Electric and magnetic potentials
Gauß’s law for magnetic flux and Faraday’s law of induction are satisfied by the electric field
Proof
Gauß’s law for magnetic flux is satisfied iff the
-field is incompressible, which is equivalent to the statement that there exists a vector potential with . Then inspecting Faraday’s law of induction, we find it holds iff
is irrotational, i.e. admits a scalar potential such that . Therefore .
The electric and magnetic fields are Gauge invariant under the transformation
Proof
Applying the identity
it immediately follows that gives the same -field as , however if we want to also be the same we require that hence
.