Calculating Capacitance of a Toroid: Formula and Proof

[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?

 

[berkeman]

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?

Welcome to the PF.

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

 

[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.

 

[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.

 

[alphysicist]

Hello! The formula for the capacitance of a toroid is given by C = (μ0*N^2*A)/(2π*r), where μ0 is the permeability of free space, N is the number of turns in the toroid, A is the cross-sectional area of the toroid, and r is the radius of the toroid. This formula can also be written as C = (μ0*N^2*π*R^2)/(2π*r), where R is the mean radius of the toroid (the average of the inner and outer radii).

To understand the proof for this formula, let's start by looking at the definition of capacitance. Capacitance is a measure of the ability of a system to store electrical charge. In the case of a toroid, this means the ability to store charge on the surface of the toroid.

Now, let's consider the electric field inside the toroid. Since the toroid is a closed loop, the electric field inside must be zero. This is because any electric field lines that enter the toroid must also exit the toroid, resulting in a net zero field inside.

However, there is still a potential difference between the inner and outer surfaces of the toroid. This potential difference creates an electric field outside the toroid, which is perpendicular to the surface. This electric field is responsible for storing charge on the surface of the toroid.

Using Gauss's Law, we can calculate the electric field outside the toroid as E = (Q/(2π*r*ε0)), where Q is the charge stored on the surface of the toroid and ε0 is the permittivity of free space.

Now, we can relate this electric field to the capacitance of the toroid. The capacitance is defined as the ratio of the charge stored to the potential difference between the surfaces. So, we can write C = Q/V.

Substituting in the expression for the electric field and rearranging, we get C = (Q/(2π*r*ε0)) * (2π*r/Q) = 1/(ε0*r).

But we also know that the capacitance is given by C = (Q/V) = (Q/(2π*R)) * (2π*R/V) = (Q/(2π*R)) * (2π*R/(2π*r)) = (Q/(2π*r)).

Comb