# EMF induced in solenoid by current in a loop.

## Homework Statement

A square loop with side-length a is positioned at the centre of a long thin solenoid, which has radius r (with r>a), length l and N turns. The plane of the loop is perpendicular to the axis of the solenoid, find $V_{emf}$ induced in the solenoid

## Homework Equations

$M=\frac{\Phi}{I}$

$\Phi=\mathbf{B}\cdot\mathbf{A}$

$V_{emf}=-M\dfrac{\mathbf{\textrm{d}}I}{\textrm{d}t}$

## The Attempt at a Solution

The magnetic field created by the current flowing through the loop is complex and the flux varies throughout the solenoid. By using the fact that the mutual inductance is the same for the coil and the solenoid we can find the emf generated in the solenoid.
If we take the magnetic field produced by a solenoid as

$\mathbf{B}=\mu_{0}nI_{sol}\:\mathbf{e}_{z}$ where $n=\dfrac{N}{L}$

Then the flux through the square loop is

$\Phi=\mathbf{B}\cdot\mathbf{A}=\mu_{0}nI_{sol} \times a^{2}=\mu_{0}na^{2}I_{sol}$

Now the mutual inductance is

$M=\dfrac{\Phi}{I_{sol}}=\mu_{0}na^{2}$

The induced emf is $I_{loop}=I_{0}\sin\omega t$

$V_{emf}=-M\dfrac{\mathbf{\textrm{d}}I_{loop}}{\textrm{d}t}=-\mu_{0}na^{2}\omega I_{0}\cos\omega t$

Is this correct?

Hi, yeah the current in the loop should be $$I_{loop}=I_{0}\sin\omega t$$.