# Magnetic field of tilted toroidal coil

Lets think about a toroidal coil. We can think that toroidal coil is set of circles. The external magnetic field of coil is zero. Then what if we tilt these circles in same direction? Is external mag field is still zero?

First of all, by tilting in that way does not really make any difference for the magnetic field outside (it is very very weak).

A technique that gives an approximate solution, is the derivation of the field outside of a toroid by Ampere's law.

$$\oint \vec B\cdot d\vec l=\mu_0I_{\rm enclosed}$$

We assume the field inside to be contained within its boundaries. So, that means there is some enclosed current inside. But if we go a little further away, we need to take in the contributions of the currents that go in a direction opposite to the currents that were going in (the current travels in a circle, but for a person viewing this from above, the current seems to be going up and then down in that same loop).

From this math, we can say that the field outside is zero, since the current contributions cancel each other away. When you tilt those wires anyhow, the field outside is really weak that it still does not make a difference.