Find the dipole moment of a uniformly magnetized material in the shape of an ellipsoid of revolution, as shown in the figure. We know that the field outside the material would be the same as that of a point dipole if we had a sphere (a=b) instead of an ellipsoid. Estimate the quadrupole moment of the geometry. Write down the integrals for the expressions, but do not spend too much time trying to evaluate the integrals. You should nevertheless check that you get the correct limiting values for a=b. | 
|
Valentine's Day special: For extra credit, analyze the love story presented in the performance of the "Magnet and the Churn" in the Gilbert and Sullivan opera Patience and explain the physics behind the broken heart.
Two insulating cylindrical shells of length L are free to rotate around their
common axis. The thinner cylinder has a radius a, moment of inertia I1
and a charge Q1, while the other cylinder has a radius b, moment of inertia I2,
and a charge Q2 uniformly distributed on them.
An external torque τext is applied on the inner cylinder, so that it starts to
rotate in the direction indicated by the arrow. The other cylinder too starts rotating
due to the elecromagnetic interaction between the two. It is expected that the angular
accelerations α1 and α2 of the two cylinders will be
constant so that the angular velocities will be
ω1 = α1 t
ω2 = α2 t .
(a) Find the magnetic fields as a function of radius inside the cylinders. Neglect
fringe effects.
(b) Find the induced electric fields (due to Faraday's Law) at the positions of the
cylinders.
(c) Relate the force (and torque) acting on the cylinders due to the induced fields.
(d) Now, relate the angular accelerations to the torques,
I1 α1 = total torque on cylinder 1
I2 α2 = total torque on cylinder 2
and solve these equations to find α1 and α2.
(e) Find the total mechanical angular momentum L in the rotating cylinders.
Is ∂L/∂t = τext satisfied?
(f) Find the total angular momentum in the electromagnetic fields by integrating the
angular momentum density εo r x ( E x B ). Note that you do not
have to include the induced electric field (due to Faraday's Law) in E. Why?
| 
|
Complete the solution to problem 10.24 in the textbook which we started in class. Please use the methodology we used in class to determine the electric field, and not use the derived general results in Griffiths.