PHY 3323
December 5, 2011
Exam #3
Â
eØn', ‚ggèllein Lakedaimonoij íti t¨de
kemeqa toØj kenwn û masi peiqìmenoi
Simonides
Spartans’ Epitaph
(1) Consider two loops of radius R which carry current I counterclockwise. Each loop is
parallel to the x-y plane, and is centered on the z axis. The top loop is at a height d/2
above the x-y plane, and the bottom loop is d/2 below the x-y plane.
a) Use the Biot-Savart Law to find the total magnetic field vector at position z on
the z axis. (30 points)
2 vanishes at z = 0.
~
b) Suppose there is a value of the separation d such that ∂ 2 B/∂z
On the basis of dimensional analysis, what must be the forms of d and the value
~ at z = 0 for this value of d? (30 points)
of B
~ at z = 0 for this value of d. (20 points)
c) Find d and the value of B
(2) A uniformly charged solid sphere of radius R carries a total charge Q, and is set
spinning with angular velocity ω about the z axis.
a) What form must the magnetic dipole moment m
~ of the sphere take on the basis of
dimensional analysis? (20 points)
b) What is the current density inside the sphere? (20 points)
c) What is the magnetic dipole moment of the sphere? (20 points)
d) What is the leading large distance form of the vector potential? (20 points)
(3) At the interface (the x-y plane) between one linear magnetic material and another the
magentic field lines bend as follows:
h
i
~
z + sin(θ1 )b
x ,
z>0
=⇒ µ1
and
B1 = −B1 cos(θ1 )b
z<0
=⇒ µ2
and
h
i
~ 2 = −B2 cos(θ2 )b
z + sin(θ2 )b
x .
B
~ in each region? (30 points)
a) What is the field H
b) Use the boundary conditions at the interface to solve for B2 and θ2 in terms of B1
and θ1 . (30 points)
~ in each region? (30 points).
c) What is the magnetization density M