General form of Maxwell equation is given by $$ \nabla_\mu F^{\mu\nu} = 4\pi J^\nu $$ where $F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu$ is the tensor of EM field. Then Maxwell equations can be rewritten as $$ \nabla_\mu F^{\mu\nu} =\nabla_\mu(\nabla^\mu A^\nu-\nabla^\nu A^\mu) = 4\pi J^\nu\ , \qquad \nabla_\mu A^\mu = 0 \to Lorentz\,\,\, gauge\ , $$ So finally we have $$ \nabla^2A^\nu=\nabla_\mu\nabla^\mu A^\nu=4\pi J^\nu $$ If we choose scalar field $\Phi$ instead of vector field $A^\nu$ then we get Poisson equation $\nabla^2\Phi = 4\pi\rho$, $\rho$ is the source term and solution of Poisson equation will be in the form of Green's function. $$ \Phi(r) = \frac{1}{4\pi} \int\frac{\rho(\vec r')d{\vec r'}}{|{\vec r}-{\vec r}'|} $$
Let us go back to our main equation $\nabla^2A^\nu = 4\pi ^\nu$. Solution of this equation will be in form of Green's function only in Cartesian coordinate $(x,y,z)$. Since we want to consider vector potential of current loop. It is convenience to write Maxwell equation either in spherical ($r,\theta,\phi$) or cylindrical ($r,\phi,z$) coordinates. If we follow Jackson's book solution for vector potential written as $$ A_\phi = \frac{1}{4\pi}\int\frac{I(r')r'^2dr'\sin\theta'd\theta'd\phi'}{|\vec r-\vec r'|} $$ but we know that $A_\phi\neq A^\phi$ in spherical coordinate. That is why square of Nabla operator will be different for $A_\phi$ and $A^\phi$ in spherical coordinate. That is why I am a bit worrying about this solution. Is it correct or not? Hint: $\nabla_\mu A^\nu = \partial_\mu A^\nu + \Gamma_{\mu\lambda}^\nu A^\lambda$ and $\nabla_\mu A_\nu = \partial_\mu A_\nu - \Gamma_{\mu\nu}^\lambda A_\lambda$ where $\Gamma_{\mu\nu}^\lambda$ is Christoffel symbol.


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