Classical Electrodynamics PHY5346

HOMEWORK 9

(November 1, 2012)

Due on Tuesday, November 20, 2012

PROBLEM 25

Consider the problem of a ring of radius a carrying a steady current I. We haveshown in class that the magnetic scalar potential for points along the z-axis may bewritten as:

M(r, =0) = 2piIc

rr2 + a2

.

(a) Use this information to compute the magnetic scalar potential M(r, ) at anarbitrary point in space satisfying r

Pl(0) .

PROBLEM 26

Consider the problem of a ring of radius a carrying a steady current I. To answerthe following questions you should use the results derived in class for the vector andmagnetic scalar potentials (they are provided here in the second page).

(a) Using the vector potential A expressed in terms of an elliptic integral, make aplot of A(r, ) (in suitable units) from r/a = 0 to r/a = 5 for the followingvalues of : =0, =pi/6, =pi/3, and =pi/2.

(b) Using the vector potential A expressed in terms of an elliptic integral, obtainthe magnetic field B in the limit of ra.

(c) Using the vector potential A expressed in terms of a sum over spherical harmon-ics, obtain the magnetic field B in the limit of ra.

(d) Using the magnetic scalar potential M , obtain the magnetic field B in the limitof ra.

PROBLEM 27 (J.D. Jackson 5.3)

A right-circular solenoid of finite length L and radius a has N turns per unit lengthand carries a current I. Show that the magnetic induction on the cylinder axis in thelimit NL is

Bz =2piNI

c(cos 1 + cos 2) ,

where the angles are defined in the figure of Problem 5.3 of J.D. Jackson.

Results for the vector and magnetic scalar potentialsfor a ring of radius a carrying a steady current I

(a) Elliptic Integrals:

A(r) = Ia

cr2 + a2

2pi0

dcos

1 cos , where 2ar sin

r2 + a2.

(b) Spherical Harmonics:

A(r) = 8pi2Ia

c

l=1

Yl,1(pi/2, 0)

2l + 1

rl

Yl,1(, 0) , where ra .