Self-inductance, mutual inductance, and the power and energy of inductors

As Feynman puts it, studying physics is not always about the ‘the great and esoteric heights’. In fact, you usually have to come down from then fairly quickly – studying physics is similar to mountain climbing in that regard 🙂 – and study ‘relatively low-level subjects’, such as electrical circuits, which is what we’ll do in this and the next post.

As I’ve introduced some key concepts in a previous post already, let me recapitulate the basics, which include the concept of the electromotive force, which is basically the voltage, i.e. the potential difference, that’s produced in a loop or coil of wire as the magnetic flux changes. I also talked about the impedance in a AC circuit. Finally, we also discussed the power and energies involved. Important results from this previous discussion include (but are not limited to):

  1. A constant speed AC generator will create an alternating current with the emf, i.e. the voltage, varying as V0·sin(ωt).
  2. If we only have resistors as circuit elements, and the resistance in the circuit adds up to R, then the electric current in the circuit will be equal to I = Ɛ/R = V/R = (V0/R)·sin(ωt). So that’s Ohm’s Law, basically.
  3. The power that’s produced and consumed in an AC circuit is the product of the voltage and the current, so P = Ɛ·I = V·I. We also showed this electrical power is equal to the mechanical power dW/dt that makes the generator run.
  4. Finally, we explained the concept of impedance (denoted by Z) using Euler’s formula: Z = |Z|eiθ, mentioning that, if other circuit elements than resistors are involved, such as inductors, then it’s quite likely that the current signal will lag the voltage signal, with the phase factor θ telling us by how much.

It’s now time to introduce those ‘other’ circuit elements. So let’s start with inductors here, and the concept of inductance itself. There’s a lot of stuff to them, and so let’s go over it point by point.

The concept of self-inductance

In its simplest form, an inductor is just a coil, but they come in all sizes and shapes. If you want to see how they might look like, just google some images of micro-electronic inductors, and then, just to see the range of applications, some images of inductors used for large-scale industrial applications. If you do so, you’re likely to see images of transformers too, because transformers work on the principle of mutual inductance, and so they involve two coils, i.e. two inductors.

Contrary to what you might expect, the concept of self-inductance (or inductance tout court) is quite simply: a changing current will cause a changing magnetic field and, hence, some emf. Now, it turns out that the induced emf is proportional to the change in current. So we’ve got another constant of proportionality here, so it’s like how we defined resistance, or capacitance. So, in many ways, the inductance is just another proportionality coefficient. If we denote it by L – the symbol is said to honor the Russian phyicist Heinrich Lenz, whom you know from Lenz’ Law – then we define it as:

L = −Ɛ/(dI/dt)

The dI/dt factor is, obviously, the time rate of change of the current, and the negative sign indicates that the emf opposes the change in current, so it will tend to cause an opposing current. That’s why the emf involved is often referred to as a ‘back emf’. So that’s Lenz’ Law basically. As you might expect, the physicists came up with yet another derived unit, the Henry, to honor yet another physicist, Joseph Henry, an American scientist who was a contemporary of Michael Faraday and independently discovered pretty much the same as Faraday: one henry (H) equals one volt·second per ampere: 1 H = 1 V·s/A.

The concept of mutual inductance

Feynman introduces the topic of inductance with a two-coil set-up, as shown below, noting that a current in coil 1 will induce some emf in coil 2, which he denotes by M12. Conversely, a current in coil 2 will induce some emf in coil 1, which he denoted by M21. M12 and M21 are also constants: they depend on the geometry of the coils, including the length of the solenoid (l), its surface area (S) and the number of loop turns of the coils (N1 and N2).

mutual inductance

The next step in the analysis is then to acknowledge that each coil should also produce a ‘back emf’ in itself, which we can denote by M11 and M22 respectively, but then these constants are, of course, equal to the self-inductance of the coils so, taking into account the convention in regard to the sign of the self-inductance, we write:

