Mutual induction is defined as the property of the coils that enables it to oppose the changes in the current in another coil. With a change in the current of one coil, the flow changes too thus inducing EMF in the other coil. This phenomenon is known as mutual induction. The circuit part that represents inductance is known as an inductor and the term inductance was found by Oliver Heaviside in the year 1886.

We can explain this with the help of an example. Consider the following diagram.

Image Credits: hyperphysics

With a smooth current in one coil, as given in the diagram, a magnetic field is formed in the other coil. That magnetic field remains unchanged and hence, according to Faraday’s law, it will not lead to any voltage creation in the secondary coil. Now, if we open the switch so as to stop the current, the magnetic field will change in the coil placed on the right-hand side, thus inducing a voltage. The coil does not support changes and hence the induced voltage will promote a flow of current in the secondary coil trying to maintain the already existing magnetic field. The fact, however, remains that the induced field will always repel the change. On interrupting the current, the switch is closed so as to allow the current to flow again. This will lead to an induced current, though flowing in the opposite direction to oppose the change in the field. This consistent formation of opposing voltages in the field is the fundamental principle in transformers and can be called mutual inductance.

## Mutual Inductance formula:

e_{m }= M (dI_{1 }/ dt)

or M = e_{m}/ (dI_{1 }/ dt)

where e_{m }is the voltage induced in the secondary coil

I_{1} is the current flowing in the primary foil.

We can use this formula when we know the value of the mutually induced emf as well as the change of current in coil two, or the neighbouring coil.

Now, if e_{m }= 1 and dI_{1} / dt = 1, then on substituting the value in the given equations, we see that the value of M, that is mutual inductance is 1 Henry.

Thus two coils have a mutual inductance of 1 henr when emf of 1 volt is induced in coil 1 and when the current flowing through coil 2 is changing at the rate of one ampere per second.

There is another formula that can be used to calculate mutual inductance. It can be written as

e_{m} = M ( dI_{1} / dt ) = d/dt (MI_{1}) . . ( 2 )

And e_{m} = N_{2} ( dφ_{12}/dt) = d/dt (N_{2}φ_{12) . . ( }3 )

On equating 1 and 2 we get,

MI_{1 }= N_{2} φ_{12 }

Or, M = N_{2} φ_{12 }/ I_{1} Henry

The above equation can be used pen the flux linkage or N_{2} φ_{12 }of one coil due to the flow of current in the other coil (I_{1}) is known.

It is to be noted that the value of mutual inductance is dependent on the three factors, namely, the proximity of 2 coils, the cross-sectional area and the number of rounds in the secondary coil.

A device that produces the effect of mutual induction is called a transformer. However, a very interesting property of the device is its ability to change voltage and current ratio only according to a simple ratio, which is determined by the input, and the output of the coil turns.

These effects are actually derived from two different and fundamental observations pertaining to physics. Those are:

- A steady current creates a steady magnetic field known as the Oersted’s law
- Faraday’s law that has been described already.

## What is a coupling coefficient?

It is the ratio of the open circuit to actual voltage ratio and the ratio that we obtain if the flux coupled from one circuit to the other circuit. It is connected to mutual inductance and is an easy way to understand the relation between specific orientations of the inductors with arbitrary inductance.

## Is mutual induction used in eddy current inspection?

The eddy currents are formed and generated in the test material mainly due to mutual induction. The test material is a wire coil through which alternating current is sent. The probe is then connected to an eddycope device. The next circuit may be a piece of any conductive material. When current is passed through a coil, it leads to the generation of a magnetic field in and all around the same coil. Then, when a probe is brought close to the conductive material, the probe’s alternating magnetic field leads to current flow in the material. These currents flow in closed loops in a plane that is at a right angle to the magnetic flux. They are called eddy currents.

These eddy currents make their own field that interacts with the primary magnetic field of the specific coil. Information on the test material can be gained by analyzing the deviations in the resistance and inductive reactance of the given coil. We can learn facts about conductivity and permeability of the material, the condition of the material etc.

** **Summary

- Mutual inductance is when the magnetic field that is generated by a coil induces a voltage in a secondary coil.
- A transformer works on the principle of mutual induction
- The formula that can be used to calculate mutual inductance is:

M = e_{m}/ (dI_{1 }/ dt)

Or, N_{2} φ_{12 }/ I_{1} Henry

4. Mutual inductance is used in eddy current inspection