Is Poisson's equation valid for a time dependent charge density? I think Poisson's equation is valid just for electrostatic fields. But I saw a paper that's used this equation for time dependent charge density. May anyone help me with this contradiction?

# [Physics] Poisson’s equation for time dependent charge

classical-electrodynamicselectromagnetism

## Best Answer

No, Poisson's equation

can bevalid for time-varying fields. The Wikipedia page on the subject says in deriving $$ \nabla^2\varphi=\frac{\rho_f}{\epsilon_0}\tag{1} $$ that (my emphasis),Where the Coulomb gauge requires $$ \nabla\cdot\mathbf A(\mathbf r,t)=0 $$ If we pick this gauge (always possible), then the (1) is valid for electrodynamics.