31.1: Mutual Inductance

Inductance is the property of a device that tells us how effectively it induces an emf in another device. In other words, it is a physical quantity that expresses the effectiveness of a given device.

When two circuits carrying time-varying currents are close to one another, the magnetic flux through each circuit varies because of the changing current in the other circuit. Consequently, an emf is induced in each circuit by the changing current in the other. Therefore, this type of emf is called a mutually induced emf, and the phenomenon is known as mutual inductance (M).

The value of M neatly encapsulates the physical properties of circuit elements and allows us to separate the physical layout of the circuit from the dynamic quantities, such as the emf and the current.

A large mutual inductance value may or may not be desirable. We want a transformer to have a large mutual inductance. But an appliance, such as an electric clothes dryer, can induce a dangerous emf on its metal case if the mutual inductance between its coils and the case is large. One way to reduce mutual inductance is to counter-wind coils to cancel the magnetic field produced.

Consider two copper coils with N₁ and N₂ turns, labeled as primary and secondary coils, placed close to each other.

A variable current I₁ is passed in the primary coil, generating a magnetic field.

Some of the magnetic field lines also pass through the secondary coil, and the changing magnetic flux induces an emf and current in the secondary coil, which can be evaluated using Faraday's law.

Also, the magnetic flux in the secondary coil is proportional to the current flowing in the primary coil.

A constant M is introduced to remove the proportionality sign, which is called mutual inductance.

Mutual inductance only depends on the geometry and orientation of both the coils, and its SI unit is the henry.

On combining and rearranging the equations, the induced emf in the secondary coil is obtained.

Now, reversing the case, the induced emf in the primary coil due to the changing current in the secondary coil can be obtained.

If the coils are in vacuum, the corresponding constants of mutual inductance of both coils are equal.