Question

A current $I = 10\sin \left( {100\pi t} \right)$ is passed in the first coil, which induces a maximum emf of $5\pi $volt in the second coil. The mutual inductance between the coils is
$\eqalign{
  & {\text{A}}{\text{. }}10mH \cr
  & {\text{B}}{\text{. }}15mH \cr
  & {\text{C}}{\text{. 25}}mH \cr
  & {\text{D}}{\text{. 2}}0mH \cr
  & {\text{E}}{\text{. 5}}mH \cr} $

Answer 1

Hint: Mutual induction is the phenomenon of production of induced emf in one coil due to a change in the current flowing through the other coil. It is numerically equal to the magnetic flux linked with one coil when a unit current passes through the other coil. Use this concept to find the mutual induction between the coils.

Formula used:
Induced emf in the secondary coil, $\varepsilon = - M\dfrac{{dI}}{{dt}}$
where $M$ is the mutual inductance between the coils, a
$\dfrac{{dI}}{{dt}}$ represents the rate of change of current passing in the coil.

Complete step by step answer:
When the current in a coil changes, the magnetic flux linked with the neighboring coil changes and induces an emf in it. This process is called mutual induction. This property of a coil is called its mutual inductance.
Mathematically, the induced emf in the secondary coil is given by:

$\eqalign{
   &\varepsilon = - M\dfrac{{dI}}{{dt}} \cr
  &\Rightarrow \left| {{\varepsilon _{\max }}} \right| = {\left| { - M\dfrac{{dI}}{{dt}}} \right|_{\max }} \cdots \cdots \cdots \cdots \left( 1 \right) \cr} $
where $M$ is the mutual inductance between the coils, a
$\dfrac{{dI}}{{dt}}$ represents the rate of change of current passing in the coil.

The negative sign in the formula comes in accordance with Lenz’s law which states that the direction of the induced current in a circuit is such that it opposes the cause or the change responsible for its production.

Given:
The current in the first coil, $I = 10\sin \left( {100\pi t} \right)$
The maximum induced emf in the second coil, ${\varepsilon _{\max }} = 5\pi $
Now, differentiating the given value of current with respect to time we get:

$\eqalign{
  & \dfrac{{dI}}{{dt}} = \dfrac{d}{{dt}}\left[ {10\sin \left( {100\pi t} \right)} \right] \cr
  & \Rightarrow \dfrac{{dI}}{{dt}} = 10 \times 100\pi \cos \left( {100\pi t} \right) \cr
  & \Rightarrow \dfrac{{dI}}{{dt}} = 1000\pi \cos \left( {100\pi t} \right) \cdots \cdots \cdots \cdots \left( 2 \right) \cr} $

But we are only concerned with the maximum emf that is induced in the coil. And the emf is maximum when$\cos \left( {100\pi t} \right)$is maximum.
i.e., $\cos \left( {100\pi t} \right) = 1$
Substituting this value in equation (2), we get:

${\left. {\dfrac{{dI}}{{dt}}} \right|_{\max }} = 1000\pi $
Putting all these values in equation (1) we have:
$\eqalign{
  & \left| {{\varepsilon _{\max }}} \right| = {\left| { - M\dfrac{{dI}}{{dt}}} \right|_{\max }} \cr
  & \Rightarrow 5\pi = M \times 1000\pi \cr
  & \Rightarrow M = \dfrac{{5\pi }}{{1000\pi }} \cr
  & \Rightarrow M = \dfrac{1}{{200}} \cr
  & \Rightarrow M = 0.005 \cr
  & \therefore M = 5mH \cr} $
Thus, the correct option is option E. i.e., the mutual inductance between the coils is $M = 5mH$.

Note:
When two coils are inductively coupled, in addition to the emf produced due to mutual induction, induced emf is set up in the two coils due to self-induction also. Self-induction is the phenomenon of production of induced emf in a coil when a current passing through it changes with time.