Is Coulomb's law valid if the electric charge is not constant in time? 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
 A: First, the Wikipedia article already says on the derivation of Gauss' law from Coulomb's law:

Note that since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law.

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.
Everything there is time-dependent and nevertheless, we get Gauss' law. However, to derive Coulomb's law, you must assume a static charge distribution and that the field of a static charge distribution is curl-free (or get that information from the Maxwell-Faraday equation).
Thus, Coulomb's law is not valid for moving charges, because deriving it from Gauss' law requires the assumption of electrostatics, and Gauss' law and Coulomb's law are not equivalent in full electrodynamics. However, Coulomb's law together with special relativity is equivalent to the full Maxwell equations, see this question.
A: Simply, no!
Light takes some time to travel, so the effect of chancing a charge at one point should be retarded further away.
If you assume such instantaneous force, it is called quasistatic approximation of Maxwell equations and can for example be used in Mie scattering theory, where a particle is much smaller than the wavelength of light.
