A time-dependent point charge q(t) at the origin, is fed by a current

(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.3

Solution 7P

Step 1 of 3:

Consider a time dependent point charge at the origin, is fed by a current .

- We need to check that, the charge is conserved by confirming that the continuity equation is obeyed.

Given

We know that,

We have,

So,

We have

So,

Continuity equation verified, so the charge is conserved.