L1 = − M11 and L1 = − M22

You will now wonder: what’s the total emf in each coil, taking into account that we do not only have mutual inductance but also self-inductance? Frankly, when I was a kid, and my father tried to tell me one or two things about this, it confused me very much. I could not imagine what happened in one coil, let alone in two coils. I had this vision of a current producing some ‘back-current’, and then the ‘back-current’ producing ‘back-current’ again, and so I could not imagine how one could solve this problem. So the image in my head was very much like that baking powder box which Feynman evokes when talking about the method of images to find the electric fields in situations with an easy geometry, so that’s the picture of a baking powder box which has on its label a picture of a baking powder box which has… Well… Etcetera. Unfortunately, my father didn’t push us to study math and, therefore, I knew that one could solve such problems mathematically – we’re talking a converging series here – but I did not know how, and that’s why I found it all very confusing.

Now I understand there’s one current only, and one potential difference only, and that the formulas do not involve some infinite series of terms. But… Well… I am not ashamed to say these problems are still testing the (limited) agility of my mind. The first thing to ‘get’ is that we’re talking a back emf, and so that’s not a current but a potential difference. In fact, as I explained in my post on the electromotive force, the term ‘force’ in emf is actually misleading, and may lead to that same erroneous vision that I had as a kid: forces generating counter-forces, that generate counter-forces, that generate counter-forces, etcetera. It’s not like that: we have some current – one current – in a coil and we’ll have some voltage – one voltage – across the coil. If the coil would be a resistor instead of a coil, we’d find that the ratio of this voltage and the current would be some constant R = V/I. Now here we’re talking a coil indeed, so that’s a different circuit element, and we find some other ratio, L = −V/(dI/dt) = −Ɛ/(dI/dt). Why the minus sign? Well… As said, the induced emf will be such that it will tend to counter the current, and current flows from positive to negative as per our convention.

But… Yes? So how does it work when we put this coil in some circuit, and how does the resistance of the inductor come into play? Relax. We’ve just been talking ideal circuit elements so far, and we’ve discussed only two: the resistor and the inductor. We’ll talk about voltage sources (or generators) and capacitors too, and then we’ll link all of these ideal circuit elements. In short, we’ll analyze some real-life electrical circuit soon, but first you need to understand the basics. Let me just note that an ideal inductor appears a zero-resistance conductor in a direct current (DC) circuit, so it’s a short-circuit really! Please try to mentally separate out those ‘ideal’ circuit components. Otherwise you’ll never be able to make sense of it all!

In fact, there’s a good reason why Feynman starts with explaining mutual inductance before discussing a little circuit like the one below, which has an inductor and a voltage source. The two-coil situation above is effectively easier to understand, although it may not look like that at first. So let’s analyze that two-coil situation in more detail first. In other words, let me try to understand the situation that I didn’t understand as a kid. 🙂

circuit with coil

Because of the law of superposition, we should add fluxes and changes in fluxes and, hence, we should also add the electromotive forces, i.e. the induced voltages. So, what we have here is that the total emf in coil 2 should be written as:

Ɛ2 = M21·(dI1/dt) + M22·(dI2/dt) = M21·(dI1/dt) – L2·(dI2/dt)

What we’re saying here is that the emf, i.e. the voltage across the coil, will indeed depend on the change in current in the other coil, but also on the change in current of the coil itself. Likewise, the total emf in coil 1 should be written as:

Ɛ1 = M12·(dI2/dt) + M11·(dI1/dt) = M12·(dI2/dt) – L1·(dI1/dt)

Of course, this does reduce to the simple L = −Ɛ/(dI/dt) if there’s one coil only. But so you see where it comes from and, while we do not have some infinite series 🙂 we do have a system of two equations here, and so let me say one or two things about it.

