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When the primary current in the spark-coil of a car changes from $4\,A$ to zero in \[10\,\mu S\], an emf of $40,000\,V$ is induced in the secondary. The mutual inductance between the primary and the secondary windings of the spark-coil will be –
A. 1 H
B. 10 H
C. 0.1 H
D. 0

Last updated date: 14th Jul 2024
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Hint:The tendency of an electrical conductor to resist a difference in the electric current flowing through it is known as inductance in electromagnetism and electronics. A magnetic field is generated around a conductor by the movement of electric current.

Complete step by step answer:
Mutual inductance occurs as one coil's magnetic field interacts with the magnetic field of another coil, causing a voltage to be induced in the opposite coil. An inductor induces an induced emf within itself as a result of the shifting magnetic field around its own turns, as we saw in the previous tutorial.

The coil's self inductance is a property of the coil. The trait of a pair of coils is mutual inductance. As the key current in the coil reduces, the induced current opposes the decay of current in the coil. Mutual inductance is given by the formula
\[e = \frac{{M\Delta i}}{{\Delta t}}\]
$e = 40000\,V$ (Given)
\[\Delta i\]= 4 A
\[\Rightarrow \Delta t\]= 10 \[\mu S\] = \[10 \times {10^{ - 6}}\] S
\[\Rightarrow 4000 = M \times \frac{4}{{10 \times {{10}^{ - 6}}}}\]
Upon Solving for M we get
\[M = \frac{{4000 \times 10 \times {{10}^{ - 6}}}}{4}\]
\[\Rightarrow M = {10^5} \times {10^{ - 6}}\]
$\therefore M = 0.1\,H$

Hence, option C is correct.

Note:As two wire coils are brought close enough together that the magnetic field of one connects with the magnetic field of the other, a voltage is produced in the second coil. Mutual inductance occurs when a voltage applied to one coil produces a voltage in another.