# [Physics] Is Coulomb’s law valid if the electric charge is not constant in time

classical-electrodynamicselectrostatics

If the electric charge is constant the law of Coulomb says that:

$E(r) = kQ/r^2$.

The question is, if $Q = Q(t)$, can I consider that:

$E(r,t) = kQ(t)/r^2$?

Update:

It appears that the first equation in Maxwell's system is the law of Coulomb put by Gauss in an equivalent mathematical form. However, in Maxwell's equations $E = E(r,t)$ not $E = E(r)$ like in the law of Coulomb. This is the reason I asked if $E(r,t) = kQ(t)/r^2$.

Source: Wikipedia

And indeed, Gauss' law does hold for general charges since Gauss' law together with Ampere's circuital law are the equations of motion for the electromagnetic field, e.g. derived from the Euler-Lagrange equations for the Lagrangian of electrodynamics $$L[A,J] = -\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu} - A_\mu J^\mu$$ for the four-current $J(\vec x,t)$ (with $J^0 \propto \rho$ and the spatial part the usual current), $A(\vec x,t)$ the four-potential and $F(\vec x,t)$ the field strength tensor.