Question

Two coils have a mutual inductance 0.005H. The current changes in first coil according to equation $I={{I}_{0}}\sin \omega t$, where ${{I}_{0}}=10A$ and $\omega =100\pi radian/\sec$. The maximum value of emf in second coil is
A. $2\pi$
B. $5\pi$
C. $\pi$
D. $4\pi$

Answer 1

Use the formula for the induced emf in a coil due to a varying current in another coil nearby. Then substitute the given values and find the induced emf in the second coil. Later, use the knowledge of trigonometry and find the maximum value of the emf.

Formula used:
${{E}_{2}}=M\dfrac{d{{i}_{1}}}{dt}$
${{E}_{2}}$ is the emf induced in the second coil
${{i}_{1}}$ is the current in the first coil
$M$ is the mutual inductance between the two coils.

Complete step by step answer:
Suppose the two coils are placed near each other. It is found that when current in one of the coils changes with time, an emf is induced in the other coil. The induced emf in the second coil due to the varying current in the first coil is given as,
${{E}_{2}}=M\dfrac{d{{i}_{1}}}{dt}$ ….. (i),
where ${{E}_{2}}$ is the emf induced in the second coil, ${{i}_{1}}$ is the current in the first coil and M is the mutual inductance between the two coils.

In this case, ${{i}_{1}}=I={{I}_{0}}\sin \omega t$ … (ii).
Substitute the values of ${{I}_{0}}$ and $\omega$ in equation (ii).
${{i}_{1}}=10\sin (100\pi t)$.
Now, substitute the values of ${{i}_{1}}$ and M in equation (i).
${{E}_{2}}=(0.005)\dfrac{d}{dt}\left( 10\sin (100\pi t) \right)$
$\Rightarrow {{E}_{2}}=(0.005)\left( {{10}^{3}}\pi \cos (100\pi t) \right)$
This means that ${{E}_{2}}=5\pi \cos (100\pi t)$ …. (iii).

From equation (iii), we can understand that the value of ${{E}_{2}}$ will be maximum when the value of the term $\cos (100\pi t)$ is maximum since $5\pi$ is a constant. The maximum value of cosine of an angle is equal to 1. Therefore, the maximum value of $\cos (100\pi t)=1$.
Substitute this value in equation (iii).
$\therefore {{E}_{2}}=5\pi (1)=5\pi$
This means that the maximum value of the emf induced in the second coil due the current in the first coil is equal to $5\pi$.

Hence, the correct option is B.

Note: If we do not know that the maximum value of cosine of an angle is equal to 1, then we can find its maximum value by differentiating the function with respect to time t and equating the derivative to zero. With this will obtain the maximum and minimum value of the function.