# Find the Mutual Inductance Between the Circular Coil and the Loop Shown in Figure. - Physics

Sum

Find the mutual inductance between the circular coil and the loop shown in figure.

#### Solution

The magnetic field due to coil 1 at the centre of coil 2 is given by

$B = \frac{\mu_0 Ni a^2}{2 ( a^2 + x^2 )^{3/2}}$

The flux linked with coil 2 is given by

$\phi = B . A' = \frac{\mu_0 Ni a^2}{2 ( a^2 + x^2 )^{3/2}}\pi a '^2$

Now, let y be the distance of the sliding contact from its left end.

Given:-

$v = \frac{dy}{dt}$

Total resistance of the rheostat = R

When the distance of the sliding contact from the left end is y, the resistance of the rheostat is given by

$r' = \frac{R}{L}y$

The current in the coil is the function of distance y travelled by the sliding contact of the rheostat. It is given by

$i = \frac{E}{\left( \frac{R}{L}y + r \right)}$

The magnitude of the emf induced can be calculated as:-

$e = \frac{d\phi}{dt} = \frac{\mu_0 N a^2 a '^2 \pi}{2 ( a^2 + x^2 )^{3/2}}\frac{di}{dt}$

$e = \frac{\mu_0 N \pi a^2 a '^2}{2 ( a^2 + x^2 )^{3/2}}\frac{d}{dt}\frac{E}{\left( \frac{R}{L}y + r \right)}$

$e = \frac{\mu_0 N \pi a^2 a '^2}{2 ( a^2 + x^2 )^{3/2}}\left[ E\frac{\left( - \frac{R}{L}v \right)}{\left( \frac{R}{L}y + r \right)^2} \right]$

emf induced,

$e = \frac{\mu_0 N \pi a^2 a '^2}{2 ( a^2 + x^2 )^{3/2}}\left[ E\frac{\left( - \frac{R}{L}v \right)}{\left( \frac{R}{L}y + r \right)^2} \right]$

The emf induced in the coil can also be given as:-

$\frac{di}{dt} = \left[ E\frac{\left( - \frac{R}{L}v \right)}{\left( \frac{R}{L}y + r \right)^2} \right]$

$e = M\frac{di}{dt} , \frac{di}{dt} = \left[ E\frac{\left( - \frac{R}{L}v \right)}{\left( \frac{R}{L}y + r \right)^2} \right]$

$M = \frac{e}{\frac{di}{dt}} = \frac{N \mu_0 \pi a^2 a '^2}{2( a^2 + x^2 )^{3/2}}$

Is there an error in this question or solution?
Chapter 16: Electromagnetic Induction - Exercises [Page 313]

#### APPEARS IN

HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 16 Electromagnetic Induction
Exercises | Q 96 | Page 313

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