### Unit 5: Electromagnetic Induction

We have learned that **stationary** electric charges produce an **electric** field, and that **moving** electric charges (electric current) produce a **magnetic** field. In this unit, we discover that the reverse is also true: changing magnetic fields can produce an electric field, or induce an electric current. This describes the phenomenon of **electromagnetic induction**, a basic principle that drives electric power generators and transformers.

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

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

- define flux;
- state Faraday's and Lenz's Laws;
- solve problems using Faraday's Law;
- define inductance and 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

At this point, our discussion of magnetism has been restricted to magnets, currents and wires that do not change over time. Yes, currents involve moving charges, but

**direct currents (DC)**produce constant magnetic fields. Now we will be concerned with magnetic fields that**change over time**. This can be caused because the source of the magnetic field is itself in motion; or because the strength of that source is changing, which happens in a current-carrying wire if you change the current (for example by connecting it to an AC power outlet).**Faraday's Law of Induction**(or**Faraday's Law**), named after Michael Faraday (1791–1867) an English scientist, refers to the basic law of electromagnetism that predicts how a time-varying magnetic field interacts with an electric circuit to produce an electromotive force (EMF). This phenomenon is known as electromagnetic induction. Magnetic induction provides the foundation for electric motors, generators, transformers, and the electric power grid.**Lenz's Law**, named after the Russian physicist Heinrich Lenz (1804–1865), states that introducing a conductor within a magnetic field will produce electricity, inducing an opposing magnetic field that repels the magnetic field producing the charge. Lenz's Law is a consequence of the conservation of energy.Let's begin by reading this introductory text.

An important quantity defined here is the magnetic flux. We can visualize magnetic flux as a measure of how many magnetic field lines intersect a given surface (for example the cross-sectional area enclosed by a wire loop). Think of a sail catching the wind, where the wind is analogous to the magnetic field. You can catch a lot of wind by turning the sail perpendicular to the wind, or you can catch almost no wind by turning the sail parallel to the wind. The amount of wind you catch with the surface area of the sail is analogous to the magnetic flux going through that area.

Watch this brief explanation of magnetic flux. In the last part, it mentions two new mathematical constructs, the

**dot product**and**integral**, which is**optional information**. Feel free to stop the video at timestamp 3:23.Next, watch this lecture demonstration by Walter Lewin until timestamp 13.30.

Watch this lecture, which puts the magnetic flux into the context of magnetic induction.

Read this text, which reviews the material from the video you just watched.

Faraday's Law always involves

**loops**that enclose a certain magnetic flux. If the loop is a conductor, then a changing magnetic flux creates a current inside the loop – even though there is no battery at all, and even if the wire has some resistance. This sounds contradictory because Ohm's Law says that you cannot have a current without a voltage; how can you have a voltage if there is no battery?The changing magnetic flux itself acts like a battery, and the voltage it generates is therefore also called electromotive force (emf), just as for "

**regular**" batteries. In other words, we have got a new type of power source for our electric circuits: the changing magnetic field is able to do work on the electrons inside the conducting loop, and if that loop contains a light bulb, you could make it light up, too. That is a transfer of energy which must come from somewhere.From the electron's point of view, that energy comes

**from the magnetic field**. But ultimately, work must have been done somewhere else to create the changing magnetic field in the first place. The law of energy conservation is never violated in this process.Watch this video to explore different factors in Faraday's Law – the law that tells us how large the generated emf will be, depending on the rate at which the magnetic flux changes.

### 5.2: Motional Emf

Earlier, we saw that when you move a bar magnet into and out of a conducting loop, an emf is generated inside the loop that will give rise to a circulating current. Another point of view can explain this effect as a consequence of the Lorentz force: just look at things from the perspective of the bar magnet! Such a change of reference frame is always permitted in physics if we assume the bar magnet is approaching at constant velocity.

An electron sitting idly inside the conducting loop will look to the bar magnet as if the electron is approaching, when in actual fact the magnet is approaching the loop. That is, in the frame of reference of the bar magnet the electron represents a moving charge with a velocity that intersects the magnetic field lines of the magnet at some angle. A moving charge will be deflected according to the Lorentz force law. And if this deflection has a component tangential to the loop it will create a current.

