Question

A conducting loop of radius r is placed concentric with another loop of a much larger radius R so that both the loops are coplanar. Find the mutual inductance of the system of the two loops. Take R >> r. [Ans: μ₀ πr²/2R]

Answer 1

This is a loop and write the note

Answer 2

Answer:

$\rm \dfrac{\mu_o \pi r^2}{2R}.$

Explanation:

Let I current be flowing in the larger loop, then the magnetic field produced by the larger loop of radius R at its center is given by

$\rm B = \dfrac{\mu_o I }{2R}.$

$\mu_o$ is the magnetic permeability of the free space.

Given that R>>r, therefore, the magnetic field through the smaller loop will be uniform throughout its area.

The two loops are coplanar, therefore, the magnetic flux linked with the smaller loop is given by

$\rm \phi = BA.$

A is the surface area of the smaller loop, $\rm A = \pi r^2.$

We know,

$\rm \phi = MI.$

M is the mututal inductance of the system of the two loops, therefore,

$\rm M = \dfrac \phi I=\dfrac{BA}{I}\\\\\text{ Putting the values of B and A, we get,}\\\\M = \dfrac 1I\dfrac{\mu_o I}{2R}\pi r^2=\dfrac{\mu_o \pi r^2}{2R}.$