Question

The mutual inductance of the system of two coils is 5mH. The current in the first coil varies according to the equation $I = {I_o}\sin (wt)$where ${I_o} = 10A$ and $w = 100\pi \dfrac{{rad}}{s}$. The value of maximum induced emf in the second coil is
(A)$2\pi V$
(B)$\pi V$
(C)$5\pi V$
(D)$4\pi V$

Answer 1

Mutual induction is defined as the property of the coils that enables it to oppose the change in the electric flux due to the change in electric current in a nearby placed coil. This change in the electric flux through one coil induces EMF in the other coil; this phenomenon is known as mutual induction. The coefficient of mutual inductance is the proportionality constant between E.M.F. induced and the rate of change of electric current with respect to time in the adjacent coil.

Formula used:
$E = M\dfrac{{dI}}{{dt}}$
Where $E$ is E.M.F., $M$ is the coefficient of mutual inductance, and $\dfrac{{dI}}{{dt}}$ is the rate of change of electric current with respect to time.

Complete step by step solution:
From the phenomena, mutual induction change in the electric current through one coil induces EMF in the other coil.
$\Rightarrow E = M\dfrac{{dI}}{{dt}}$
Where $E$is the E.M.F.(Electro-Motive Force), $M$ is the coefficient of mutual inductance, and $\dfrac{{dI}}{{dt}}$ is the rate of change of electric current with respect to time.
Given,
$M = 5mH$ and $I = 10\sin (100\pi t)A$
$5mH = 5 \times {10^{ - 3}}H$
From above we can say that,
$\Rightarrow E = M\dfrac{{dI}}{{dt}}$
$\Rightarrow E = 5 \times {10^{ - 3}} \times \dfrac{{d(10\sin (100\pi t))}}{{dt}}$
(We know that $\dfrac{{d(\sin ax)}}{{dx}} = a\cos ax$
$\Rightarrow E = 5 \times {10^{ - 3}} \times (10 \times 100\pi \times \cos (100\pi t))$
For ${E_{\max }}$ $\cos (100\pi t) = 1$ as we want the maximum value and $\cos \theta \in [ - 1,1]\forall \theta \in R$
$\Rightarrow {E_{\max }} = 5 \times {10^{ - 3}} \times (10 \times 100\pi )$
$\Rightarrow {E_{\max }} = 5\pi V$

Hence, the correct answer is option (C) $5\pi V$

Additional information:
Mutual induction is the phenomena behind the working of an electric transformer, induction cooktop, induction furnace, and many more.
When inductors are placed in series the inductance of the system is also influenced by the coefficient of mutual inductance of the coils in the system.

Note:
In self-induction, E.M.F. is induced due to the change in electric current in the same coil in which E.M.F. is induced. But in the case of mutual induction, E.M.F. is induced due to the change in electric current in the coil adjacent to the one in which E.M.F. is induced.