Can you have electric current without a circuit?

Question

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"?

Answer 1

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!

Answer 2

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.

Answer 3

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: $$\rho = \frac{\Delta q}{\Delta x\Delta y\Delta z}$$