Download Homework No. 02 (Spring 2015) PHYS 420: Electricity and Magnetism II

Homework No. 02 (Spring 2015)
PHYS 420: Electricity and Magnetism II
Due date: Tuesday, 2015 Feb 10, 4.30pm
1. (10 points.) Qualitatively sketch the electric field lines of a point charge placed (off
centered) inside a conducting cylinder. Next, sketch the electric potential of a point
charge inside a conducting cylinder. Show both the constant z cross section and constant
x cross section.
2. (50 points.) Consider a point charge placed on the axis of a perfectly conducting cylinder.
(a) Using the connection between the electric potential and Green’s function,
φ(r) = q G(r, r0 ),
(1)
and the Green function for a perfectly conducting cylinder, derived in class, determine the electric potential to be
Z ∞
q
K0 (ka)
dk ikz
φ(r) =
K0 (kρ) −
e
I0 (kρ) ,
(2)
2πε0 −∞ 2π
I0 (ka)
where r = (ρ, φ, z) and the position of the point charge r0 is chosen to be the origin.
Here a is the radius of the cylinder.
(b) Verify that the potential satisfies the boundary condition
φ(a) = 0
(3)
on the inner surface of the conducting cylinder.
(c) Using the relation E = −∇φ evaluate the electric field on the inner surface of the
conductor to be
Z
q 1 ∞ dk eikz
E(a) = ρ̂
.
(4)
2πε0 a −∞ 2π I0 (ka)
Note that the electric field is normal to the inner surface of the cylinder.
(d) Using Gauss’s theorem we can argue that the induced charge on the surface of a
conductor is given using
,
(5)
σ(φ, z) = ε0 n̂ · E
surface
where n̂ is normal to the surface of conductor. Thus, determine the induced charge
density on the inner surface of the cylinder to be
Z ∞
z
eit a
q
.
(6)
dt
σ(φ, z) = − 2 2
4π a −∞ I0 (t)
1
(e) By integrating over the surface of the cylinder determine the total induced charge
on the cylinder. Thus, find out if its magnitude is less than, equal to, or greater
than, the charge q.
2