The first thing to note is that it is not so difficult to show that M21 is equal to M12, so we can simplify and write that M21 = M12 = M. Now, while I said ‘not so difficult’, I didn’t mean it’s easy and, because I don’t want this post to become too long, I’ll refer you to Feynman for the proof of this M21 = M12 = M equation. It’s a general proof for any two coils or ‘circuits’ of arbitrary shape and it’s really worth the read. However, I have to move on.

The second thing to note is that this coefficient M, which is referred to as the mutual inductance now (so singular instead of plural) depends on the ‘circuit geometry’ indeed. For a simple solenoid, Feynman calculates it as

M = −(1/ε0c2)·(N1·N2)·S/l,

with the length of the solenoid, S its surface area (S), and N1 and Nthe respective number of loop turns of the two coils. So, yes, only ‘geometry’ comes into play. [Note that’s quite obvious from the formula because a switch of the subscripts of Nand Nmakes no difference, of course!]. Now, it’s interesting to note that M is the same for, let’s say, N= 100 and N2 = 10 and for N= 20 and N2 = 50. In fact, because you’re familiar with what transformers do, i.e. transforming voltages, you may think that’s counter-intuitive. It’s not. The thing with the number of coils does not imply that Ɛ1 and Ɛ2 remain the same. Our set of equations is Ɛ1 = M·(dI2/dt) – L1·(dI1/dt) and Ɛ= M·(dI1/dt) – L2·(dI2/dt), and so Land Lclearly do vary as Nand N2 vary! So… Well… Yes. We’ve got a set of two equations with two independent variables (Iand I2) and two dependent variables (Ɛ1 and Ɛ2). Of course, we could also phrase the problem the other way around: given two voltages, what are the currents? 🙂

Of course, that makes us think of the power that goes in and out of a transformer. Indeed, you’ll remember that power is voltage times current. So what’s going on here in regard to that?

Well… There’s a thing with transformers, or with two-coil systems like this in general, that is referred to as coupling. The geometry of the situation will determine how much flux from one coil is linked with the flux of the other coil. If most, or all of it, is linked, we say the two coils are ‘tightly coupled’ or, in the limit, that they are fully coupled. There’s a measure for that, and it’s called the coefficient of coupling. Let’s first explore that concept of power once more.

Inductance, energy and electric power

It’s easy to see that we need electric power to get some current going. Now, as we pointed out in our previous post, the power is equal to the voltage times the current. It’s also equal, of course, to the amount of work done per second, i.e. the time rate of change of the energy W, so we write:

dW/dt = Ɛ·I

Now, we defined the self-inductance as L = −Ɛ/(dI/dt) and, therefore, we know that Ɛ = −L·(dI/dt), so we have:

dW/dt = −L·I·(dI/dt)

What is this? A differential equation? Yes and no. We’ve got not one but two functions of time here, W and I, and, while their derivatives with respect to time do appear in the equation, what we need to do is just integrate the two sides over time. We get: W = −(1/2)·L·I2. Just check it by taking the time derivative of both sides. Of course, we can add any constant, to both sides in fact, but that’s just a matter of choosing some reference point. We’ll chose our constant to be zero, and also think about the energy that’s stored in the coil, i.e. U, which we define as:

U = −W = −(1/2)·L·I

Huh? What’s going on here? Well… It’s not an easy discussion, but let’s try to make sense of it. We have some changing current in the coil here but, obviously, some kind of inertia also: the coil itself opposes the change in current through the ‘back emf’. It requires energy, or power, to overcome the inertia. We may think of applying some voltage to offset the ‘back emf’, so we may effectively think of that little circuit with an inductor and a voltage source. The voltage V we’d need to apply to offset the inertia would, obviously, be equal to the ‘back emf’, but with its sign reversed, so we have:

V = − Ɛ = L·(dI/dt)

