6.4 Faraday's Law of Induction

From the experimental observations, Faraday arrived at a conclusion that an emf is induced in a coil when magnetic flux through the coil changes with time. Experimental observations discussed in Section 6.2 can be explained using this concept.

The motion of a magnet towards or away from coil $C_1$ in Experiment 6.1 and moving a current-carrying coil $C_2$ towards or away from coil C 1 in Experiment 6.2, change the magnetic flux associated with coil $C_1$ . The change in magnetic flux induces emf in coil $C_1$ . It was this induced emf which caused electric current to flow in coil $C_1$ and through the galvanometer. A plausible explanation for the observations of Experiment 6.3 is as follows: When the tapping key K is pressed, the current in coil $C_2$ (and the resulting magnetic field) rises from zero to a maximum value in a short time. Consequently, the magnetic flux through the neighbouring coil $C_1$ also increases. It is the change in magnetic flux through coil $C_1$ that produces an induced emf in coil $C_1$ . When the key is held pressed, current in coil $C_2$ is constant. Therefore, there is no change in the magnetic flux through coil $C_1$ and the current in coil $C_1$ drops to zero. When the key is released, the current in $C_2$ and the resulting magnetic field decreases from the maximum value to zero in a short time. This results in a decrease in magnetic flux through coil $C_1$ and hence again induces an electric current in coil $C_1$ *. The common point in all these observations is that the time rate of change of magnetic flux through a circuit induces emf in it. Faraday stated experimental observations in the form of a law called Faraday’s law of electromagnetic induction.

The law is stated below.

The magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit.

Mathematically, the induced emf is given by

$$\epsilon =-\frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{B}}}{\mathrm{d}t}$$

The negative sign indicates the direction of ε and hence the direction of current in a closed loop. This will be discussed in detail in the next section.

In the case of a closely wound coil of N turns, change of flux associated with each turn, is the same. Therefore, the expression for the total induced emf is given by

$$\epsilon =-N\frac{\mathrm{d}{\mathrm{\Phi }}_B}{\mathrm{d}t}$$

The induced emf can be increased by increasing the number of turns N of a closed coil.

From Eqs. (6.1) and (6.2), we see that the flux can be varied by changing any one or more of the terms $\mathbf{B}$, $\mathbf{A}$ and $\theta$ . In Experiments 6.1 and 6.2 in Section 6.2, the flux is changed by varying $\mathbf{B}$. The flux can also be altered by changing the shape of a coil (that is, by shrinking it or stretching it) in a magnetic field, or rotating a coil in a magnetic field such that the angle $\theta$ between $\mathbf{B}$ and $\mathbf{A}$ changes. In these cases too, an emf is induced in the respective coils.