How is potential difference defined across a resistor with time varying current From this discussion How can we define a potential for a moving charge? we know that we cannot define a  scalar potential (as in electrostatics) in the case of moving charges as described by G. Smith:

You cannot describe the electromagnetic field of a moving charge as the gradient of a potential. If you could, the curl of the electric field would be zero, which would imply that the time derivative of the magnetic field would be zero. This is clearly false

How then can we define a potential difference across  a resistor with time varying current?
 A: 
How then can we define a potential difference across a resistor with time varying current?

Basically we just assume that we can.
Circuit theory is an approximation to Maxwell’s equations which relies on three assumptions:

*

*the distances are small enough and the time scales large enough that we can treat electromagnetic effects as instantaneous rather than propagating at c.


*there is no net charge on any component.


*there is no magnetic flux outside of any component.
With those three assumptions the vector potential from Maxwell’s equations becomes zero and only the scalar potential remains. And also the Coulomb gauge can be used (up to a simple additive constant) for that potential. While the statement by G. Smith is absolutely correct, the errors introduced by simply using the Coulomb potential anyway go to zero as deviations from these assumptions go to zero. Thus circuit theory is a well defined approximation to Maxwell’s equations and in that approximation the potential difference across a resistor is well defined.