Now, it helps to think of what a current really is: it’s about electric charges that are moving at some velocity v because some force is being applied to them. As in any system, the power that’s being delivered is the dot product of the force and the velocity vectors (that ensures we only take the tangential component of the force into account), so if we have moving charges, the power that is being delivered to the circuit is (F·v)·n. What is F? It’s obviously, qE, as the electric field is the force per unit charge, so E = F/q. But so we’re talking some circuit here and we need to think of the power being delivered to some infinitesimal element ds in the coil, and so that’s (F·v)·n·ds, which can be written as: (F·ds)·n·v. And then we integrate over the whole coil to find:power

Now, you may or may not remember that the emf (Ɛ) is actually defined as the line integral ∫ E·ds line, taken around the entire coil and, hence, noting that = F/q, and that the current I is equal to I = q·n·v, we got our power equation. Indeed, the integrand or kernel of our integral becomes F·n·v·d= q·E·n·v·d= I·E·ds. Hence, we get our power formula indeed: P = V·I, with V the potential difference, i.e. the voltage across the coil.

I am getting too much into the weeds here. The point is: we’ve got a full and complete analog to the concept of inertia in mechanics here: instead of some force F causing some mass m to change its velocity according to Newton’s Law, i.e. F = m·a = m·(dv/dt), we here have a potential difference V causing some current I to change according to the V = L·(dI/dt) law.

This is very confusing but, remember, the same equations must have the same solutions! So, in an electric circuit, the inductance is really like what the mass is in a mechanics. Now, in mechanics, we’ll say that our mass has some momentum p = m·v, and we’ll also say that its kinetic energy is equal to (1/2)m·v2. We can do the same for our circuit: potential energy is continuously being converted into kinetic energy which, for our inductor, we write as U = (1/2)·L·I2.

Just think about by playing with one of the many online graphing tools. The graph below, for example, assumes the current builds up to some maximum. As it reaches its maximum, the stored energy will also max out. Now, you should not worry about the units here, or the scale of the graphs. The assumption is that I builds up from 0 to 1, and that L = 1, so that makes U what it is. Using a different constant for L, and/or different units for I, will change the scale of U too, but not its general shape, and that shape gives you the general idea.

powerThe example above obviously assumes some direct current, so it’s a DC circuit: the current builds up, but then stabilizes at some maximum that we can find by applying Ohm’s Law to the resistance of the circuit: I = V/R. Resistance? But we were talking an ideal inductor? We are. If there’s no other resistance in the circuit, we’ll have a short-circuit, so the assumption is that we do have some resistance in the circuit and, therefore, we should also think of some energy loss to heat from the current in the resistance, but that’s not our worry here.

The illustration below is, perhaps, more interesting. Here we are, obviously, applying an alternating current, and so the current goes in one and then in the other direction, so I > 0, and then I < 0, etcetera. We’re assuming some nice sinusoidal curve for the current here (i.e. the blue curve), and so we get what we get for U (i.e. the red curve): the energy goes up and down between zero and some maximum amplitude that’s determined by the maximum current.

power 2

So, yes, it is, after all, quite intuitive: building up a current does require energy from some external source, which is used to overcome the ‘back emf’ in the inductor, and that energy is stored in the inductor itself. [If you still wonder why it’s stored in the inductor, think about the other question: where else would it be stored?] How is stored? Look at the graph and think: it’s stored as kinetic energy of the charges, obviously. That explains why the energy is zero when the current is zero, and why the energy maxes out when the current maxes out. So, yes, it all makes sense! 🙂

Let’s now get back to that coupling constant.

