Question

An e.m.f of 5 millivolt is induced in a coil when in a nearby placed another coil, the current changes by 5 ampere in 0.1 second. The coefficient of mutual induction between the two coils will be
A. 1 Henry
B. 0.1 Henry
C. 0.1 millihenry
D. 0.001 millihenry

Answer 1

Hint: Mutual induction is the property of two coils by virtue of which each opposes any change in the magnitude of current flowing through the other by inducing an emf in itself, provided magnetic flux of one coil is linked with other.
The coefficient of mutual induction, M, between two coils is given by:
\[M=-\dfrac{e}{{di}/{dt}\;}\]
Where e is the emf of one coil and \[{di}/{dt}\;\] is the rate of change of current in the other coil.

Complete answer:
Let us consider the given data:
Induced emf in one coil, \[e=5\text{ mV}\]
Change in current, \[di=5\text{ A}\]
Time-interval in which current changes, \[dt=0.1\text{ s}\]
Substituting the given values in the mutual induction formula, we get,
\[\begin{align}
  & M=-\dfrac{5\text{ mV}}{{5\text{ A}}/{0.1\text{ s}}\;} \\
 & \Rightarrow M=-\dfrac{5\text{ mV}}{50\text{ A}{{\text{s}}^{-1}}} \\
 & \Rightarrow M=-0.1\text{ mH} \\
\end{align}\]
The negative sign indicates that the direction of emf induced in the coil is always such that it opposes any change in current in the other coil.
Therefore, the magnitude of coefficient of mutual inductance is 0.1 millihenry.

Hence, option C is the correct answer.

Note:
It is important to know that the mutual inductance of a pair of coils, solenoids, etc., depends on their separation as well as their relative orientation.
Mutual inductance between two coils depends upon the geometry of two coils and their geometrical orientation with respect to each other. To find mutual inductance for a given arrangement, we assume a current flowing through one of the coils and find the flux through the other coil. Then using the formula\[\phi =MI\], mutual inductance M can be calculated.