The lesson from this change of viewpoint is that induction is all about

**relative**motion. It could be that field lines through a loop are changing because a bar magnet is being moved, or because the loop is being moved. Faraday's Law of induction offers a way to**unify**many of these effects, giving a single equation that can explain multiple different phenomena where voltages are generated.One of these phenomena is the Hall effect which we discussed earlier. Read this text, which discusses how we can view the Hall effect as a manifestation of magnetic induction. More generally, the text studies motion in a magnetic field, which is stationary relative to the earth, producing what we loosely call motional emf.

The paradigm of motional emf is illustrated for a conducting rod rolling on two rails connected to a battery. Watch this video, which explores this scenario further.

Here is a sanity check for the formula used above: it is good to remind ourselves that an emf is just a voltage, so it must also have the unit of electric potential, or voltage. If you recall that potential is just potential energy per unit charge, we can figure out that a volt is the same as one joule per coulomb. Watch this video to see how this unit arises from the formula for the motional emf in terms of the magnetic field.

Watch this video, which discusses how to get started on a problem that involves an airplane flying through the Earth's magnetic field. Note that one of the slides accidentally reads "hand hand rule" when it should say "right-hand rule".

To conclude, we should address the question of how fundamental Faraday's Law of induction really is. After all, many of the examples you have seen appear to suggest that you can always find an "

**alternative**" explanation that does not rely on Faraday's Law directly, but instead on the Lorentz force law for the deflection of moving charges in a magnetic field.The answer is that both laws are essential, and you cannot always replace one by the other.

The Lorentz Law is indispensable because it explains how magnetic fields interact with charged objects, creating an actual force. Faraday's Law does not talk about forces at all, only about emf (which is

**not actually a force**despite its name). In a circuit, the electromotive force is a potential difference which is always accompanied by an**electric field**. But fields are**would-be forces**, not actual forces. This means that Faraday's Law would predict the appearance of electric fields even in loops that consist purely of insulating material without any movable charges!This may sound like a rather useless prediction, but it is not. We will see later that electromagnetic waves can propagate through air (and even vacuum) precisely because of this ability of a changing magnetic flux to induce loops of electric field even when there are no currents anywhere in sight. This is why Faraday's Law is one of the fundamental laws of electrodynamics.

### 5.3: Eddy Currents

An important part of applying Faraday's Law of induction is to find the correct direction of the emf, and hence the induced current in a conducting loop. To help us with this, we look to a useful rule that only concerns the directions, but not the magnitude of the induction:

**Lenz's Law**. It is based on the fact that any circulating current is itself the source of a magnetic field.Lenz's Law says that when a current is induced by a changing magnetic flux, the direction in which that current circulates is always such that the magnetic field generated by that circulation itself opposes the change in magnetic flux that is being imposed on the loop.

That is, if the magnetic flux happens to be increasing, the induced current will generate a magnetic field that by itself would create a negative flux. If the magnetic flux happens to be decreasing, the induced current will be directed just right so that it produces a positive magnetic flux.

If you care to anthropomorphize the situation, Lenz's Law makes induced currents behave "stubbornly" – always opposing what is being done to them. If you prefer a physics analogy, the behavior has a lot in common with friction forces. Read this text, which explains how this effect is being applied in everyday life.

Many machines use magnetic brakes from the small to the gigantic. The basic idea is that when a conductor moves through a magnetic field, currents are induced that resist the motion. We call these

**eddy currents**. 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 tends 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?

### 5.4: Electric Generators

Electric generators transform mechanical energy into electrical energy, typically by electromagnetic induction via Faraday's Law. For example, a generator may consist of a gasoline engine, attached to a system of coils and/or magnets, which turns a crankshaft.

Read this text, which explains how the rotation of a loop (or coil) in a magnetic field causes the magnetic flux through the cross section to vary periodically, and that is what gives rise to an emf (voltage) which periodically changes direction. In other words, the "

**natural**" output of a generator is an**AC voltage**.Watch this animation of a generator. Note that you can run the interactive simulation in this video yourself if you have a desktop computer. Go to https://phet.colorado.edu/en/simulation/generator.

The reason why electromagnetic generators produce AC voltage is that the turning wire loop has the magnetic flux going through its cross section in opposite directions every half turn.

