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Physics PHYS 354
Electricity and Magnetism II
Problem Set #8
1.
Rotating Sphere
Total charge Q is uniformly distributed over the surface of a sphere of radius R, as
shown in the figure below, so that the charge density is

Q
4R 2
The sphere rotates about the z axis with angular velocity . The charge is fixed to
the surface of the sphere and rotates with it, so that there is a surface current,

K  R sin ˆ .
z


P
r


R
r
y

x
a)
Write down the electrostatic field inside and outside the sphere.
b)
Calculate the magnetostatic field inside and outside the sphere. Assume a


constant field B in the z direction inside, and the field of a dipole m at


the center outside. Choose B and m to satisfy the boundary conditions at
r R .
c)
Now suppose  is slowly increased from zero. The magnetic field found
in (b) will change with time and induce an electric field throughout space
in the ˆ direction. Calculate the induced electric field on r R as a
function of  .
d)
The field found in (c) produces a torque opposing increases in .
Calculate it. Suppose the sphere has a moment of inertia I about the z
axis, and is acted on by an external torque G. Write down the equation of
motion that determines  and show that the sphere behaves as if it had an
additional moment of inertia:
I magnetic 
2 kQ2 R
.
9 c2
B2 3
 2 0 d x , the rate energy is stored in the magnetic field. Show
that 67% of the energy is stored inside the sphere, and 33% outside. Show
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that the total is  2 I magnetic .
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e)
Calculate
f)
Calculate the angular momentum stored in the field,


 
 
L  x  0 E B d 3 x .
 
 
Clearly, since E 0 inside, L 0 inside, so there is no contribution to this
integral from inside the sphere.
Note: in parts (e) and (f), use the static fields found in (a) and (b).
2.
Energy Conservation
A very long solenoid consists of thin wire tightly wound in a cylinder of radius R
with n turns per unit length. The wire carries current I. The resistance of the wire
is negligible so current flows with no external E.M.F. Mechanical equilibrium is
maintained by balancing the force due to a tension in the wire against the
magnetic force. Now R is allowed to increase slowly (n is unchanged).
a)
Show that I is proportional to R 2 .
b)
Find how the energy stored in the magnetic field changes.
c)
Demonstrate how energy conservation works in this situation.
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