Download Example 1.4 Consider an infinitely long cylinder with charge density

Example 1.4
Consider an infinitely long cylinder with charge density ρ,
dielectric constant ε 0 and radius r0 . What is the electric field in
and around the cylinder?
L
r0
ρ
r
ε0
Solution
Because of the cylinder symmetry one expects the electric field
to be only dependent on the radius, r. Applying Gauss's law one
finds:
E ⋅ A = E 2π rL =
Q ρπ r 2 L
=
for r < r0
ε0
ε0
and
E ⋅ A = E 2π rL =
Q ρπ r02 L
=
for r > r0
ε0
ε0
where a cylinder with length L was chosen to define the surface
A, and edge effects were ignored. The electric field then equals:
E (r) =
ρr
ρ r02
for r < r0 and E ( r ) =
for r > r0
2ε 0
2ε 0 r
The electric field therefore increases within the cylinder with
increasing radius as shown in the figure below. The electric
field decreases outside the cylinder with increasing radius.
09/ 04/ 02
© B. Van Zeghbroeck, 1998
E
ρ r0
2ε 0
r
0
0
r0