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Mutual induction is the phenomenon of inducing e.m.f in a coil due to rate of change of current or change in magnetic flux linked with nearby coil.

Consider a primary coil P connected to a battery through a key and another coil called secondary coil ‘s’ connected to a galvanometer is placed near the primary coil as shown in the figure. A key is pressed the current through primary coil begins to increase so that magnetic field around P increases as a result magnetic flux linking with secondary coil also changes. Due to it, e.m.f. is induced in the secondary coil. Hence current flows through the secondary coil which is indicated by the deflection in the galvanometer. This phenomenon of inducing e.m.f is called mutual inductance.

It is found that the magnetic flux linked with the secondary coil is directly proportional to the current flowing though the primary coil.

\begin{align*} \text {i.e} \: \phi _s &\propto I_p \\ \phi _s &= M\: I_p \dots (i) \\ \end{align*}

where M is proportionality constant called coefficient of mutual induction.

From Faraday’s law of electromagnetic induction,

\begin{align*} E &= -\frac {d\phi }{dt}, \\ \text {so} \: E_s &=\frac {d\phi _s}{dt} \\ \text {or,} \: E_s &= - \frac {d}{dt}(M\:I_p) \\ \therefore E_s &= -M \frac {dI_p}{dt} \dots (ii) \\ \text {If} \: -\frac {dI_p}{dt} = 1, \: \text {then equation}\: (ii)\: \text {becomes} \\ E_s = M \\ \end{align*}

Thus, the coefficient of mutual induction is defined as the e.m.f.induced in the secondary when the rate of change of current in the primary is unity. SI unit of M is Henry. The negative sign shows that M and e.m.f. are opposite in sign.

Consider two solenoids S_{1} and S_{2} such that solenoid S_{2} completely surrounds the solenoid S_{1}. The two solenoids are so closely wound that they have the same area of cross-section A. Let N_{1} and N_{2} be the total number of turns of solenoids S_{1} and S_{2} respectively.

Let current I_{1 }flows through solenoids S_{1}. Then magnetic field inside the solenoid S_{1} is given by

\begin{align*} B_1 &= \mu_0n_1I_1 \\ &= \mu_0\frac {N_1}{l}I_1 \\ \text {where}\: n_1 &= \frac {N_1}{l} \\\therefore \text {Magnetic flux linked with each turn of solenoid}\: S_2 \: \text {is} \\ \phi _2 &= N_2(B_1A) = N_2\mu_0 \frac {N_1}{l}I_1A \\ \text {or,} \: \phi_2 &= \mu_0 \frac {N_1N_2I_1A}{l} \dots (i) \\ \text {But}\: \phi_2 &= M_{12}I_1 \dots (ii) \\ \end{align*}

where \(M_{12}\) is the mutual inductance when current changes in solenoid S_{1} to change the magnetic flux linking with solenoid S_{2} or M_{12} is the mutual inductance of S_{1} with respect to S_{2}.

\begin{align*} \text {From equation}\: (i)\: \text {and} \: (ii,)\: \text {we get} \\ M_{12} I_1 &= \frac {\mu_0N_1N_2I_1A}{l} \\ \therefore M_{12} &= \frac {\mu_0N_1N_2A}{l}\\ \dots (iii) \\ \text {Similarly,} \\ M_{21} &= \frac {\mu_0N_1N_2A}{l}\\ \dots (iv) \\ \end{align*}

where M_{21} is the mutual inductance when current changes is solenoid S_{2} to change the magnetic flux linking with Solenoid S_{1} or mutual inductance of S_{2} with respect to S_{1}.

\begin{align*} \text {From equation}\:(iii)\: \text {and} (iv), \: \text {we get} \\ M_{12} = M_{21} = M \end{align*}

Thus the mutual inductance between the two coils is same, no matter which of the two coils carries the current.

$$\therefore M = \frac {\mu_0N_1N_2A}{l} $$

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, et al.*Principle of Physics*. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. *A text Book of Physics*. Kathmandu: Surya Publication, 2003.

Mutual induction is the phenomenon of inducing e.m.f in a coil due to rate of change of current or change in magnetic flux linked with nearby coil.

The coefficient of mutual induction is defined as the e.m.f.induced in the secondary when the rate of change of current in the primary is unity.

The mutual inductance between the two coils is same, no matter which of the two coils carries the current.

$$\therefore M = \frac {\mu_0N_1N_2A}{l} $$

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