17.17 Mutual Induction

17.17 Mutual Induction

Consider two coils close to each other as shown in Fig. (17.20). One coil connected with a battery through a switch S and a rheostat, is called Primary coil and the other one connected to the galvanometer is called Secondary coil. If the current in the primary is changed by varying the resistance of the rheostat, the magnetic flux in the surrounding region changes. Since the secondary coil is in magnetic field of the primary, the changing flux also links with the secondary. This causes an induced emf in the secondary.

The phenomenon in which a changing current in one coil induces an emf in another coil is called mutual induction. According to Faraday's law, the emf induced in the secondary coil $\epsilon_{s}$ is proportional to the rate of change of flux $\Delta\phi_{s} / \Delta t$ passing through it and is given by

where $N_{s}$ is the number of turns in the secondary coil. Let $\phi_{s}$ represents flux passing through the secondary coil. The net flux passing through the secondary coil of $N_{s}$ loops is $N_{s} \phi_{s}$. As this net flux is proportional to the magnetic field produced by the current $I_{p}$ in the primary and the magnetic field itself is proportional to $I_{p}$, therefore $N_{s} \phi_{s} \propto I_{p}$

where $M$ is proportionality constant and is called Mutual inductance of the two coils, given by $M = N_{s} \dfrac{\phi_{s}}{I_{p}}$ The mutual inductance $M$ depends upon the following factors,

1. Number of turns of both the primary and secondary coils.
2. Area of cross-section of the two coils.
3. Magnetic permeability of medium between the coils.
4. Nature of material on which two coils are wound.
5. Distance between two coils.
6. Orientation between the primary and secondary coils.

(Fig. (17.20): Circuit diagram to find Mutual induction between two coils.)

By Faraday's law, the emf in the secondary coil is given by the rate of change of flux through the secondary coil r as: $\epsilon_{s} = -N \dfrac{\Delta\phi_{s}}{\Delta t}$ Using Eq. (17.40), we have: $\epsilon_{s} = - \dfrac{\Delta N_{s} \phi_{s}}{\Delta t}$ $\epsilon_{s} = - \dfrac{\Delta M I_{p}}{\Delta t}$

Which shows that emf induced in the secondary coil is proportional to time rate of change of current in the primary. The negative sign indicates the fact that the induced emf is in such a direction that it opposes the change of current in the primary coil. The magnitude of mutual induction $M$ is

Mutual induction can be defined as ratio of average emf induced in the secondary to the time rate of change of current in the primary. The SI unit of mutual inductance $M$ is $\mathrm{V\,s\,A^{-1}}$, which is called henry after Joseph Henry. One henry is the mutual inductance of the pair of coils in which the rate of change of current of one ampere per second in the primary causes an induced emf of one volt in the secondary. Mutual inductance finds many applications in various devices such as transformers, electric motors, and generators etc. It also plays a key role in digital signal processing and is utilized in devices like pacemakers and metal detectors.

(Fig. 17.21: Human heart)

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