How do I find the current density vector in an electromagnet that has a time-varying current?

Question

I know that the current $I$ is a scalar quantity, and to calculate current density $J$ , the cross sectional area $A$ is needed, to give us $J = IA$ .
$J$ also has other relations involving conductivity and charge density. However, in an electromagnet, I am not able to understand how to find the direction of current density easily. It is moving with varying current (sine current) in a loop.

Can I use displacement/time approach for a quarter of a loop? I have no idea what should the time of traversal be. More importantly, how do I bring $I$ into use, since that is the most important expression I have.

It may look like a stupid question to many of you, but I simply don't have a choice. Professors here are only surface level in knowledge in things that matter, this being an average university. I try to read basic physics as much as I can, but its clearly not enough. Hence, I request for any real help that you can provide.

Answer 1

Approximating your wires as infinitely thin, In calculating the magnetic field you need to use Jdv Which has units

$\frac{Q}{m^3} \frac{m}{s} m$

= $\frac {Q}{s} m$

This is thensame as Current * dl

Where dl is the displacement differential of your chosen curve

this is obviously a simplication derivation using units. The real derivation is more complex , But for another way is that in calculating the fields, Dv can be split into da dl and the 1/R^2 are independent of da as da is infinitely thin, so$ \iiint Jdv $becomes$ \int \iint J da dl = \int I dl $

Where I is vector current, and dl is a scalar, If I and dl are in the same direction then I can say its (Scalar current * vector dl)

so in calculating the magnetic field of a solenoid you can just use Idl instead of Jdv, so you don't need to worry about come complex vector current density.You can simply worry about SCALAR current.

Obviously dl is whatever curve I want my current to be, which is done with a specific curve parameterisation.

For a helix ( solenoid shape) a parameterisation you could use would be

$ r = Rcos(\theta)\hat i + Rsin(\theta)\hat j + \frac{\theta}{2\pi} \hat k$

This would span a spring shape that one full turn goes over a distance 1 in the k direction

then to find dr it's$ r'(\theta)d\theta$

Look up biot savart law

Answer 2

You'll notice $|\vec{J}|$ and $I$ have the same units, unlike $\rho$ and $Q$ by contrast. You must integrate $\rho$ over all space to get Q, but not $I$.

The continuity equation says that $\nabla \cdot\vec{J} = \frac{d\rho}{dt}$. This is 0 for a normal circuit (though if there are capacitors it may be non-zero locally, net zero overall), so $I(t) = |\vec{J}(t)|$.

The direction of current in a circuit is, obviously, variable with position (and/or time). All you can generally say about the circuit as a whole is clockwise/counter clockwise.