(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 terms 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 momentum $m = N I A$

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Solution

(a)
According to Ampere's circuital law, the line integral of magnetic field induction along a closed curve is equal to the total current passing through the surface enclosed in the closed curve times the permeability of the medium.
$\oint \to B . \to d l = μ_{o} I_{e n c l o s e d}$

Applying ampere's law for the given toroid,
$B (2 π r) = μ_{o} N I$
But, $N = 2 π r n$
$B = μ_{o} n I$

(b)
The observer sees south pole as shown in the attached figure.
Magnetic moment due to a loop is given by:
$m = i A$
For N turns, $i = N I$
$m = N I A$

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