Electromagnetic Induction || Lenz law || Faraday’s Laws of Electromagnetic Induction || MOTIONAL ELECTROMOTIVE FORCE || INDUCTANCE || MUTUAL INDUCTANCE || SELF-INDUCTANCE || Notes class 12 Chapter 6 ||

                               ELECTROMAGNETIC INDUCTION

                      Faraday’s Laws of Electromagnetic Induction 

1. An emf is induced in a loop when the number of magnetic field lines i.e., magnetic flux passing through the loop is changing. 

2. Magnitude of the emf ‘e’ induced in a conducting loop is equal to the rate at which the magnetic flux through that loop changes with time.

                                                                Lenz’s Law 

The direction of induced emf (i.e., polarity of induced emf) and hence the direction of induced current in a closed circuit is to oppose the cause due to which they are produced. For example, if the flux is increasing, induced emf (and hence induced current) will try to decrease the flux and vice-versa. It is based on energy conservation law. 

Methods to Change the Magnetic Flux Here are the general methods by which we can change the magnetic flux through a coil. 

(i) Change the magnitude B of the magnetic field within the coil. 

(ii) Change either the total area of the coil or the portion of that area that lies within the magnetic field (for example, by expanding the coil or sliding it in or out of the field. 

(iii) Change the angle between the direction of the magnetic field B and the plane of coil (For example, by rotating the coil so that field B is first perpendicular to the plane of the coil and then is along that plane).

                Applications of Lenz’s law shall become more clear by carefully studying the following examples (1) A conducting loop is kept in a uniform magnetic field B directed into the plane of paper. The magnetic field starts increasing with time. As the magnetic field increases, flux B.A starts increasing. Therefore, an induced current will flow into the loop. 

The current is induced due to increasing field, therefore, it will try to decrease the field. This can happen if induced current produces an outward field in the loop. Therefore, an anticlockwise current appears in the loop. 

                           MOTIONAL ELECTROMOTIVE FORCE

 Let us consider a straight conductor moving in a uniform and time independent magnetic field. Fig. (c) shows a rectangular conductor PQRS in which the conductor PQ is free to move. The rod PQ is moved towards the left with a constant velocity v as shown in the figure. Assume that there is no loss of energy due to friction. PQRS forms a closed circuit enclosing an area that changes as PQ moves. It is placed in a uniform magnetic field B which is perpendicular to the plane of this system. If the length RQ = x and RS , the magnetic flux B enclosed by the loop PQRS will be 

It is also possible to explain the motional emf expression in Eq. (iii) by invoking the Lorentz force acting on the free charge carriers of conductor PQ. Consider any arbitrary charge q in the conductor PQ. When the rod moves with speed v, the charge will also be moving with speed v in the magnetic field B. The Lorentz force on this charge is qvB in magnitude, and its direction is towards Q. All charges experience the same force, in magnitude and direction, irrespective of their position in the rod PQ.

   INDUCED ELECTRIC FIELD DUE TO A TIME VARYING MAGNETIC FIELD

 Consider a conducting loop placed at rest in a magnetic field B  . Suppose, the field is constant till t = 0 and then changes with time. An induced current starts in the loop at t = 0. The free electrons were at rest till t = 0 (we are not interested in the random motion of the electrons.) The magnetic field cannot exert force on electrons at rest. Thus, the magnetic force cannot start the induced current. The electrons may be forced to move only by an electric field. So we conclude that an electric field appears at time t = 0.

This electric field is produced by the changing magnetic field and not by charged particles. The electric field produced by the changing magnetic field is non electrostatic and non conservative in nature. We cannot define a potential corresponding to this field. We call it induced electric field. The lines of induced electric field are closed curves. There are no starting and terminating points of the field lines. 

                                                    INDUCTANCE 

Inductance is a measure of the ratio of the flux to the current I. Consider two coils, the current through one coil sets up a magnetic field both in the second coil as well as through itself.

                                         MUTUAL INDUCTANCE 

Consider two long co-axial solenoids each of length l. We denote the radius of the inner solenoid S1 by r1 and the number of turns per unit length by n1 . The corresponding quantities for the outer solenoid S2 are r 2 and n2 respectively.

 

We set up a time varying current I2 through S2 . This sets up a time varying magnetic flux through S1.

                                

                                      SELF-INDUCTANCE 

It is also possible that emf is induced in a single isolated coil due to change of flux through the coil by means of varying the current through the same coil. This phenomenon is called self-induction. In this case, flux linkage through a coil of N turns is proportional to the current through the coil where constant of proportionality L is called self inductance of the coil. It is also called the coefficient of self-induction of the coil. When the current is varied, the flux linked with the coil also changes and an emf is induced in the coil.