6.6. MOTIONAL ELECTROMOTIVE FORCE

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.

In this section, we start by visualizing a straight conductor (a metal rod) that is moving within a magnetic field. Imagine the rod named PQ moving towards the left while a magnetic field, indicated by B, is acting perpendicular to the movement. This setup creates a closed circuit, which changes its area as the rod moves. When the area of a circuit changes in a magnetic field, it leads to the generation of electromotive force (emf), known as motional emf.

Think about riding a bicycle with a small generator attached to the wheel. As you pedal and the wheel spins, the movement causes a conductor to move through the magnetic field inside the generator, producing electricity just like the movement of the rod PQ creates an emf. This principle is leveraged in things like dynamo systems used to power lights on bicycles.

If the length RQ = x and RS = l, the magnetic flux $\Phi$ enclosed by the loop PQRS will be $\Phi = B l x$. Since x is changing with time, the rate of change of flux $\Phi$ will induce an emf given by: $e = - \frac{d\Phi}{dt} = - \frac{d(Blx)}{dt} = -Blv.$

The flux through the loop changes as the rod moves, which we can express mathematically. The induced emf is based on this changing flux, and it's given by the equation $e = -Blv$. Here, B is the magnetic field strength, l is the length of the rod, and v is the velocity with which the rod moves. The minus sign indicates the direction of the induced current, in accordance with Lenz's Law, which states the induced current opposes the change in magnetic flux.

Imagine a water wheel that turns in a flowing river. As the wheel spins faster (analogous to the rod moving quickly through the magnetic field), it captures more energy from the water, leading to increased energy being produced. Similarly, the faster the rod moves, the greater the induced emf due to the changing area it sweeps in the magnetic field.

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.

The concept of Lorentz force comes into play when we look specifically at how individual electrical charges within the conductor respond to the magnetic field as the rod moves. Each charge experiences a force due to the magnetic field, which results in a separation of charges—leading to the induced emf. The work done to move these charges is related to the induced emf, as the induced potential is essentially the energy per charge produced in the circuit.

You can think of this like how pushing a child on a swing causes the swing to move. If you push harder (representing greater speed or more energy), the swing moves higher. Similarly, the more force exerted on the charges in the conductor due to the Lorentz force, the more emf is produced, like the energy exerted on that swing results in higher movement.

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 $F = q (E + v \times B) = qE$ since v = 0. Thus, any force on the charge must arise from the electric field term E alone.

Here we explore a different scenario where the conductor remains stationary, but the magnetic field it is situated in changes. When the conductor does not move, the electric field generated needs to be addressed. Faraday demonstrated that during a changing magnetic field, an electric field is produced that exerts a force on the charges within the stationary conductor, thereby leading to induced emf.

Think about a doorbell. When you push the button (the changing action), it creates an electrical signal that rings the bell (induced emf) even though the bell itself doesn’t move. Similarly, a changing magnetic field creates an 'electric push' on stationary charges, inducing an emf in the conductor.

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