Electromagnetic Induction
The term electromagnetic induction constitutes two phenomena. The first involves a current that is induced in a conductor moving relative to magnetic field lines. The second involves the generation of an electric field associated with a time-varying magnetic field.
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The observations of Michael Faraday are illustrated in the following cases:
(i) Change in Field Strength.
When a bar magnet moves relative to a Loop of wire, there is an induced current in the loop (see Fig. 1 ) when the magnet and the loop are stationary, nothing happens.
| Fig.(1) (a) When the north pole moves toward the loop a current flows in anitclockwise sense-as-seen from the side on which the magnet is located.
(b) When the north pole moves away from the loop, the current flows in the clockwise sense. (c) When the south pole of the magnet moves toward the loop, the current flow in the clockwise sense. (d) When the south pole of the magnet moves away from the loop, the current flows in the anticlockwise sense. |
These results (shown in Fig. 1) are unchanged when the magnet is kept at rest and the loop is moved instead. The magnitude and direction of the induced current depends on the relative velocity of the coil and magnet.
| Let us consider two coils at rest, as shown in Fig. 2. The primary coil is connected in series with a battery and a switch, whereas the secondary coil is connected to a galvanometer. When the switch in the primary circuit is closed, the galvanometer in the secondary deflects for an instant. As long as the primary current stays constant (switch remain closed), nothing happens. When the switch is opened, the meter again has a momentary deflection. | |
| (ii) Change in Area
Consider a circular coil made of flexible wire lying with its plane perpendicular to a uniform constant magnetic field B, as shown in Fig. 3. When opposite ends of the diameter are suddenly pulled apart, thereby reducing the area enclosed by the loop, an induced current is produced. |
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(iii) Change in Orientation. Let us consider the case when the magnitude of the field strength and area of the coil remains constant. When the plane of the coil is rotated relative to the direction of the field, an induced current is produced as long as the rotation lasts.
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FARADAY’S LAW
In order to explain the above results Faraday introduced the concept of magnetic flux ΦB, which is defined as
The SI unit of magnetic flux is the weber (Wb)
1 Wb = (1 T) ( 1m2)
The generation of an electric current in a circuit implies the existence of an emf. Faraday stated,
The induced emf along any closed path is proportional to the rate of change of magnetic flux through the area bounded by the path.
The three terms represent the contributions from the rate of change of B, A and θ, respectively to the rate of change of flux. In a given situation, more than one of these may contribute.
The negative sign in the last term signifies that an increase in θ leads to a decrease in flux.
Note that the induced emf is not confined to a particular point. It is distributed around the loop.
LENZ’S LAW
The effect of the induced emf is such as to oppose the change in flux that produces it.
Fig. 5 (a) As the magnet approaches the loop, the positive flux through the loop increases. The induced currents sets up an Induced Magnetic Field, Bind whose (negative) flux opposes this change. The direction of Bind is opposite to that of external field Bext due to the magnet.
(b) When the flux through the loop decreases as the magnet moves away fmo the loop, the flux due to the induced magnetic field tries to maintain the flux through the loop.
In order to incorporate Lenz’s law in equation (1) the modern statement of Faraday’s law of electromagnetic induction is
The negative sign indicates that the induced emf opposes the change in magnetic flux that produces it.
If the single loop is replaced by a coil of N turns and same flux fB passes through every turn, then the net emf induced is given by
Lenz’s law is closely related to the law of conservation of energy and is actually a consequence of this general law of nature. As the north pole of the magnet moves toward the loop(see Fig. 5 a) a north pole appears on the upper surface of the loop which opposes the motion of the N-pole of the bar magnet. Thus, in order to move the magnet toward the loop with a constant velocity an external force is to be applied. The work done by this external force gets transformed into electric energy which produces induced current in the loop.
| Illustration-1
A metal rod of length l slides at constant velocity v on conducting rails, placed in a uniform and constant magnetic field B perpendicular to the plane of the rails as shown in figure. A resistance R is connected between the two ends of the rail. |
(a) Find the current in the resistor
(b) Find the power dissipated in the resistor
(c) Find the mechanical power needed to pull the rod.
Solution
In this case flux varies due to the change of area. Let x be the instantaneous distance moved by the rod, then the flux through the area enclosed is
The flux is increasing because the area is increasing. The induced emf opposes the increase in flux, which means that the induced magnetic field is opposite to the external field. Thus, the induced current in the circuit is anticlockwise.
The magnitude of the current is
(b) The electric power dissipated in the resistor is
| (c) Because of the induced current flowing in it, the rod experiences a force = I ´ due to the external field. The force F is directed opposite to v. In order to move the rod at constant velocity an external force Fext = must be applied to the right. |
The mechanical power supplied by the external agent is
| Illustration-2
A square loop of side l moves at constant velocity v perpendicular to a uniform magnetic field as shown in the Fig. 7. Starting at the time at which it enters the field until it leaves the field, make plots of the variation in the flux through the loop and the emf induced in it as functions of time. |
Solution
| Illustration-3
A circular coil of radius 5 cm consists of exactly 250 turns. A magnetic field is directed perpendicular to its plane (as shown in Fig. 8) is increasing at a rate of 0.6 T/s. If the resistance of the coil is 8W, find the current induced in the coil.
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Solution
| Illustration-4
A circular loop of area A and resistance R rotates with an angular velocity w about an axis through its diameter as shown in |
Solution
The instantaneous magnetic flux through the loop is
ΦB = BA cos θ
Since θ = ωt, therefore, ΦB = BA cos ωt
From Faraday’s law, equation (3) is
The induced current is
Both the induced emf and the current vary sinusoidally. The amplitude of the emf is BAω and that of the current is
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