Week 8

Electromangetic Induction

Recall that when a wire loop is placed in a magnetic field that it will feel a torque,

tau = mu(m) x B if a current I is run through it. Here mu(m) = IA in the direction normal to the face of the loop (as determined by the right hand rule).

Due to the torque, the loop spins in the magnetic field. This is the basis for electrical motors.

[show motor] We supply energy in the form of a current. The energy in the current is converted into mechanical energy, the spinning of the loop. The spinning motion can then be used to perform useful tasks, drills, slot cars, fans, ... . The natural question to ask is whether the reverse of this process is also true: If I supply mechanical energy to a loop (by spinning it using wind power, water power, ... ), will this energy be converted into current? That is, can I generate electricity in this manner? We investigate this question in the next few days.

Faraday's Induction Law

We see that the generator produces an EMF ===> there is a transfer of energy between the mechanical motion and the electromagnetic fields.

[show generator] How does this work?

[demonstrate coil, magnet, and voltmeter]

If I hold the magnet stationary, there is no induced EMF. If, however, I move the magnet field toward the loops, an EMF is induced! Depending upon which pole I point toward the loop, the polarity of the EMF will be different. The key is that when I hold the magnet stationary, the magnetic which threads the loop is steady, i.e, dB/dt = 0. When I move the magnet toward or away from the loop, an EMF is generated, i.e., dB/dt nonzero ===> induced EMF!

However, is this all that is happening? No, consider two loops with different sizes. For the same dB/dt, the larger loop (larger A) will have a larger induced EMF.

Phi(m) = B dot A = BA cosine (theta)

where theta is the angle made by the B-field and the normal to the face of the surface. The units of the magnetic flux are known as Webers (Wb = T m^2). Given the definition of the magnetic flux, it has been shown that the induced EMF is given by

dB/dt is nonzero
the area A can change either through a physical change in the area or by a change in theta, the orientation of the field and the loop

Why the "minus" sign?

Well, what happens is that the current (EMF) is induced in the sense that it opposes the change in the magnetic flux. Huh? Let's see.

Lenz Law

What happens as the field is increased? The flux threading the loop increases in strength. Lenz's Law says that a current is induced in the loop which generates a B-field which opposes the change in the B-field. So, since the original field points to the left and increases, the current is generated in the sense that it produces a field which points to the right. By the right hand rule, this means that the current must flow in the CCW-sense as viewed from the right. If the field is made smaller, the opposite occurs in that a current is induced to compensate for the weaker field. In this case, the current is induced to flow in the CW-sense! Lenz's Law is a consequence of energy conservation.

Examples

Phi(m) = 1.2x10^(-3) T x 25/10^4 m^2 = 3 x 19^(-6) Wb

22. A flat circular coil of 10 turns and diameter 10 cm is located in an uniform magnetic field of 0.20 T that passses through the coil area at an angle of 45 degrees. What will be the voltage, on average, across the terminals if the plane of the coil is smoothly rotated in a time 0.10 s? Indicate which terminal has the higher potential. What will the voltage be when the coil comes to a stop?

Flux:
- Beginning: Phi1 = 0.20 T x 10 x pi x (0.05 m)^2 x cos 45 = 0.011 Wb
- End: Phi2 = 0.0
- delta Phi/delta t = - 0.11 Wb/s
The induced EMF on average is then EMF = 0.11 Volts. The flux, initially, was positive, upward (by assumption). Since the change made the flux go to zero, the current must be induced in the sense that it will generate a flux in the same sense, i.e., a current will flow in the CCW-sense as viewed from the right. When the coil comes to a stop, the voltage will go to zero.