6.6 Motional Electromotive Force
The arm PQ is moved to the left side, thus decreasing the area of the rectangular loop. This movement induces a current I as shown.
Let us consider a straight conductor moving in a uniform and time- independent magnetic field. Figure 6.10 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 = l, the magnetic flux \(\Phi_{B}\) enclosed by the loop PQRS will be
\[ \Phi_{\mathrm{B}}=B l x \]
Since x is changing with time, the rate of change of flux \(\Phi_{B}\) will induce an emf given by:
\[ \begin{aligned} \varepsilon &=\frac{-\mathrm{d} \Phi_{B}}{\mathrm{~d} t}=-\frac{\mathrm{d}}{\mathrm{d} t}(B l x) \\ &=-B l \frac{\mathrm{d} x}{\mathrm{~d} t}=B l v \end{aligned} \]
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.
It is also possible to explain the motional emf expression in Eq. (6.5) 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. The work done in moving the charge from P to Q is,
\[ \mathrm{W}=q \mathrm{vBl} \]
Since emf is the work done per unit charge,
\[ \begin{aligned} \varepsilon &=\frac{W}{q} \\ &=B w \end{aligned} \]
This equation gives emf induced across the rod PQ and is identical to Eq. (6.5). 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
\[ \mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})=q \mathbf{E} \]
since v = 0. Thus, any force on the charge must arise from the electric field term E alone. Therefore, to explain the existence of induced emf or induced current, we must assume that a time-varying magnetic field generates an electric field. However, we hasten to add that electric fields produced by static electric charges have properties different from those produced by time-varying magnetic fields. In Chapter 4, we learnt that charges in motion (current) can exert force/torque on a stationary magnet. Conversely, a bar magnet in motion (or more generally, a changing magnetic field) can exert a force on the stationary charge. This is the fundamental significance of the Faraday’s discovery. Electricity and magnetism are related.