# How/why does the magnetic field in a circuit play a role in the Joule heating of a resistor?

A bit of background: I only recently learned that, contrary to common notion, the energy that heats the resistor in a simple battery+resistor circuit is not transported through the wire, but instead through the magnetic and electric field, described via the Poynting vector. For the most part the energy transfer still happens along the wire, but just outside of it, due to surface charges on the wire that create an E field, and the magnetic field generated by the wire running to and from the resistor.

But, here's my actual question: The common explanation for Joule heating in resistors is that electrons get scattered on the ion lattice and thus impart thermal energy. However, there are two kinks:

• Apparently the contribution of the kinetic energy of the electrons to the Joule heating is a tiny percentage of the total energy transferred. After all, it's the electrical and magnetic field that transports the energy according to the Poynting vector.
• So, what IS then the mechanism of energy transfer? Specifically, since the magnetic field is a key factor in the Poynting vector, it strikes me that it must then also play a key role in the energy transfer. How does this energy contained in the fields "magically" get transformed?

Interestingly, I found a Feynman lecture where he complains about that very lack of understanding of the fundamentals (him including), but finding information about this is very hard.

EDIT: The closest I have found in the matter is this paper:

https://redirect.viglink.com/?format=go&jsonp=vglnk_149119210942012&key=6afc78eea2339e9c047ab6748b0d37e7&libId=j11la2fn010009we000MAf1ea2vw1&loc=https%3A%2F%2Fwww.physicsforums.com%2Fthreads%2Fconceptual-understanding-for-voltage-in-a-circuit.908478%2Fpage-3&v=1&out=https%3A%2F%2Fwww.google.com%2Furl%3Fsa%3Dt%26source%3Dweb%26rct%3Dj%26url%3Dhttp%3A%2F%2Fwww.if.ufrj.br%2F~dore%2FFis3%2FPoynting-AJP.pdf%26ved%3D0ahUKEwjU86mcm4fTAhXKvxQKHWOZDT8QFggdMAA%26usg%3DAFQjCNEDKJDnZfVJRAuvXhdWDtJPhVmZDQ%26sig2%3DbiqYhpyUsgx4vtoA4-6wHw&ref=https%3A%2F%2Fwww.physicsforums.com%2Fforums%2Fgeneral-physics.111%2F&title=Conceptual%20Understanding%20for%20Voltage%20in%20a%20Circuit%20%7C%20Page%203%20%7C%20Physics%20Forums%20-%20The%20Fusion%20of%20Science%20and%20Community&txt=https%3A%2F%2Fwww.google.com%2Furl%3Fsa%3Dt%26amp%3Bsou...AuvXhdWDtJPhVmZDQ%26amp%3Bsig2%3DbiqYhpyUsgx4vtoA4-6wHw

But sadly it still glosses over the gap to the microscopic explanation of the energy transfer.

## 1 Answer

If anyone really thinks that the heating of a resistor has anything to do with the magnetic and/or electric fields in on or around the wires connected to the resistor then it would be trivial to devise some laboratory experiments to test this. For example one could use the same value resistor and have vastly different physical arrangements of the wires, magnetic shielding, variation in E-field from externally applied field from other isolated conductors, etc. I predict that the heating in the resistor will be the same in all cases and calculated by Ohm's Law.