What happens when two coils are brought together and current in one coil is changed? In self-inductance we learn how a coil behaves if current is changed in the same coil, but in mutual inductance we learn how the second coil behaves if current changes in the first coil and vice versa.
In Figure 1 two coils are put together. We first change current in the first coil. Let the current in the first coil at a particular instant be \(i_1\). Note that this current changes with time. We all know that if a current changes in the first coil, the magnetic field of that coil also changes. If the magnetic field is changing, the flux is changing and according to Faraday's law, the induced emf is the negative of the rate of change of magnetic flux.
So, the emf is induced in the second coil due to the change in current in the first coil. Let the second coil has \(N_2\) turns and the magnetic flux through this coil is \(\Phi_{B2}\). The experimental evidences suggest that the magnetic flux through one coil is proportional to the current in the another coil.
The magnetic flux \(\Phi_{B2}\) is the average flux through each turn of the coil 2. The magnetic field inside the coil is not uniform and the magnetic flux through each turn is the average magnetic flux. So, the total magnetic flux through the coil 2 is \(N_2\Phi_{B2}\). And the magnetic flux through coil 2 is directly proportional to the current in coil 1, that is
\[N_2\Phi_{B2} = M_{21}i_1 \tag{1} \label{1}\]
where the constant \(M_{21}\) is the mutual inductance of coil 2 with respect to coil 1. We will define mutual inductance later once we know that the mutual inductance of coil 1 with respect to coil 2 is the same as the mutual inductance of coil 2 with respect to coil 1.
Note that we could simply add a constant say \(a\) for the portability constant and simply write \(\Phi_{B2} = a\,i\) without \(N_2\) but it is more convenient to include \(N_2\) in the above equation. Differentiating Equation \eqref{1} with respect to \(t\), you get
\[N_2\frac{d\Phi_{B2}}{dt} = M_{21}\frac{di_1}{dt}\]
but according to Faraday's law if \(\mathcal{E}_2\) is the emf induced in the coil 2 due to the change in current in coil 1, you get
\[\mathcal{E}_2 = -N_{2}\frac{d\Phi_{B2}}{dt}\]
So, we have
\[\mathcal{E}_2 = -M_{21}\frac{di_2}{dt}\]
Similarly if the current \(i_2\) in coil 2 changes and causes change in magnetic flux in coil 1, the emf induced in coil 1 is
\[\mathcal{E}_1 = -M_{12}\frac{di_2}{dt}\]
where \(\mathcal{E}_1\) is the emf induced in coil 1 and \(M_{12}\) is the mutual inductance of coil 1 with respect to coil 2.
It has been found that the mutual inductance of coil 2 with respect to coil 1 is equal to that of coil 1 with respect to coil 2, that is \(M = M_{21} = M_{12}\). The mutual inductance \(M\) can be defined as the induced emf in one coil per unit rate of change of current in another coil. So, the mutual inductance is common to both coils, that's what the word mutual means. We can rewrite the equations we just derived as
\[\mathcal{E}_1 = -M\frac{di_2}{dt}\]
\[\mathcal{E}_2 = -M\frac{di_1}{dt}\]
And we can obtain the expression of mutual inductance from Equation \eqref{1} as
\[M = \frac{N_2\Phi_{B2}}{i_1} = \frac{N_1\Phi_{B1}}{i_2}\]
Mutual inductance depends on the geometry of the two coils, that is, it depends on the shape, size, orientation, number of turns and separation of the two coils. If the separation of the two coils increase, the less magnetic field reaches another coil and less mutual inductance. If any magnetic material is present, it also depends on that magnetic material. Note that it does not depend on the current on either coil.
In car ignition coil, two coils used, one with fewer turns with smaller diameter and another with large number turns with larger diameter. Several thousand volts can be easily obtained by a simple 12V battery. When the current from this battery is changed in the smaller coil with only a few turns, a very high voltage appears across another coil with large number of turns. That high voltage is applied to spark plug to ignite the fuel mixture. Even a battery of 1.5V can generate thousands of volts!
These simple coils are extremely dangerous! So, do not try to make these coils if you are not trained to work with high voltages!
One problem that can arise due to mutual inductance is that the unwanted emf can be induced in the nearby coils, so to mitigate this problem we place coils farther apart so the influence of changing magnetic field of one coil on another coil is negligible.
A pair of coils with different turns can be used to increase or decrease voltage. A device that increases or decreases the voltage to a necessary value is called transformer. And the transformer is not different from the pair of coils. Of course a ferromagnetic material is used as its core and the core is more refined to reduce the effects of eddy currents.
