The humble toroid

I was asked months ago how a toroid removes noise from an audio cable. This is absurdly complicated so if the technical stuff eludes you, come back tomorrow. I'll avoid equations, but there is going to be some math.

In short, a toroid is a ferrite doughnut. Sometimes they are made of maganese-zinc alloy, or nickel-zinc. I can't really stop there because that wouldn't be interesting or arcane. So hold on to that doughnut idea. I'll come back to it. First we have to visit the 1800s long before Krispy Kreme.
In 1820 Hans Oersted discovered that electric currents can create magnetic fields in conductive material. Seems simple now, but this was hot news then. Andres Ampere got very excited about this. He began determining the relationship of electric currents to magnetic fields. Ampere went on to write a fundamental law of electromagnetism, Ampere's law states that the magnetic field around an electrical element is proportional to the electrical current that is it's source. This was groundbreaking and also happened to be wrong. He wasn't totally wrong, more incomplete. But he was wrong enough that there also exists an equation referred to by the catchy name of "Maxwell's correction to Ampere's law."

So in any circuit the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop. The magnetic field depends only on the amount of current enclosed by the loop and not on how that current is distributed. In other words, the magnetic field outside the wire depends on the amount of current, not on the diameter of the wire.

Regardless the equation governs the properties of a magnetic field in a solenoid (coil of wire) a straight wire, the field inside the wire and yes even the humble toroid. Yes, we're back to the doughnut now. It is a ring of ferrite, aka iron with wire coiled around it making an inductor. This is a resonant circuit because of its self-capacitance.

This inductor acts as though it includes a parallel capacitor, because of its closely spaced windings. These adjacent windings have different electrical potential, but also lie in each others magnetic fiends. These different potential behave like the plates in a capacitor and storing charge. (Storing charge makes it like a capacitor. ) In twisted pair cable the two currents, in the toroid are equal and opposite and therefore null themselves so no radiation takes place at the cable. The two wires are thus balanced. It becomes a balun. This is how it reduces RFI.