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I'm trying to understand Maxwell's 4th equation, which is

$$ \bigtriangledown \times B = \mu_0j + \mu_0\epsilon_0\frac{\partial E}{\partial t} $$

I understand that $\bigtriangledown \times B$ corresponds to curl of the magnetic vector field. I know $\frac{\partial E}{\partial t}$ is another 3d vector representing the change in a vector field over time.

But what I'm trying to understand, is $j$. Is it just change in charge density over time? Can a stray electron moving through space be understood to have current, or does it only have meaning in circuits? And if the latter, what precisely is required for a circuit?

Essentially, what is meant by "current"?

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  • $\begingroup$ There are bulk conductors and plasmas with vortex currents etc. $\endgroup$ – user137289 Nov 27 '20 at 21:45
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In the equation j can be zero it usually comes in a circuit, but a electron beam also results in current, and charge with phases through an area is a current, but a stray e would be a very, vera short time current!

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  • $\begingroup$ What exactly does current mean then? $\endgroup$ – SarcasticSully Nov 27 '20 at 21:54
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The current is given by

$$I=\frac{dq}{dt}$$

i.e, the amount of charge flowing through a certain region per unit time. Current density

$$j = \frac{I}{A}$$

or the amount of charge per unit time per unit area.

Can a stray electron moving through space be understood to have current, or does it only have meaning in circuits?

It is a charge moving through space per unit time, so yes. It can be considered a current.

And if the latter, what precisely is required for a circuit?

An electric circuit is any path for which transmitting an electric current is possible. An electric circuit must include a device that gives an EMF (electromotive force), or energy for the charged particles constituting the current to flow.

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  • $\begingroup$ "Current" has two meanings in the context of electrical circuits. One meaning is a phenomenon: current is the motion of electrical charge. Second is the physical quantity, defined as in the other answers. So any motion of charge is a current. And can be associated with the physical quantity current (or intensity of the current). $\endgroup$ – nasu Nov 27 '20 at 22:14
  • $\begingroup$ How does $I = \frac{dq}{dt}$ work? In a circuit, isn't $\frac{dq}{dt}$ equal to 0, since just as much charge is entering any given region of the circuit as is leaving? $\endgroup$ – SarcasticSully Nov 27 '20 at 22:28
  • $\begingroup$ Sorry. I was surprised because you were interpreting Maxwell's equations but are having a hard time understanding simple current. I cannot explain this any more simpler. Please click here which explains the concept of current in depth and starting from the beginning. Cheers. $\endgroup$ – joseph h Nov 27 '20 at 23:01
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But what I'm trying to understand, is j. Is it just change in charge density over time?

No. If the density of charge is constant for example, it doesn't mean that current is zero. Only its divergence is zero. $$\frac{\partial \rho}{\partial t} + \nabla\mathbf{.j} = 0$$

Essentially, what is meant by "current"?

Current means a flow of charge carriers, electrons in a conductor, electrons or holes in a semiconductor, ions in a solution and so on.

The density of current (in the $x$ direction for example) can be understood as a density over a spacetime volume, where $\Delta x$ is replaced by $\Delta t$: $$j_x = \frac{\Delta q}{\Delta t\Delta y\Delta z}$$

Density of charge carriers on the other hand can be regarded as a density of current through time: $$\rho = \frac{\Delta q}{\Delta x\Delta y\Delta z}$$

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