The coupling constant

We can apply our reasoning to two coils. Indeed, we know that Ɛ1 = M·(dI2/dt) – L1·(dI1/dt) and Ɛ= M·(dI1/dt) – L2·(dI2/dt). So the power in the two-coils system is dW/dt = Ɛ1·I1 + Ɛ2·I2, so we have:

dW/dt = M·I1(dI2/dt) – L1·I1·(dI1/dt) + M·I2·(dI1/dt) – L2·I2·(dI2/dt)

= – L1·I1·(dI1/dt) – L2·I2·(dI2/dt) + M·I1(dI2/dt)·I2·(dI1/dt)

Integrating both sides, and equating U with −W once more, yields:

U = (1/2)·L1·I1+ (1/2)·L2·I2+ M·I1·I2

[Again, you should just take the time derivative to verify this. If you don’t forget to apply the product rule for the M·I1·I2 term, you’ll see I am not writing too much nonsense here.] Now, there’s an interesting algebraic transformation of this expression, and an equally interesting explanation why we’d re-write the expression as we do. Let me copy it from Feynman so I’ll be using his fancier L and M symbols now. 🙂

explanation coupling constant

So what? Well… Taking into account that inequality above, we can write the relation between M and the self-inductances L1 and L2 using some constant k, which varies between 0 and 1 and which we’ll refer to as the coupling constant:

formula coupling constant 2

We refer to k as the coupling constant, for rather obvious reasons: if it’s near zero, the mutual inductance will be very small, and if it’s near one, then the coils are said to be ‘tightly coupled’, and the ‘mutual flux linkage’ is then maximized. As you can imagine, there’s a whole body of literature out there relating this coupling constant to the behavior of transformers or other circuits where mutual inductance plays a role.

The formula for self-inductance

We gave the formula for the mutual inductance of two coils that are arranged as one solenoid on top of the other (cf. the illustration I started with):

M = −(1/ε0c2)·(N1·N2)·S/l

It’s a very easy calculation, so let me quickly copy it from Feynman:

calculation solenoid

You’ll say: where is the M here? This is a formula for the emf! It is, but M is the constant of proportionality in front, remember? So there you go. 🙂

Now, you would think that getting a formula for the self-inductance L of some solenoid would be equally straightforward. It turns out that that is not the case. Feynman needs two full pages and… Well… By now, you should now how ‘dense’ his writing really is: if it weren’t so dense, you’d be reading Feynman yourself, rather than my ‘explanations’ of him. 🙂 So… Well.. If you want to see how it works, just click on the link here and scroll down to the last two pages of his exposé on self-inductance. I’ll limit myself to just jotting down the formula he does obtain when he’s through the whole argument:

solenoid formula

See why he uses a fancier L than ‘my’ L? ‘His’ L is the length of the solenoid. 🙂 And, yes, r is the radius of the coil and n the number of turns per unit length in the winding. Also note this formula is valid only if L >> R, so the effects at the end of the solenoid can be neglected. OK. Done. 🙂

Well… That’s it for today! I am sorry to say but the next post promises to be as boring as this one because… Well… It’s on electric circuits again. 😦

Induced currents

In my two previous posts, I presented all of the ingredients of the meal we’re going to cook now, most notably:

  1. The formula for the torque on a loop of a current in a magnetic field, and its energy: (i) τ = μ×B, and (ii) Umech = −μ·B.
  2. The Biot-Savart Law, which gives you the magnetic field that’s produced by wires carrying currents:

B formula 2

Both ingredients are, obviously, relevant to the design of an electromagnetic motor, i.e. an ‘engine that can do some work’, as Feynman calls it. 🙂 Its principle is illustrated below.

motor

The two formulas above explain how and why the coil go around, and the coil can be made to keep going by arranging that the connections to the coil are reversed each half-turn by contacts mounted on the shaft. Then the torque is always in the same direction. That’s how a small direct current (DC) motor is made. My father made me make a couple of these thirty years ago, with a magnet, a big nail and some copper coil. I used sliding contacts, and they were the most difficult thing in the whole design. But now I found a very nice demo on YouTube of a guy whose system to ‘reverse’ the connections is wonderfully simple: he doesn’t use any sliding contacts. He just removes half of the insulation on the wire of the coil on one side. It works like a charm, but I think it’s not so sustainable, as it spins so fast that the insulation on the other side will probably come off after a while! 🙂

Now, to make this motor run, you need current and, hence, 19th century physicists and mechanical engineers also wondered how one could produce currents by changing the magnetic field. Indeed, they could use Alessandro Volta’s ‘voltaic pile‘ to produce currents but it was not very handy: it consisted of alternating zinc and copper discs, with pieces of cloth soaked in salt water in-between!

