(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. 

Ampere’s Law states that the line integral of magnetic field  around any closed path in vacuum is  times the total current through the closed path.

That is,    

Toroid is a hollow circular ring on which a wire of large number of turns is closely wound.

Let’s consider an air-cored toroid with center O. 

We have, 
 

which is the magnetic field due to a toroid carrying current.

b) Given, current is flowing in the clockwise direction for an observer who is on the left side of the solenoid. Implies, left face of the solenoid is the South Pole and right face acts as the North Pole. The magnetic field lines are directed from south to north, inside the bar magnet. Hence, the magnetic field lines are directed from left to right in the solenoid. The figure below illustrates the direction of flow of current inside the solenoid. 
 

Magnetic moment of single current carrying loop = IA

Therefore,

Magnetic moment due to the whole solenoid is, m = N(IA)

Where, 

N is the number of turns of solenoid,

I is the current flowing through the loop, and

A is the area of the loop.