We will now apply Ampere circuital law to calculate magnetic field of a toroid
A toroidal solenoid is a hollow circular ring with a large number of turns of a wire carrying current wound around the ring
Suppose we have to find the magnetic field B at a point P inside the toroid as shown below in figure
In this case amperion loop would be a circle through point P and concentric inside the toroid
By symmetry field will have equal magnitude at all points of this circle and this field is tangential to every point in the circle
Thus
If there are total N number of turns ,net current crossing the area bounded by the circle is NI where I is the current in the toroid
using Ampere law
Thus we see that field B varies with r i.e. field B is not uniform over the cross-section of the core because the path l=2πr is longer at the outer side of the section then at the inner side
Imagine a concentric circle through point P^{'} outside the toroid
The net current passing through this circular disc is zero ,since the current NI passes in and same current passes out. Thus using Ampere's circuital law, the field B=0 outside the toroid