# 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