PHY 3323
December 6, 2010
Exam #3
παράτ α τ η ζωή σoυ
αγάπα τ η σχoλή σoυ
και διάβαζǫ πoλύ
Greek Anarchist Slogan
(1) A semicircular wire, lying in the lower half of the x-y plane with radius R and centered
at the origin, carries a steady current I in the counterclockwise direction. This problem
concerns the magnetic field at a position (in the upper half plane) ~r = R cos(θ)b
x+
R sin(θ)b
y . Of course the semicircular wire must be hooked up to some other wires to
complete the circuit but we want just the effect of the semicircular wire.
~ r) take from dimensional analysis? (10 points)
a) What form must B(~
~ r). (20 points)
b) Write down an explicit integral expression for 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? (10 points)
b) What is the current density inside the sphere? (10 points)
c) What is the magnetic dipole moment of the sphere? (10 points)
d) What is the leading large distance form of the vector potential? (10 points)
(3) This problem concerns the analogy between electrostatics and magnetostatics.
a) Write down the equations of electrostatics with no free charges. (10 points)
b) Write down the equations of magnetostatics with no free currents. (10 points)
c) Use the analogy to find the magnetic field inside a uniformly magnetized sphere
~ 0 , given that the electric field inside a uniformly polarized
with magnetization M
~0 is E
~ = − P~0 . (10 points)
sphere with polarization P
3ǫ0
d) Use the analogy to find the magnetic field inside a sphere of linear magnetic material
~ 0 , given that the
with susceptibility χm in an otherwise uniform magnetic field B
electric field inside a sphere of linear dielectric material with susceptibility χe in
~ 0 is E
~ = 3E~ 0 . (10 points)
an otherwise uniform electric field E
3+χe
e) Use the analogy to find the average magnetic field over sphere, due to steady
currents within the sphere, given that the average electric field over a sphere, due
~ ave = −~p 3 , where R is the radius of
to constant charges within the sphere, is E
4πǫ0 R
the sphere and p~ is the electric dipole moment of the sphere. (10 points)