Problem 1:

During some time interval Δt, a fat wire of radius a, carries a constant current I, uniformly distributed over its cross-sectional area.  A narrow gap in the wire, of width w << a, forms a parallel-plate capacitor.
(a)  Find the electric field E in the gap at a distance s < a from the axis and the time t.  (Assume the charge is zero at t = 0). 
(b)  Find the magnetic field B in the gap at a distance s < a from the axis.

Problem 2:

An infinite straight wire carries the current I(t) = 0 for t < , I(t) = I₀ for t > 0, that is, a constant current I₀ is turned on abruptly at t = 0.  Find the scalar and vector potentials in the Lorentz gauge.

Problem 3:

Submit this problem on Canvas as Assignment 4.  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. 

The vector potential A(r,t) of an oscillating dipole p at the origin is A(r,t) = -ik p exp(i(kr - ωt))/(4πε₀rc).
Note:  complex notation, the real part matters.
(a)  Let p = p k.  Use this vector potential to calculate the magnetic field.
(b)  Use Maxwell's equations to calculate the accompanying electric field.
(c)  Find the fields in the radiation zone.

Problem 4

(a)  From Maxwell's equations, derive the conservation of energy equation,
(∂u/∂t) + ∇∙S = -E∙j,
where u is the energy density and S is the Poynting vector.  Rewrite this equation in integral form and explain why it is a statement of energy conservation.
(b)  A long, cylindrical conductor of radius a and conductivity σ carries a constant current I.  Find S at the surface of the cylinder and interpret your result in terms of conservation of energy.