Jackson 1.7 Homework Problem Solution
Dr. Christopher S. Baird
University of Massachusetts Lowell
PROBLEM:
Two long, cylindrical conductors of radii a1 and a2 are parallel and separated by a distance d, which is
large compared with either radius. Show that the capacitance per unit length is given approximately by
C≈π ϵ0
(
ln d
a
)
−1
where a is the geometrical mean of the two radii. Approximately what gauge wire (state diameter in
millimeters) would be necessary to make a two-wire transmission line with a capacitance of 1.2 x 10-11
F/m if the separation of the wires was 0.5 cm? 1.5 cm? 5.0 cm?
SOLUTION:
Place the center of cylinder 1 at the origin and the center of cylinder 2 at x = d. Place a total charge per
unit length -Q/L on cylinder 1 and +Q/L on cylinder 2. We first pretend cylinder 2 is not present and
solve by superposition. Without cylinder 2, the field lines from cylinder 1 are radially outwards. This
symmetry allows us to use Gauss's law in integral form. Draw a cylindrical Gaussian surface centered
on cylinder 1 and enclosing it.
∮E1
⋅nda=qenc
ϵ0
E1=−Q
2πr L ϵ0
r
The second cylinder by itself will create a similar field, only shifted to the right:
E2=Q
2π
∣
x−d
i
∣
2Lϵ0
(x−d
i)
So that the total field is:
E=−Q
2π( x2+y2)Lϵ0
(x
i+y
j)+ Q
2π (( x−d)2+y2)Lϵ0
(( x−d)
i+y
j)
The potential is then:
∇ Φ= Q
2π(x2+y2)Lϵ0
(x
i+y
j)− Q
2π (( x−d)2+y2)Lϵ0
(( x−d)
i+y
j)
