(a) Let A(x) be a vector function of a scalar argument. Find the conditions that make A(k · r − ckt) a legitimate Coulomb gauge vector potential in empty space. Compute E(r, t) and B(r, t) explicitly and show that both are transverse.

(b) Consider a Coulomb gauge vector potential A_C(r, t) = u(r, t)a where u is a scalar function and a is a constant vector. What restrictions must be imposed on u(r, t) and a, if any? Find the associated electric and magnetic fields.

(c) Specialize (b) to the case when u(r, t) = u(k · r − ckt). Show that cB = k̂ × E.

(d) Consider a Lorenz gauge potential A_L(r, t) = u(r, t)s where u is a scalar function and s is a constant vector. What restrictions must be imposed on u(r, t) and s, if any? Find the associated electric and magnetic fields.

(e) Specialize (d) to the case when u(r, t) is the same plane wave as in part (c). Show that the electric and magnetic fields are exactly the same as those computed in part (c).