Using the concept of induction, we can also answer a question you may have had when we discussed motors and electromagnets: if you connect a

**solenoid**(or**coil**) directly to a battery, isn't that just a short circuit that will drain the battery really quickly? For an electromagnet operating with DC current, that is indeed a problem, so we have to limit the current it can draw from the battery (e.g. using resistors). But in a motor, the coil does not act the same way. It does not drain the battery even if connected directly to it.The difference is that the coil in the motor does not see a supply of continuous direct current while it is turning. To keep the rotation going, the current through the coil is reversed every half turn, so there is, in fact, an alternating current in the rotating coil. Now we know that any current produces a magnetic field, and that makes the motor into an electromagnet.

However, unlike the DC electromagnet, we are now dealing with a magnet whose magnetic field is constantly changing. And according to Faraday's Law, this changing field will in turn cause magnetic induction – leading to an emf. But what is the loop in which this emf will appear? It is the same coil that produced the changing magnetic field in the first place. So any electromagnet whose current is changing will produce an emf inside of itself!

Read this text, which explains how Lenz's Law dictates the direction of the induced emf and always opposes the voltage applied to the coil to get the current going in the first place. What does this means for a motor coil? As soon as it starts rotating, it will not act like a short circuit anymore because it will oppose the varying current flowing through it.

Watch this lecture, which accompanies our textbook. We will explore the other topics of this lecture in more detail next.

### 5.5: Transformers

As their name implies, transformers change voltages from one value to another – we use the term voltage rather than emf, because transformers have internal resistance. Many cell phones, laptops, video games, and power tools and small appliances have a transformer built into their plug-in unit that transforms 120 V or 240 V AC into the voltage the device uses.

Engineers use transformers at several points in the power distribution systems. Power is sent for long distances at high voltages, because less current is required for a given amount of power, and this means less line loss. Since high voltages can be quite dangerous, we use transformers to produce lower voltage at the user's location.

What makes transformers different from motors and generators is the forms of energy that are involved.

**Generators**transform mechanical to electrical energy; motors transform electrical to mechanical energy – and**transformers**transform electrical to electrical energy.You may say that this is not really an energy transformation at all. But observe that the two coils of a transformer make up parts of two completely separate electric circuits, and still energy is able to transfer from one circuit to the other. This happens via an intermediate form of energy, stored in the magnetic field lines that permeate both coils simultaneously.

### 5.6: Electrical Safety

Let's take a moment to review two hazards of electricity. A

**thermal hazard**occurs during electrical overheating. A**shock hazard**occurs when electric current passes through a person.Read this text to review some systems and devices that prevent electrical hazards.

### 5.7: Inductance

Induction refers to when changing magnetic flux induces an emf. For example, transformers induce a desired voltage and current with little loss of energy. Inductance (or more specifically self-inductance) refers to the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor, and when the magnetic field changes there will be an opposing induced emf according to Faraday's Law. The back emf of a motor is an example of this effect.

**Mutual inductance**refers to the effect Faraday's Law of induction has from one device on another, such as the energy a primary coil transmits to the secondary coil in a transformer.Read this text, which explains how when we look at a coil as a circuit element, we can eliminate the magnetic field from Faraday's Law by using the fact that the magnetic field is always proportional to the current that created it in the first place. The result is that the emf across the coil is given by . Here, is the change in the current during time , and is a proportionality constant called the

**inductance**.With the concept of inductance, we have turned coils into a circuit element by hiding all the details of the coil's geometry and magnetic properties in the single quantity , the inductance. This is analogous to how we turned capacitors into circuit elements by hiding their details in the

**capacitance (C)**. Knowing these characteristic constants is sufficient to calculate how**inductors**(our new name for coils) and capacitors behave in**electrical circuits**.

### 5.8: RC, RL, and RLC Circuits

An

**RC circuit**is a series connection of resistance and capacitance. This circuit stores energy in the form of an electric field. An**RL circuit**is a series combination of resistance and inductance which stores energy in the form of magnetic energy. An**RLC circuit**is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel.In our discussion of capacitors, we encountered the exponential function when we described how the current and voltage behave during the process of charging and discharging. Read this text where we encounter the exponential function once again – this time in the process of turning the current through an inductor coil on and off.

