Complex-valued current density?

Question

I'm reading in several places (e.g., here), that

the current density is supposed to be in the form $\mathbf{j}=j(r, z) \exp(i\omega t) \,\mathbf{e}_\varphi, [...]$

This implies that the current density has complex values, while I had been under the impression that it, as an observable quantity (the electric current per unit area), is always real-valued.

How to consolidate the two views?

Related: https://physics.stackexchange.com/q/80028/2451 , http://physics.stackexchange.com/q/44959/2451 and links therein. — Qmechanic, Jun 21 '17 at 16:59
Also related, and with essentially the same answer: What is the physical significance of the imaginary part when plane waves are represented as $e^{i(kx-\omega t)}$? — Emilio Pisanty, Jun 21 '17 at 17:11

score 5 · Accepted Answer · edited Jun 22 '17 at 16:05

The current is real-valued like electromagnetic fields. The complex formalism is largely used in electromagnetism when you have to face oscillating quantities because it is easier to handle that cosines / sines.

You can do calculation using the complex notation and then take the real part of the expression to come back to the physical quantities.

In your case the current is $\mathbf{j}=\Re( j(r, z) \exp(i\omega t))\,\mathbf{e}_\varphi$. Taking real part of a complex quantity in electromagnetism it is usually implicit.

Complex-valued current density?

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