Introduction:
So far, you have studied essentially two topics: 1) electric fields caused by stationary charges and 2) magnetic fields produced by moving charges. Now we look at electric fields arising from changing magnetic fields - induced currents.
Faraday's Law of Induction:
...the emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux...
Remember that you can find the magnetic flux by using an integral, although if the field is constant and perpendicular to the area, this becomes simply BA:
We are often dealing with the fields produced by circular loops and solenoids, so you should remember that you have formulas for these from the last mini-lecture and Chapter 30. If the flux due to just one loop is computed, and there are N loops, the formula for the emf becomes
The emf is really the voltage, so if there's a voltage induced across the coil and assuming a resistance R of the coil, the current in the coil will be given by Ohm's Law: V = IR. (You can work without N until the end and then multiply by N, or multiply by N from the beginning.)
You should first study Example 31.1 on page 910 and then my examples: Time-Variable Magnetic Flux and Flux Approximation.
Motors and Generators:
Figure 31.8 depicts a conducting bar forced to move with velocity v in a constant field B. In accordance with our convention, the x's represent the tails of the magnetic field vectors so that the field is into the page. The force is applied to the right and so the bar moves to the right. But there are charges (electrons and protons) in the bar; they are moving because the bar is moving -to the right. So these moving charges in a magnetic field experience a force according to
Doing the cross product (rotating the velocity into the field using your hand and noting that the thumb points up) we get a current that is up - from bottom to top. Serway shows that the current is caused by a voltage - an emf - that is given by
(I use an upper case L because the lower case Serway uses would get confused with the number one, and my fonts are limited.)
This isn't terribly interesting. After all, how many times do you move a bar across a couple of rails in a magnetic field? But the point here is that the flux is changing even though the field is not! The flux changes because the bar moves, and the area (which is part of the flux calculation) changes!
We can get the flux to change by doing something else. Look at Figure 31.18 on page 920. As the loop rotates, the dot product of the area vector (labeled "normal" in the figure) and the field varies! So here again we have a rate of change of the flux and thus a generated emf (or voltage). But now it's more useful because this is essentially a generator, if the loop is forced to rotate by an external force such as falling water. Conversely, if we apply a time-varying current to the loop (run AC through it), we can get motion out of it - a motor! If the forced rotation of N loops has a constant angular velocity omega, the emf generated is
You should study Example 31.10 on page 920 and my example: The Armature of a Generator.
In Figure 31.17 (page 919) Serway shows how the loop rotating at a constant angular speed produces a variable voltage. If DC is desired, a split ring commutator can be used as shown in Figure 31.19 (page 921). Doesn't look much like DC, does it? Actually, this is called "pulsed DC". The design of motors and generators gets much more complex and we can't get into it here, but multiple poles are used. Only the smallest motors have external fields caused by permanent magnets as shown; most motors and generators utilize electromagnets.
Lenz's Law:
Essentially, Lenz's Law states that "the result opposes the cause". Serway says it a little more precisely at the bottom of page 914. Lenz's law is a quick way to determine the direction of an induced current or field.
Look at Figure 31.12 (page 915). The cause is the bar magnet, moved to the right through the coil. The effect is the current induced in the coil. Here's another use of the right hand rule: "Curl your fingers so that they curl around the coil in the direction of the (conventional) current. Your thumb will give the direction of the field. Notice that the field opposes the magnet - repels it? (It's not strong enough to force equilibrium, though.) Can you see that, if the magnet were moved in the opposite direction, the current would reverse (to try to keep the magnet from moving away).
There are two more possibilities: reverse the magnet so that the south pole is on the right and 1) move it to the right or 2) move it to the left. Can you come up with the current in each case?
You should study Example 31.7 on page 916.
Here's a diagrammed explanation of how a generator works.
