PHY 3323
December 7, 2009
καὶ τάχα δὴ
Exam #3
α κoύoυσι βo ώντ ων τω̂ν στ ρατ ιωτω̂ν
,
Θάλατ τ α θάλατ τ α καὶ παρǫγγυ ω̂ντ ων.
Xenophon, Anabasis, IV.7
(1) Consider a circular loop of radius R which lies in the x-y plane. Suppose that the loop
carries a current I directed counterclockwise as viewed from above.
a) Write an explicit integral expression for the x component of the magnetic field at
position ~r = (0, y, z). (10 points)
b) Write an explicit integral expression for the y component of the magnetic field at
position ~r = (0, y, z). (10 points)
c) Write an explicit integral expression for the z component of the magnetic field at
position ~r = (0, y, z). (10 points)
d) Evaluate the integral for Bx in part a). If you could not get the integral at least
evaluate the leading multipole contribution to the x component. (5 points)
(2) Consider a spherical shell of radius R which carries a surface charge density σ and is
rotating with angular velocity ~ω . Recall that the vector potential at position ~r inside
the sphere is,
~ = 1 µ0 Rσ(~ω ×~r) .
A
3
For ~r outisde the sphere the vector potential is,
~ = 1 µ0 Rσ(~ω ×~r) R33 .
A
3
r
a) Find the magnetic field inside the sphere in any coordinate system of your choice.
(10 points)
b) Find the magnetic field outside the sphere in any coordinate system of your choice.
(10 points)
c) Find the magnitude of the total force exerted on the top half of the sphere. (10
points)
(3) Consider a sphere of radius R and magnetic permeability µ. It turns out that a
magnetic dipole m
~ at the origin induces the following magnetic field inside the sphere,
~ b
r )b
r −m]
~
µ 2µ0 −2µ m
~
~ = µ [3(m·
B
+
4π
4π 2µ0 +µ R3 .
r3
a) What is the magnetization inside the sphere? (15 points)
b) What are the bound volume current and the bound surface current? Don’t worry
about the possibility of a delta function at the origin. (15 points)
c) Suppose you are told that the field outside the sphere is that of a dipole with
moment m
~ ′ . What is m
~ ′ ? (15 points)