Question: Q. 1. Define mutual inductance between a pair of coils. Derive an expression for the mutual inductance of two long coaxial solenoids of same length wound one over the other.
R [O.D. I, II, III 2017, 2014]
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Solution:
Ans. Definition of mutual inductance
1 Derivation of mutual inductance for two long solenoids
(i) Mutual inductance, is numerically equal to the induced emf in the secondary coil when the current in the primary coil changes by unity.
Alternatively,
Mutual inductance in numerically equal to the magnetic flux linked with one coil/secondary coil when unit current flows through the other coil/ primary coil.
(ii)
Let a current $I_{2}$, flow in the secondary coil
$$ \therefore \quad B_{2}=\frac{\mu_{0} N_{2} i_{2}}{l} $$
$\therefore$ Flux linked with the primary coil
$$ \begin{aligned} & =\frac{\mu_{0} N_{2} N_{1} A_{1} i_{2}}{l} \ & =M_{12} i_{2} \end{aligned} $$
Hence,
$$ \begin{aligned} M_{12} & =\frac{\mu_{0} N_{2} N_{1} A_{2}}{l} \ = & \mu_{0} n_{2} n_{1} A_{1} l \ & \quad\left(n_{1}=\frac{N_{1}}{l} ; n_{2}=\frac{N_{2}}{l}\right)^{1 / 2} \end{aligned} $$
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