We see that the current through an inductor always shows a kind of "

**sluggishness**", in the sense that it delays the onset of a current when it is turned on, and perhaps even more surprisingly it also delays the turning-off of a current. This sluggishness is characterized by a**time constant**which is directly proportional to the inductance. You can loosely understand the inductance as an electronic equivalent to**inertia**, if you take electric current to be analogous to velocity and electric voltage to be analogous to mechanical force.Keep this analogy in mind when you read this section, which explores how inductors and capacitors respond when they are placed in an

**AC circuit**. First, an inductor is connected to a voltage that varies in time according to a**sine function**. This is just the type of voltage that would be produced by a generator. Measuring the current through the inductor coil, you find that it does not reach its maximum value when the voltage is at its maximum. Instead, there is a delay caused by the electronic inertia mentioned above, which is to say –**the inductance**.In the second part of the text, the same analysis is shown for a capacitor. In this case, the roles of voltage and current are

**reversed**. The current periodically reaches a**maximum**, but the voltage across the capacitor reaches its maximum with a**delay**. The reason is that a large voltage can only build up when the capacitor has accumulated a large stored charge. But the accumulation of charge must be**preceded**by a charging current. So we**first**need a large current and**then**get a large voltage.In addition to the delay between current and voltage, there is also the question of how

**large**the**peaks**of current and voltage become, whenever they happen. Read this text, which introduces a new concept to characterize the relative heights of the current and voltage peaks, called the**reactance**.It is the generalization of the concept of resistance, in that the reactance is defined as the ratio of voltage to current – but because both are variable we must take their rms values. Recall that rms stands for

**root-mean-square**and represents an average of a time-varying quantity. In fact, you can equivalently get the reactance by**dividing**the**peak voltage**by the**peak current**.The reactance of a capacitor depends on the capacitance, and the reactance of an inductor depends on the inductance. Both also depend on the

**frequency**of the**sinusoidally**-varying voltage. Because inductors act like they have inertia, they will produce less peak current for an input voltage that changes direction rapidly, making the ratio of voltage to current (i.e. their reactance) larger for high frequencies.Capacitors behave the other way around: their reactance becomes smaller at high frequencies. This is because when the charging current changes direction rapidly, the capacitor never gets enough time to charge fully, which never allows the voltage to grow to large values. As a result, the ratio of voltage to current is small.

Even though reactances have the same units as the familiar resistance we know from Ohm's Law, there is one big difference: it is the time delay between voltage and current that we see (in different directions) for inductors and capacitors – but there is no such delay for "

**normal**" resistors.In the text below we consider the most complex circuit of this course: a combination of resistor, capacitor and inductor in

**series**. The circuit diagram does not look complex, but the difficulties are hidden in the delay between current and voltage. Because that delay is affected in opposite ways by capacitors and inductors, and not at all by a resistor, the combination of all three produces a delay that depends on the parameters L, R and C.Fortunately, when such a circuit is driven by a sinusoidally-varying voltage, it behaves a lot like a

**mechanical oscillator**driven by a harmonically-varying force. In particular, the current and voltages will all respond by oscillating at the same frequency as the driving voltage. And just as we saw for a driven oscillator, there is a specific driving frequency at which the circuit responds particularly strongly.To understand what is going on, you can think about the AC voltage supply as the

**analogue**of a mechanical driving force and the peak current through the circuit as the analogue of the maximum speed of the oscillator. Also remember that the inductor acts as if it gives inertia to the current that is trying to flow.On the other hand, the capacitor acts analogously to a

**spring**. This is because whenever the current through the capacitor stops, it is done charging and has therefore reached its maximum voltage across the plates. This voltage is analogous to the restoring force of a spring. And if you imagine when an oscillating spring reaches its maximum restoring force, that is precisely when it turns around so that its speed is momentarily zero. Zero speed is the analogue of a stopped current.With these mechanical analogies, it should come as no surprise that a circuit consisting of capacitor and inductor will exhibit the phenomenon of

**resonance**. This is indeed the case: when the**frequency**of the AC voltage supply hits an**optimal value**, the circuit will show a current oscillating back and forth with especially large**amplitude**.Read this text, which comes to this conclusion using another concept which generalizes the idea of resistance even further (and has the same units):

**impedance**. It is still defined as the ratio of peak voltage to peak current, just allowing for arbitrary delays between when those peaks occur. The resonance phenomenon is identified by an impedance that becomes very small (or zero). This corresponds to very large currents in relation to the voltage.