# Currents in an isolated metal exposed to an alternating uniform electric field

For example, let's look at a metal sphere between a capacitor's (infinite) plates. Let's run an AC current through the capacitor, resulting in an oscillating electric field.

The charges will move in the sphere in accordance to the external field (and try to keep the field in the sphere 0). From the resulting local electric field configuration, I can find the sphere's charge density and (hopefully) some estimation to the currents.

Now, in addition to the mentioned currents, there should also be present some induced eddy currents, right? (due to time-varying electric field -> time-varying magnetic field -> eddy currents).

The induced eddy currents are a bit confusing to me. How could I estimate them?

(I need the currents to eventually estimate the heat generation in the metal object.)

Maybe in a first very rough approach you could consider the metal sphere as being an oscillating charge hence radiating fields like a Hertzian dipole or an antenna. It will indeed induce currents inside the sphere in addition to the already existing from the sinusoidal field generated by the capacitor.

By the way the field created by this kind of hertzian dipole is :

$E(x,t)=-\frac{\omega^{2}qx_{0}}{\epsilon_{0}c^{2}r}sin(\theta)exp^{i\omega(t-r/c))}$

$\theta$ being the angle made with the dipole axis, hence it equals 0 along the motion of the sphere.

I am not sure it will help but in any case an oscillator charge like your sphere is a small antenna that must be radiating electromagnetic waves.

Yes, you can capacitively couple to a conductor and induce currents.

Those currents, though, are not flowing in a circuit, but are constrained by the conducting sphere to only flow back-and-forth on the surface (because a conductor shields its interior from electric fields, to a good approximation).

So, you need to know the skin depth that applies for the material of interest, at the frequency of interest, and treat each element of the spherical surface as a small bit of sheet metal, in order to derive a resistor-heating effect in each area element, after determining how much charge movement (current density) is present.

As for eddy currents-- those are part of the skin depth calculation, so if you use standard formulae, (like are valid for a cylinder conductor) you'll be assuming those rather than calculating them.