σ1 φ2 d a =
ρ2 φ1 d v +
σ2 φ1 d2 a
V
S
V
S
(the energy of ρ1 , σ1 in the potential φ2 is the same as the energy of ρ2 , σ2 in the potential
φ1 ).