# Displacement and conduction currents

#### TrickyDicky

One thing that I don't intuitively get about these currents is that in the conduction current the electric field is supposed to be static at each point of the conductor(there is no assumption of current intensity change), whilst the displacement current is defined as a varying electric field, the assumption is made of building up and decreasing electric fields, both current types producing a magnetic field and both being computed and measured in terms of Amperes(C/s) i.e. a steady current intensity.
Why is not the conduction current also thought of as a time-varying E field?

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#### Dale

Mentor
I guess that you mean in Ampere's law $c\nabla\times\mathbf{B}=\mathbf{J}+\partial_t\mathbf{E}$ it seems like $\mathbf{J}$ and $\partial_t\mathbf{E}$ are on completely equal footing. So why do we speak of $\partial_t\mathbf{E}$ as a sort of $\mathbf{J}$ but never seem to speak of $\mathbf{J}$ as a sort of $\partial_t\mathbf{E}$. Is that what you are asking?

#### TrickyDicky

I guess that you mean in Ampere's law $c\nabla\times\mathbf{B}=\mathbf{J}+\partial_t\mathbf{E}$ it seems like $\mathbf{J}$ and $\partial_t\mathbf{E}$ are on completely equal footing. So why do we speak of $\partial_t\mathbf{E}$ as a sort of $\mathbf{J}$ but never seem to speak of $\mathbf{J}$ as a sort of $\partial_t\mathbf{E}$. Is that what you are asking?
Yes. That's what I'm basically asking.

I know it can be answered by saying that it is unnecessary since the problem that Maxwell detected in Ampere's law is solved with the part about the displacement current resembling a conduction current, and moving charges are enough to justify the magnetic field appearance anyway in Ampere's law, but I would say symmetry reasons and logic seem to demand that regular currents be understood in terms of time-varying E-fields too.
Historical reasons made that when Maxwell wrote the equations the distinction had to be made between the 2 types of current, my question is what are the modern reasons to keep it, assuming there is not some deep conceptual error in my logic(wouldn't be surprising ), in which case I'd like to have it explained.

#### Dale

Mentor
I don't see any error in your reasoning nor do I have a good explanation. It does indeed seem like an odd artifact of terminology.

On the other hand, treating current as a time varying E field would be strange when you have a conductor with a static E field, but it would seem that you should be able to do just that.

A capacitor is often cited as an example where the changing E is in some sense understandable as a current. I wonder if there is a simple scenario where the reverse is true.

#### Dale

Mentor
One thought that I just had. In more concise formulations of Maxwell's equations the problem goes away. You get instead equations like $\partial_{\mu}F^{\mu\nu}=J^{\nu}$ where the E and B terms are together and the J is on its own. In that formulation you probably wouldn't call a component of the left hand side a displacement current, so the fact that you don't call a current a displacement E-field goes away too.

#### TrickyDicky

Interesting comments, thanks. The tensorial formulation certainly helps seeing it in a different light.

Being personally slanted towards the GR view, I tend to consider static fields as useful but idealized approximations, and to see all fields as dynamical.