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

\[ \varepsilon=-\frac{\mathrm{d} \Phi_{\mathrm{B}}}{\mathrm{d} t} \]

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

\[ \varepsilon=-N \frac{\mathrm{d} \Phi_{B}}{\mathrm{~d} t} \]

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.