How can I calculate eddy currents in a changing magnetic field?

[mienaikoe]

I searched a lot of google for calculations on eddy currents, and got a lot of things that describe how eddy currents work, but almost nothing about how to quantify the induced currents and the resulting magnetic field. Does anyone here know much about where to start?

In particular, I'm looking to quantify the induced eddy currents in a flat aluminum plate with a changing, perpindicular magnetic field.

 

[Philip Wood]

Maybe it would help to look at a simple special case: a disc. By symmetry, the eddy currents will follow circular paths centred on the centre, O, of the disc. The emf acting in a circular path of radius r will, from Faraday’s law be given by

$$\varepsilon = \frac{d\Phi}{dt} = \pi r^2 \dot B$$

But if the current density is J around a path of radius r, we have

$$\varepsilon = \rho 2 \pi r J$$, in which $\rho$ is the resistivity.

So $$J = \frac{r \dot B}{2\rho}$$.

Say if this isn't clear.

 

[mienaikoe]

This Helps a Ton and it's very clear! thank you.

I'm still a little confused about how to derive the induced magnetic field. The current density that you've given is now a function of radius. Does this mean we can use Ampere's law to model the Magnetic Field as a function of radius?

$$J=\frac{r \dot B}{2\rho}$$
$$\oint B \cdot \partial L = \mu_0 I_{enc}$$
$$I_{enc} = \int J \cdot \partial A = J \pi r^2$$
$$\int \int_0^{2\pi} Br \cdot \partial \theta \cdot \partial r = \pi \mu_0 \int Jr^2 \cdot \partial r$$

Or is this just bad calculus?

 

[Philip Wood]

Ampère's law

 

[Philip Wood]

There are only a few cases where Ampère's law can be used to find B. These are cases in which there's enough symmetry for B to be effectively the same all along a particular integration path. Looking at your post, it seems that you don't have a particular path in mind. And the bad news is that for these circular eddy currents, as for a single circular loop of wire, there is no path along which B is constant. Ampère's law, beautiful though it is, can't help.

In fact the general problem of finding B at points in the vicinity of the disc, as for a circular loop, is very difficult. The only easy cases are for points on the axis of the disc, and, simplest of all, at its centre.

To find the field at the centre of the disc (of thickness b, say), think if it as made up of annuli of cross-sectional area b dr. Then the current in an annulus is Jbdr. But from the Biot-Savart law we know that the field at the centre of a ring carrying current I is $\mu$₀I/2r. Using the J from my previous post, and integrating for a disc of radius a, I find

$$B_{ind} = \frac{\mu_0 ba \dot B}{4\rho}$$

A neat result, I thought. But I'm prone to slips...

 

[mienaikoe]

this makes a good amount of sense. As for the overall behavior of the areas outside the center, though, is it safe to say that the induced B field is greatest at the center, falls to zero where the Magnetic Field Source ends, and then goes a bit negative before returning to zero?

(Positive being the direction dictated by Lenz's Law)

 

[Philip Wood]

Hadn't thought of the field ending; was thinking of the whole disc being subjected to a uniform normal field. But if the field 'covered' only an inner part, D_B, of the disc, I wouldn't expect the induced field to drop to zero at exactly the edge of D_B. Disc annuli outside D_B will still have emfs induced in them, because changing flux will still be linked with them. The emf will be
$$\varepsilon = \pi r_B^2 \dot B$$
in which r_B is the radius of D_B.
But beyond D_B, the current density will fall because of the increasing value of 2$\pi$r, and at some point, I think that B will indeed drop to zero and then reverse, as would happen for an ordinary current-carrying loop.

Thanks for such an interesting question.

 

[mienaikoe]

You've been a tremendous help. Thank you!