Generalization To Electrodynamics
When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), it is not possible to describe the electric field simply in terms of a scalar potential V because the electric field is no longer conservative: is path-dependent because ∇ × E ≠ 0 (Faraday's law of induction).
Instead, one can still define a scalar potential by also including the magnetic vector potential A. In particular, A is defined to satisfy:
where B is the magnetic field. Because the divergence of the magnetic field is always zero due to the absence of magnetic monopoles, such an A can always be found. Given this, the quantity
is a conservative field by Faraday's law and one can therefore write
where V is the scalar potential defined by the conservative field F.
The electrostatic potential is simply the special case of this definition where A is time-invariant. On the other hand, for time-varying fields, note that
unlike electrostatics.
Note that this definition of V depends on the gauge choice for the vector potential A (the gradient of any scalar field can be added to A without changing B). One choice is the Coulomb gauge, in which we choose ∇ · A = 0. In this case, we obtain
where ρ is the charge density, just as for electrostatics. Another common choice is the Lorenz gauge, in which we choose A to satisfy
Read more about this topic: Electric Potential