Coulomb Gauge vs. Lorenz Gauge

Hi there,

I am trying to solve Problem 10.5 from Griffiths Introductionto Electrodynamics. The problem says:

Which of the potentials in Ex. 10.1, Prob. 10.3, and Prob. 10.4 are in the Coulomb gauge? Which are in the Lorenz gauge? (Notice that these gauges are not mutually exclusive.)

From Ex. 10.1: V = 0, **A** = [u_0*k*(ct - |x|)^{2}]/4c

From Prob. 10.3: V = 0, **A** = (-1/4*pi*epsilon_naught)*(qt/r^{2}) r_hat

From Prob. 10.4: V = 0, **A** = A_naught*sin(kx - wt) y_hat, where A_naught, w and k are constants

I know that for Coulomb Gauge: Div. **A** = 0

And for Lorenz Gauge: Div. **A** = - epsilon_naught*u_naught**d*V/*d*t.

So I applied these conditions on **A** to see if the potential fell under Coulomb or Lorenz Gauge. For Ex. 10.1 and Prob. 10.4 I got Div. **A** = 0. Hence I said that they fell under **Coulomb Gauge**. BUT the solutions says they fall under **Lorenz Gauge as well**.

1. I do not understand why?

2. How do we calculate if a **scalar** potential is in Coulomb or Lorenz Gauge?

Thanks