(a) State Ampere’s circuital law. Use this law to obtain the expression for the magnetic field inside an air cored toroid of average radius ‘r’, having ‘n’ turns per unit length and carrying a steady current I.

(b) An observer to the left of a solenoid of N turns each of cross section area ‘A’ observes that a steady current I in it flows in the clockwise direction. Depict the magnetic field lines due to the solenoid specifying its polarity and show that it acts as a bar magnet of magnetic moment m = NIA.

(a) Amperes circuital law in electro magnetism is analogous to Gauss law in electrostatics. This law states that “The line integral of resultant magnetic field along a closed plane curve is equal to μ₀ time the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant. Thus where permeability of free space and lens is is the net current enclosed by the loop.

A toroid is a hollow circular ring on which a large number of turns of wire are closely wound. Consider an air-cored toroid (as shown at right side) with centre O.

Given:

r = Average radius of the toroid
I = Current through the solenoid .
n = Number of turns per unit length
to determine the magnetic field inside the toroid, we consider three Amperian loops (loop 1, loop 2 and loop 3).

According to Ampere’s circuital law, we have (Total current)

Total current for loop 1 is zero because no current is passing through this loop.

So, for loop 1;

(Total current)

For loop 3:

According to Ampere’s circuital law, we have,

(Total Current)

Total current for loop 3 is zero because no current is coming out of this loop is equal to net current going inside the loop.

For loop 2:

The total current flowing through the toroid is NI, where N is the number of turns.

----- (i)

Now and are in same direction

-------- (ii)

Comparing (i) and (ii) we get;

Number of turns per unit length id given by

This is the expression for magnetic field inside air-cored toroid.
(b) Given that the current flows in the clockwise direction for an observer on the left side of the solenoid. This means that left face of the solenoid acts as South Pole and right face acts as North Pole. Inside a bar magnet the magnetic field lines are directed from south to north. Therefore, the magnetic field lines are directed from left to right in the solenoid.
Magnetic moment of single current carrying loop is given by
m = LA
where
I = Current flowing through the loop A = Area of the loop
So, Magnetic moment of the whole solenoid is given by
M = Nw’ = N (IA)