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Which of the potentials in the following is in the Coulomb gauge? Which is in Lorenz Gauge?( Notice that these gauges are not mutually exclusive.)

This is what the solution gave:

[tex]V)(\vec r, t)=0,\; and\; \vec A(\vec r,t)=-\hat r \frac{qt}{4\pi\epsilon_0r^2}[/tex]

[tex]\nabla\cdot \vec A=-\frac{qt}{4\pi\epsilon_0)} \nabla\left(\frac{\vec r}{r^2}\right)=-\frac{qt}{\epsilon_0)}\delta^3(\vec r)\Rightarrow\; \nabla\cdot\vec A\neq 0[/tex]

I disagree:

[tex]\nabla\cdot\vec A=\frac{1}{r^2}\frac{\partial (r^2A_r)}{\partial r}+\frac{1}{r\sin\theta}\frac{\partial (A_{\theta}\sin\theta)}{\partial \theta}+\frac{1}{r\sin\theta}\frac{\partial A_{\phi}}{\partial \phi}[/tex]

[tex]\Rightarrow\;\nabla\cdot\vec A=-\frac{1}{r^2}\frac{\partial \left(r^2\frac{qt}{4\pi\epsilon_0 r^2}\right)}{\partial r}= -\frac{1}{r^2}\frac{\partial \left(\frac{qt}{4\pi\epsilon_0}\right)}{\partial r}=0[/tex]

Please help

Thanks