# Symmetry argument for a toroid?

When using Ampere's law for a toroid (in the toroid and around a circular path) please can someone explain the symmetry argument (or an alternative argument) which allows us to assume the field is tangental. I cannot see why the field would not have a radial component, i.e. point in one particular direction as we have a larger 'density' of current on one side (closer to the centre of the toroid) then the other. Also does such an argument apply to any point in the toroid or only to those lying in the plane containing the axis of the toroid and its centre?

• Could you maybe reproduce the text or the argument you don't understand in more detail? – Void Mar 13 '15 at 13:20
• @Void The only texts that I can find go something like this 'By symmetry, B is tangent to this circle [a loop coaxial to the toroid, in the toroid] and has constant magnitude at every point if the circle.' I can't find anywhere that goes more in detail about this symmetry argument and to me there seems no reason why B should be tangental. – Quantum spaghettification Mar 13 '15 at 20:41
• Maybe this one: rotate the toroid of Pi around any axis in the the plane containing points in the outer circle of the toroid (I don't know if it's clear what I mean: turn it upside down). You obtain a toroid with the current circulating in the opposite direction. If there were as radial component to the field (say pointing outwards) it will be unaffected by this rotation. This is absurd because then the sum of the two current configurations yields a zero current configuration creating a non-zero radial outward pointing field. Note that the rotation instead invert the tangent component. – giulio bullsaver Mar 14 '15 at 11:54
• Then no non-zero radial component is allowed – giulio bullsaver Mar 14 '15 at 11:54
• @giuliobullsaver This is a very nice argument, if you put it as an answer I will accept it and award you the bounty. – Quantum spaghettification Mar 14 '15 at 12:08

## 1 Answer

The system is a toroidal coil and we want to show that in the center of the coil (not of the torus) there is no radial component for the magnetic field.

Suppose that there exists such a component in a point p. By cylindric symmetry we then know that it exists and is constant all along the circumference of the torus, at the center of the coil.

Now if we rotate the system by 180 degree around any axis in which lies this circumference, the radial component in any point remains unchanged, but now the currents flow in the opposite direction.

By linearity of the fields we know that if we sum two configurations of currents/charge, they generate the sum of the fields. But in this case then we have two configurations whose sum is zero while their fields do not sum op to zero. This is the absurd that forbids the radial component.

Note that instead the tangential component changes sign under the above rotation, and so it is allowed to exist.