The question really could be written so: when we instantly apply a potential-difference to the ends of a resistor, how does the resistor respond over time, in order to eventually exhibit a constant current-density throughout its interior?
A "quick" intro is found in the 1999 paper by Chabay and Sherwood, author of undergrad physics text MATTER AND INTERACTIONS: A unified treatment..." The main conclusion is that, for the final DC case, the surface-charge is distributed in circles of constant surface-charge density, i.e. a resistor is like a stack of charged rings. (For the DC case, all charge-imbalance must exist only on the conductor surface.)
But for the full description, we really should use FEM finite-element simulators with animated field-diagrams, since each charged volume in the wire will experience a capacitance to distant Earth ...and also exhibit a capacitance to every other charged volume in the wire!
One method is to initially assume that the wire has zero resistance, and can be modeled as a thin hollow tube (pure skin-effect, all transient and no long-term DC.)
When the voltage-drop is first applied to a long thin conductor, EM waves dance back and forth along this hollow tube, creating the rings of charge and shoving them around, until they settle down and produce a uniform current density on the surface. This occurs at the speed of light, over fractions of nanoseconds.
Hmmm, rather than a wire in space, maybe it would simplify things to first examine the center-conductor of a piece of coax cable. That way the resistor becomes a simple transmission line, with all the capacitance only existing between the wire surface and the nearby inner surface of the shield-braid. In that case the wire becomes a looooong inductor, like a long coil, but with numerous capacitors connected between the wire surface and the nearby Earth-potential.
Then, once we know the nanosecond behavior of voltage-waves on this conductor, then for real conductors having finite resistance, next the current "oozes inside" over long microseconds. All metals are EM-shields of course, so the initial e-fields and currents are blocked from affecting the interior of the metal wire. The elements of the wire exhibit L/R time constants, with slower response for lower resistance. Or in other words, charge "has inertia," and the applied e-field can only, after a significant delay, create a current within the interior of the wire.
Or said differently, the speed of light inside copper is quite slow, on the order of tens of M/S, and "skin effect" is really about the electrical energy being able to "leapfrog" quickly across the space outside the metal surface, and only slowly to propagate inwards ...as if the wire was really an "onion layer" of concentric pipes. At first, the entire current only arises on the outside pipe. For a thick copper wire, an entire millisecond may pass by before the current becomes uniform throughout the metal. Yet the first initial current-pattern became established millions of times faster, in roughly a nanosecond.
See, we really really need animated diagrams!
But so few people are interested in this niche-topic, such things have never been done before. And with good reason. At MIT, a Dr. Belcher tried to introduce it to thir large undergrad EM course, with field-simulations giving a view of the internal workings of all components. Rumor is, the undergrad students rose up en masse and forced the administration to go back to the old way: obscure walls of equations, with none of these easily-understood animated pictures! To me the situation appears like that of medical doctors in the year 1300, if their med school suddenly started teaching courses in plain English (or French!) Nooooo! Then just anyone could understand the material! It turns the complex and obscure into "Physics for Poets!" BRING BACK THE LATIN RIGHT DAMN QUICK, OR THE STUDENT BODY WILL MARCH IN AND HANG THE LOT OF YOU!
Heh.
Here's the residue of the late-1990s MIT "T.E.A.L" project by JW Belcher and crew...
ALso: https://web.mit.edu/fnl/vol/162/belcher.htm
