Physics 318
Electromagnetism
R.G. Palmer
2/18/08
Test 1
1. If A(x) → 0 as |x| → ∞, and ∇ · J = 0, and there are only static currents, then we know
A(x) =
µ0
4π
Z
V
J (x0 ) 3 0
dx
|x − x0 |
Show, only with above, that:
a.
µ0
B(x) = ∇ × A =
4π
Z
V
J (x0 ) × (x − x0 ) 3 0
dx
|x − x0 |3
b.
∇·A=0
c.
∇2 A = −µ0 J .
2. [∼Jackson 2.4]
A point charge q is placed a distance d > R from the center of an equally charged, isolated, conducting
sphere of radius R.
a. Inside of what distance from the surface of the sphere is the point charge attracted rather than
repelled by
√ the charged sphere? Show that the answer d/R = φ where φ is the “golden ratio”,
φ = (1 + 5)/2.
b. What is the limiting value of the force of attraction when the point charge is located a distance a
(= d − R) from the surface of the sphere, if a R? [Answer: F = −q 2 /(16πε0 a2 ), i.e. image force]
c. What are the results for part (b) if the charge on the sphere is twice (or half) as large as the point
charge, but still the same sign? Is it the same charge λq on the sphere for any real λ?