Question

A rectangular conductor, L M N O, is placed in a uniform magnetic field $\vec{B}$ , directed perpendicular to the plane of the conductor, as shown in Fig. $9.58$ . Obtain an expression, for the emf induced in the arm M N, when the arm is moved towards the left (parallel to itself) with a speed v. Use the above expression to find the emf induced between the ends of an axle, of length L, of a railway carriage, travelling on level ground, with speed v. Assume the value of earth's magnetic field at the place to be B and the angle of dip to be $θ$

Accepted Answer

Here uniform magnetic field $= \vec{B}$ , Length $= L$ , Speed $= v$ , angle of deep $= θ$ .

Flux induced $ϕ =$ $B . A = B A \cos (θ)$

Emf induced in moving rod $M N$ is $E = |\frac{d ϕ}{d t}| = B v L$

shows a rectangular conductor $L M N O$ in which the conductor LM is free to move. The rod MN 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. LMNO forms a closed circuit enclosing an area that changes as MN moves.

The magnetic flux $ϕ_{B}$ enclosed by the loop LMNO will be $ϕ_{B} = B L x$

Since $x$ is changing with time, the rate of change of flux $ϕ_{B}$ will induce an emf given by:

where we have used dx/dt = –v which is the speed of the conductor PQ. The induced emf Blv is called motional emf. Thus, we are able to produce induced emf by moving a conductor instead of varying the magnetic field, that is, by changing the magnetic flux enclosed by the circuit.

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 MN. The work done in moving the charge.

$W = q v B l$

Since emf is the work done per unit charge,

$E = \frac{w}{q}$ $= B L v$
$E = B l v$

This equation gives emf induced across the rod MN and is identical. We stress that our presentation is not wholly rigorous. But it does help us to understand the basis of Faraday’s law when the conductor is moving in a uniform and time-independent magnetic field. On the other hand, it is not obvious how an emf is induced when a conductor is stationary and the magnetic field is changing – a fact which Faraday verified by numerous experiments. In the case of a stationary conductor, the force on its charges is given by

Important Questions on Electromagnetic Induction

Learn and Prepare for any exam you want !