### Unit 5: Electromagnetic Induction

You learned that stationary electric charges produce electric field, and moving electric charges (that is, electric current) produce magnetic field. In this unit, you will find out that the reverse is also true: changing magnetic flux produces electric field, or induces electric current. This is the phenomenon of the electromagnetic induction, which is a basic principles in such devices as generators of electric power, electric motors, and transformers.

**Completing this unit should take you approximately 13 hours.**

Upon successful completion of this unit, you will be able to:

- state Faraday's and Lenz's laws;
- solve problems using Faraday's law;
- define inductance and explain how it affects the change of current in a circuit;
- analyze RC, RL, and RCL circuits; and
- compare and contrast electromagnetic generators and motors.

### 5.1: Faraday's Law

Watch this lecture, pausing to take notes, before moving on to the reading below. Please test your understanding of this lecture by attempting the problems #3, #11 and #12 from this problem set. Check your solutions here.

Read the following sections: "Faraday's Law," "Lenz's Law," "Magnetic Induction," "Motional EMF," and "Eddy Currents." Also, work through Examples 9.1-9.3 before looking at the solutions. Make sure you understand not only the solutions but also how to approach solving the problems so that you can obtain the solutions yourself. You will be responsible for being able to solve problems of this type on the final exam.

This demonstration illustrates the relationship between magnetic flux and size and orientation of the surface through which the magnetic field is passing. Note that the relationship is particularly easy to observe when constant magnetic fields parallel to the field lines can be seen. Here, a constant flux is simply a constant number of field lines penetrating the wire loop. Although the relationship is more difficult to observe when variable magnetic fields are used in the demonstration, the relationship is still the same: magnetic flux is the number of field lines penetrating the wire loop. Pick a value for the radius of the loop and calculate the magnetic flux through the loop in the two cases when the field is uniform: (0,0, 1) and (1, 1, 1), Do your results agree with the demonstration?

Many machines use magnetic brakes from the small to the gigantic. The analysis of the effect is simple in some ways and quite tricky in others. There are two ways of looking at magnetic braking, both of which are mentioned in the demonstration write-up. The basic idea is that when a conductor moves through a magnetic field, currents are induced that resist the motion. These are called Eddy currents, and the reduction in the kinetic energy of the conductor is equal to the resistive heating caused in the conductor by the induced Eddy currents. Another approach to explaining magnetic braking is that the Lorentz force acting on the electrons in the moving conductor acts to move the electrons outward, and the Lorentz force associated with that outward motion in the applied magnetic field serves to slow down the moving conductor. These two ways of thinking about magnetic braking are equivalent; that is, they make the same predictions. A couple of questions for reflection: If the conductor is a perfect conductor (no resistance, but not a superconductor), is there any braking effect? Also, can a magnetic brake by itself bring a moving conductor to a complete stop? Why, or why not?

Use this worksheet as a guide to explore the demonstration.

This demonstration is modeled after a classic classroom demonstration. A conducting ring is placed atop an electromagnet, and a large pulse of current passes through the electromagnet. As seen in the lectures, readings, and demonstrations above, a current is induced in the conducting ring in a direction that opposes the formation of the magnetic field. As usual, this is to minimize the total energy of the system. The result is that the current in the ring generates a magnetic field with the opposite sign as that of the electromagnet. Opposed magnetic fields repel, so the ring launches into the air. Why does nothing happen when we use the split ring?

### 5.2: Inductance

Read these sections: "Mutual Inductance," "Self-Inductance," and "Energy Stored in an Inductor." Work through the Examples 10.1 and 10.2 before looking at the solutions. Make sure you understand not only the solutions but also how to approach solving the problems so that you can obtain the solutions yourself. You will be responsible for being able to solve problems of this type on the final exam.

This demonstration provides examples for both capacitors and inductors, but work only with the inductors for now. Treat this demonstration as a laboratory experiment in which you measure the inductance of various geometries and confirm from theory that the values are correct. Get the electromagnetic field energy generated by applied current from the demonstration, and confirm them by direct calculation based on the theoretical relations listed here and in the lectures and readings above. First, calculate the inductance of the coil, using the formula given in the description in the demonstration, or the readings above. Then, calculate current through the inductor using Ohm’s Law, and magnetic field created by this current. Finally, calculate the magnetic energy stored in the inductor, and the magnetic energy density. Again, confirm your result.

You can also explore this demonstration conceptually, to get a feel for how the magnetic field inside the coil and magnetic energy density are affected by the geometry of the inductor, resistance, and applied voltage. Vary each of the parameters in turn and observe the changes in

*B*and*ρ*. Why does not the magnetic field depend on the radius of the coil?

### 5.3: RC, RL and RCL Circuits

Read this chapter carefully (on pages 713-727). Answer the Self-Check questions in the text (answers on page 1011). Think about the Discussion Questions and Examples, and work out problems #1-8 on the page 729.

This demonstration shows the graphs of the voltage and current in the circuit containing a resistor, and inductor, and a capacitor connected in series, driven by the alternating voltage. The frequency of the alternating voltage supplied by the battery will determine the phase shift

*φ*between the voltage and the current. Pick the values of the parameters and calculate the phase shift using the formula in the demonstration. Then, try to vary the frequency until you make the phase shift 0 (this might be difficult to accomplish, but try to make it as small as you can), and confirm that this frequency is close to the resonant frequency of the circuit.

### 5.4: Electromagnetic Generators and Motors

Watch this lecture series, pausing to take notes, before moving on to the reading below.

Read the following sections: "The Alternating Current Generator," "The Direct Current Generator," "The Alternating Current Motor," and "The Direct Current Motor." Work through Examples 9.4 and 9.5 before looking at the solutions. Make sure you understand not only the solutions but also how to approach solving the problems so that you can obtain the solutions yourself. You will be responsible for being able to solve problems of this type on the final exam.

This is a demonstration of one of the simplest electromagnetic motors ever invented. Before looking it up, figure out just how it works. (Hint: Lorentz force.)

The Faraday Disk is a simple electric generator. Before looking it up, figure out how it works.

### Unit 5 Assessment

Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment