Capacitance of a toroid

1. Nov 10, 2012

F.ono

Hi, I am new here.
I need to know the formula for the capacitance of a toroid of circular cross section.
I found this: http://deepfriedneon.com/tesla_f_calctoroid.html

But I need to show the proof for the formula. Can anyone help?

2. Nov 11, 2012

Staff: Mentor

Welcome to the PF.

What is the application? The capacitance of windings on a core depends on several things. Is this for schoolwork?

3. Nov 14, 2012

Enthalpy

Do you mean the capacitance between a toroidal electrode and the rest of the Universe? It's probably what the linked website gives, but this must be very difficult to prove!

It's completely out of reach for a normal engineer.

Because it's a 3D problem I doubt a conformal transformation will give the answer.

The most promising analytical method would be to put charges uniformly on a perfectly thin wire - within the toroid but smaller than the circle that generates it. IF you're lucky, the equipotentials will be toroids of circular cross sections.

Either adjust the diameter of the circle that carries the charges to fit both r and R of your toroid, or take an arbitrary diameter and take the toroid that fits only R/r, then scale the capacitance as R.

Then, you remember the charge you put on the circle, compute the potential between the location where your desired toroid is and an infinite distance, and the quotient gives you the capacitance.

You may add somewhere: because the potential at the toroid is the same as the one created by the virtual wire, the potential, as an analytical function, is the same in all the space surrounding this delimitation, hence it's the same solution blah blah blah.

If the equipotentials are not circular toroids I've no idea. This one is adapted from the symmetrical 2-wire transmission line.

4. Nov 16, 2012

Enthalpy

The virtual thin wire that carries the charge has a diameter bigger than the toroid, not smaller, my mistake. It's somewhere within the toroid, nearer to the external surface when the toroid is thicker.

Does the analytical solution smell like elliptic functions?

The original method is for bifilar propagation lines, where two parallel thin wires carrying charges create cylindrical equipotential surfaces, hence the same electrostatic field at the cylindrical conductors.