Question

Using Ampere's circuital Law obtain the formula for the magnitude of magnetic field due to current in a toroid.

Answer 1

1. For the open space interior to the toroid.

Length of the loop 1 L1=2πr1  

 As the loop encloses no current, so I=0  

 Applying Ampere's circuital law,  

                 B1L1=μ0I  

 Or           B1×2πr1=μ0×0  

 Or           B1=0  

2. For points inside the toroid  

Let B be the magnitude of the magnetic field along the Amperean loop 2 of radius r.  

  Length of loop 2, L2= 2πr  

If N is the total number of turns in the toroid and I the current in the toroid,then total current enclosed by the loop 2=NI

Applying Ampere’s circuital law,  

     B×2πr=μ0×NI  

or        B=μ0NI2πr  

If r be the average radius of the toroid and n the number of turns per unit length, then  

          N=2πrn  

   ∴B=μ0nI

3. For exterior to the toroid, I and B will be 0.