Application of Ampere's Law - Magnetic Field of a Toroid

Magnetic Field of a Toroid:

We can use Ampere’s Law to examine the case of another device that produces a uniform magnetic field. In this case, we take advantage of our analysis of the solenoid to ask what happens if we bend a solenoid into a circle so that the ends join. The new configuration still has approximately zero field in the regions outside the volume contained by the coils, but the field inside that volume is again approximately uniform if the distance between the coils is small compared to the size of the coils. This device is a toroid.

Ampere's Law applied to a toroid. Note that Amperian loops which lie wholly outside the volume contained by the toroidal coils experience no magnetic field.

We expect the magnetic field to have circular symmetry about the center of the toroid using the same reasoning as for the solenoid. Hence, we expect it to be most useful to use circular paths for evaluating the Ampere integral of magnetic field and path. For any circular path whose area is not intersected by the coils, the magnetic field is zero and the current penetrating the area is, by definition, zero.