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 \(\mathrm{C}_1\) in Experiment 6.1 and moving a current-carrying coil \(\mathrm{C}_2\) towards or away from coil \(C_1\) in Experiment 6.2, change the magnetic flux associated with coil \(\mathrm{C}_1\). The change in magnetic flux induces emf in coil \(C_1\). It was this induced emf which caused electric current to flow in coil \(\mathrm{C}_1\) 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 \(\mathrm{C}_2\) (and the resulting magnetic field) rises from zero to a maximum value in a short time. Consequently, the magnetic flux through the neighbouring coil \(\mathrm{C}_1\) also increases. It is the change in magnetic flux through coil \(\mathrm{C}_1\) that produces an induced emf in coil \(\mathrm{C}_1\). When the key is held pressed, current in coil \(\mathrm{C}_2\) is constant. Therefore, there is no change in the magnetic flux through coil \(\mathrm{C}_1\) and the current in coil \(\mathrm{C}_1\) drops to zero. When the key is released, the current in \(\mathrm{C}_2\) 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 \(\mathrm{C}_1\) and hence again induces an electric current in coil \(\mathrm{C}_1{ }^*\). 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
\(
\varepsilon=-\frac{\mathrm{d} \Phi_B}{\mathrm{~d} t} \dots(6.3)
\)
The negative sign indicates the direction of \(\varepsilon\) 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
\(
\varepsilon=-N \frac{\mathrm{~d} \Phi_B}{\mathrm{~d} t} \dots(6.4)
\)
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 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.

* Note that sensitive electrical instruments in the vicinity of an electromagnet can be damaged due to the induced emfs (and the resulting currents) when the electromagnet is turned on or off.

Example 6.1: Consider Experiment 6.2. (a) What would you do to obtain a large deflection of the galvanometer? (b) How would you demonstrate the presence of an induced current in the absence of a galvanometer?

Solution: (a) To obtain a large deflection, one or more of the following steps can be taken: (i) Use a rod made of soft iron inside the coil \(C_2\), (ii) Connect the coil to a powerful battery, and (iii) Move the arrangement rapidly towards the test coil \(C_1\).
(b) Replace the galvanometer by a small bulb, the kind one finds in a small torch light. The relative motion between the two coils will cause the bulb to glow and thus demonstrate the presence of an induced current.
In experimental physics one must learn to innovate. Michael Faraday who is ranked as one of the best experimentalists ever, was legendary for his innovative skills.

Example 6.2: A square loop of side 10 cm and resistance \(0.5 \Omega\) is placed vertically in the east-west plane. A uniform magnetic field of 0.10 T is set up across the plane in the north-east direction. The magnetic field is decreased to zero in 0.70 s at a steady rate. Determine the magnitudes of induced emf and current during this time-interval.

Solution: The angle \(\theta\) made by the area vector of the coil with the magnetic field is \(45^{\circ}\). From Eq. (6.1), the initial magnetic flux is
\(
\begin{aligned}
& \Phi=B A \cos \theta \\
& =\frac{0.1 \times 10^{-2}}{\sqrt{2}} \mathrm{~Wb}
\end{aligned}
\)
Final flux, \(\Phi_{\min }=0\)
The change in flux is brought about in 0.70 s. From Eq. (6.3), the magnitude of the induced emf is given by
\(
\varepsilon=\frac{\left|\Delta \Phi_B\right|}{\Delta t}=\frac{|(\Phi-0)|}{\Delta t}=\frac{10^{-3}}{\sqrt{2} \times 0.7}=1.0 \mathrm{mV}
\)
And the magnitude of the current is
\(
I=\frac{\varepsilon}{R}=\frac{10^{-3} \mathrm{~V}}{0.5 \Omega}=2 \mathrm{~mA}
\)
Note that the earth’s magnetic field also produces a flux through the loop. But it is a steady field (which does not change within the time span of the experiment) and hence does not induce any emf.

Example 6.3: A circular coil of radius \(10 \mathrm{~cm}, 500\) turns and resistance \(2 \Omega\) is placed with its plane perpendicular to the horizontal component of the earth’s magnetic field. It is rotated about its vertical diameter through \(180^{\circ}\) in 0.25 s. Estimate the magnitudes of the emf and current induced in the coil. Horizontal component of the earth’s magnetic field at the place is \(3.0 \times 10^{-5} \mathrm{~T}\).

Solution: Initial flux through the coil,
\(
\begin{aligned}
\Phi_{\mathrm{B} \text { (nittal) }} & =B A \cos \theta \\
& =3.0 \times 10^{-5} \times\left(\pi \times 10^{-2}\right) \times \cos 0^{\circ} \\
& =3 \pi \times 10^{-7} \mathrm{~Wb}
\end{aligned}
\)
Final flux after the rotation,
\(
\begin{aligned}
\Phi_{\mathrm{B}(\text { fnal) }} & =3.0 \times 10^{-5} \times\left(\pi \times 10^{-2}\right) \times \cos 180^{\circ} \\
& =-3 \pi \times 10^{-7} \mathrm{~Wb}
\end{aligned}
\)
Therefore, estimated value of the induced emf is,
\(
\begin{aligned}
\varepsilon & =N \frac{\Delta \Phi}{\Delta t} \\
& =500 \times\left(6 \pi \times 10^{-7}\right) / 0.25 \\
& =3.8 \times 10^{-3} \mathrm{~V} \\
I & =\varepsilon / R=1.9 \times 10^{-3} \mathrm{~A}
\end{aligned}
\)
Note that the magnitudes of \(\varepsilon\) and \(I\) are the estimated values. Their instantaneous values are different and depend upon the speed of rotation at the particular instant.