Problem 1:

Submit this problem on Canvas as Assignment 9.  If you used an AI as a Socratic tutor, submit a copy of your session leading to your solution and reflect on your session.  If you did not need any help or worked with another student, explain your reasoning, do not just write down formulas. 

A point electric dipole located at the origin has a time-dependent dipole moment given by
p(t) = p₀ cos(ωt) k.

In the far field region this gives
E(r, t) =μ ₀p₀ω²/(4πr) sinθ cos(ωt − kr) e_θ,
B(r, t) = μ₀p₀ω²/(4πrc) sinθ cos(ωt − kr) e_φ,
or electric and magnetic field.
(a)  Calculate the time-averaged power radiated per unit solid angle dP/dΩ .
(b)  Integrate over all angles to find the total radiated power.
(c)  Discuss the angular distribution of the radiation and where the radiation is strongest.

Problem 2:

A proton is accelerated through a voltage of 1000 V and enters a magnetic field perpendicular to its direction of travel.  The magnetic field strength is 1.5 Tesla.  What electric field (direction and magnitude) is needed in this region to compensate for the deflection due to the magnetic field?

Problem 3:

A thin spherical shell of radius R carries a uniform surface charge density σ.  The shell rotates with constant angular velocity ω = ωk.

(a)  Find the surface current density K(θ, φ) on the sphere.
(b)  Compute the magnetic dipole moment m of the rotating shell.
(c)  Use the magnetic dipole approximation to find the magnetic field B at a point far
from the sphere (r >> R).
(d) Compare the magnitude of the field at the equator and the pole.

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.