Assignment 8

Problem 1: 

A pulsar is a neutron star with mass M ≈ 1.4*M_sun ≈ 2.8 × 10³⁰ kg and radius R ≈ 10 km.  The star rotates with angular velocity ω and has a magnetic moment m, which is, in general, not parallel to the rotation axis.
(a)  Describe the radiation emitted by the pulsar, and find the total radiated power, assuming that the angle between the magnetic moment and the rotation axis is α, as in the figure.
(b)  Find the "spin-down rate" (i.e. find ω(t)) of the pulsar, assuming that energy loss is due to radiation only.

Problem 2:

A spherical shell of radius R carrying a uniform surface charge σ is set spinning at angular velocity ω.

(a)  Find the vector potential A that it produces at point P located at a distance s, s > R from the center of the sphere.
(b)  Find the magnetic field B outside the sphere.
(c)  How does the result for A change if r < R?
(d)  Find B inside the sphere.

Problem 3:

Assume you have two very small spheres with radius a, made of lih material with magnetic susceptibility Χ_m > 0.
The spheres are separated by a distance  R >> a .
Assume that the magnetic dipole moment m of each sphere is induced by the magnetic field due to the magnetic dipole moment of the other sphere.
Choose your coordinate system so that sphere 1 is located at the origin and its magnetic dipole moment points in the z-direction.
(a)  Where do you have to place sphere 2, so that the dipole moments of the two spheres are aligned?
(b)  Where do you have to place sphere 2, so that the dipole moments of the two spheres are anti-aligned?

Problem 4:

At t = 0 a particle of mass m and charge q moves in the xy-plane in a circular orbit of radius R in a uniform magnetic field B = Bk.  The particle will loose kinetic energy through radiation.  Assume that the energy loss per revolution is small compared to the total energy of the particle. 
(a)  Neglecting radiation, what is the magnitude of the particles momentum at t = 0?  Does your answer depend on the speed of the particle (relativistic or non-relativistic)?
(b)  Now assume the particle is moving non-relativistically.  Derive an expression for the time it takes the particle to loose half of its kinetic energy.