I'm reteaching electrodynamics to myself on a more rigorous footing, and I'm trying to prove to myself that setting the divergence of the vector potential is justified using a gauge shift. I could use the Helmholtz theorem to do this, but the problem with this from my perspective is that I haven't actually justified the full version of the theorem, only the weaker version which requires that a vector function decay to zero faster than 1/r at infinity. This isn't a problem for the field quantities (since all physical fields decay like 1/r^2); nevertheless, they do pose a problem for the potential quantities (which in general will not even decay). Basically, given a vector potential [tex]\vec{A}[/tex], I want to show that fixing the divergence of the gauge-shifted potential [tex]\vec{A}\prime[/tex] to some scalar function [tex]D[/tex] is equivalent to adding the gradient of some some scalar function [tex]\phi[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\nabla \cdot \vec{A}' = D[/tex]

[tex]\nabla \cdot (\vec{A} + \nabla \phi) = D[/tex]

[tex]{\nabla}^2 \phi = D - \nabla \cdot \vec{A}[/tex]

Since the right-hand side is just some function of position, proving that the divergence can be adjusted by adding the gradient of a scalar amounts to proving that Poisson's equation has a solution for an arbitrary source term. No boundary conditions are specified, so I would expect that there are actually an infinite number of solutions; however, I cannot prove this. Does anyone have any insights? Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Existence of solution to Poisson's equation

**Physics Forums | Science Articles, Homework Help, Discussion**