Physics 318
Electromagnetism
R.G. Palmer
2/9/09
Test 1
1. Consider a three-dimensional vector field V (x) such that
∇·V =s
(1)
∇×V =c
(2)
with, of course ∇ · c = 0, and s and c are bounded.
Show that if
V = −∇φ + ∇ × A
where (r ≡ |x − x0 |)
Z
s(x0 ) 3 0
1
dx
4π
r
Z
1
c(x0 ) 3 0
A(x) =
dx
4π
r
φ(x) =
then V satisfies eq (1) and (2).
You can assume ∇2 1r = −4πδ(r), and any formulas and theorems from Jackson’s endnotes.
2. A long right circular cylinder (a rod) of dielectric constant κ of radius a has its axis coincident with the
z and its ends at z = −L/2 and z = L/2 with L a. There is a previously uniform electric field E 0
along the cylinder (i.e. E 0 = E0 k̂). It may not be what you first think.
a. Show that the bound volume-charge ρb is zero.
b. Find the fields E, D, and P at the center (x = y = z = 0).
c. Find the field E, D, and P just inside and outside the end of the cylinder along the axis (x = y = 0,
z = 21 L± .
d. Make two sketches for E and D, and add the bound charges (+, −) on at least one of your sketches.
3. A toroidal coil consists of a circular ring, or “donut,” around which a long wire is wrapped. The winding
is uniform and tight enough so that each turn can be considered a closed loop. The coil has radius a,
N turns per unit length, and the current I.
Find the magnetic field B at all points, both inside and outside the coil. You must show the direction
of the field as well as the magnitude.