- #36
etotheipi
I believe all of the equations Skilling quotes are correct, so long as you replace ##\mathbf{E}_s## with ##\mathbf{E}##, and ##\mathbf{E}_m## with ##\mathbf{f}_s##. I have no idea why he is treating the force of the battery as an electric field.
Furthermore, for reasons that @Dale notes in the comment thread of the Split Electric Fields insight, ##\mathbf{f}_s## is conservative (i.e. derivable from a chemical potential) and thus cannot be the non-conservative component of the electric field.
Whilst you can certainly always perform the Helmholtz decomposition of ##\mathbf{E} = \mathbf{E}_{cons} + \mathbf{E}_{n-cons}##, and come up with formulae like $$\nabla \times \mathbf{E}_{n-cons} = -\partial_t \mathbf{B}$$I don't see how this is ever too helpful. More importantly, such a decomposition is irrelevant in the example of a static battery where there is only one conservative electric field.
Furthermore, for reasons that @Dale notes in the comment thread of the Split Electric Fields insight, ##\mathbf{f}_s## is conservative (i.e. derivable from a chemical potential) and thus cannot be the non-conservative component of the electric field.
Whilst you can certainly always perform the Helmholtz decomposition of ##\mathbf{E} = \mathbf{E}_{cons} + \mathbf{E}_{n-cons}##, and come up with formulae like $$\nabla \times \mathbf{E}_{n-cons} = -\partial_t \mathbf{B}$$I don't see how this is ever too helpful. More importantly, such a decomposition is irrelevant in the example of a static battery where there is only one conservative electric field.