Question

(a) Explain the meaning of the term mutual inductance. Consider two concentric circular coils, one of radius $r_1$ and the other of radius $r_2(r_1<r_2)$ placed coaxially with centres coinciding with each other. Obtain the expression for the mutual inductance of the arrangement.
(b) A rectangular coil of area A, having number of turns N is rotated at 'f' revolutions per second in a uniform magnetic field B, the field being perpendicular to the coil. Prove that the maximum emf induced in the coil is 2 $\pi fNBA$.

Answer 1

(a) Mutual inductance is the property of a pair of coils due to which an e.m.f. is induced in one of the coils, due to the change in the current in the other coil.

Let a current $I_2$ flow through the outer circular coil.
The magnetic field at the centre of the coil is
$B_2=\frac{{\mu }_0{I}_2}{2r_2}......\left(i\right)$

As the inner coil placed co-axially has very small radius, therefore, $B_2$ may be take as constant over its cross-sectional area. Hence, flux associated with inner coil is
${\varphi }_1=\pi r_1^2{B}_2$

$=\pi r_1^2\frac{{\mu }_0{I}_2}{2r_2}$ [From (ii)]

$=\left(\frac{{\mu }_0\pi r_1^2}{2r_2}\right)I_2$

$=M_{12}{I}_2$

$\therefore M_{12}=\frac{{\mu }_0\pi r_1^2}{2r_2}$

Now, $M_{21}=M_{12}=\frac{{\mu }_0\pi r_1^2}{2r_2}$

(b) Let N be the number of turns of the rectangular coil and A be its cross-sectional area placed in a magnetic field B, then, the magnetic flux linked with the coil,

$\varphi =NBAcos\theta$

The induced emf,
$e=-\frac{-d\varphi }{dt}$

$e=-\frac{d\varphi }{dt}=(-NBA(-sin\theta \left)\frac{d\theta }{dt}\right)$
$=NBA.sin\theta \left(2\pi f\right)\left[\begin{array}{c}\because \theta =\omega t\\ \frac{d\varphi }{dt}=\omega =2\pi f\end{array}\right]$

For maximum induced e.m.f.

$sin\theta =1$

$\therefore e=NBA\left(2\pi f\right)$