A time-dependent point charge q(t) at the origin, ρ(r,t)=q(t)δ3(r), is fed by a current J(r,t)=−(1/4π)(q˙/r2)r^, where q˙≡dq/dt.
(a) Check that charge is conserved, by confirming that the continuity equation is obeyed.
(b) Find the scalar and vector potentials in the Coulomb gauge. If you get stuck, try working on (c) first.
(c) Find the fields, and check that they satisfy all of Maxwell's equations.