[Physics] Poisson’s equation for time dependent charge

classical-electrodynamicselectromagnetism

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?

Best Answer

No, Poisson's equation can be valid 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),

The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the Coulomb gauge is used. In this more general context, computing $\varphi$ is no longer sufficient to calculate $\mathbf E$, since $\mathbf E$ also depends on the magnetic vector potential $\mathbf A$, which must be independently computed. See Maxwell's equation in potential formulation for more on $\varphi$ and $\mathbf A$ in Maxwell's equations and how Poisson's equation is obtained in this case.

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.

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