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?
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.