Now, while the Biot-Savart Law goes back to 1820, it took another decade to find out how that could be done. Initially, people thought magnetic fields should just cause some current, but that didn’t work. Finally, Faraday unequivocally established the fundamental principle that electric effects are only there when something is changingSo you’ll get a current in a wire by moving it in a magnetic field, or by moving the magnet or, if the magnetic field is caused by some other current, by changing the current in that wire. It’s referred to as the ‘flux rule’, or Faraday’s Law. Remember: we’ve seen Gauss’ Law, then Ampère’s Law, and then that Biot-Savart Law, and so now it’s time for Faraday’s Law. 🙂 Faraday’s Law is Maxwell’s third equation really, aka as the Maxwell-Faraday Law of Induction:

×E = −∂B/∂t

Now you’ll wonder: what’s flux got to do with this formula? ×E is about circulation, not about flux! Well… Let me copy Feynman’s answer:

Faraday's law

So… There you go. And, yes, you’re right, instead of writing Faraday’s Law as ×E = −∂B/∂t, we should write it as:

emf

That’s a easier to understand, and it’s also easier to work with, as we’ll see in a moment. So the point is: whenever the magnetic flux changes, there’s a push on the electrons in the wire. That push is referred to as the electromotive force, abbreviated as emf or EMF, and so it’s that line and/or surface integral above indeed. Let me paraphrase Feynman so you fully understand what we’re talking about here:

When we move our wire in a magnetic field, or when we move a magnet near the wire, or when we change the current in a nearby wire, there will be some net push on the electrons in the wire in one direction along the wire. There may be pushes in different directions at different places, but there will be more push in one direction than another. What counts is the push integrated around the complete circuit. We call this net integrated push the electromotive force (abbreviated emf) in the circuit. More precisely, the emf is defined as the tangential force per unit charge in the wire integrated over length, once around the complete circuit.

So that’s the integral. 🙂 And that’s how we can turn that motor above into a generator: instead of putting a current through the wire to make it turn, we can turn the loop, by hand or by a waterwheel or by whatever. Now, when the coil rotates, its wires will be moving in the magnetic field and so we will find an emf in the circuit of the coil, and so that’s how the motor becomes a generator.

Now, let me quickly interject something here: when I say ‘a push on the electrons in the wire’, what electrons are we talking about? How many? Well… I’ll answer that question in very much detail in a moment but, as for now, just note that the emf is some quantity expressed per coulomb or, as Feynman puts it above, per unit charge. So we’ll need to multiply it with the current in the circuit to get the power of our little generator.

OK. Let’s move on. Indeed, all I can do here is mention just a few basics, so we can move on to the next thing. If you really want to know all of the nitty-gritty, then you should just read Feynman’s Lecture on induced currents. That’s got everything. And, no, don’t worry: contrary to what you might expect, my ‘basics’ do not amount to a terrible pile of formulas. In fact, it’s all easy and quite amusing stuff, and I should probably include a lot more. But then… Well… I always need to move on… If not, I’ll never get to the stuff that I really want to understand. 😦

The electromotive force

We defined the electromotive force above, including its formula:

emf

What are the units? Let’s see… We know B was measured not in newton per coulomb, like the electric field E, but in N·s/C·m, because we had to multiply the magnetic field strength with the velocity of the charge to find the force per unit charge, cf. the F/q = v×equation. Now what’s the unit in which we’d express that surface integral? We must multiply with m2, so we get N·m·s/C. Now let’s simplify that by noting that one volt is equal to 1 N·m/C. [The volt has a number of definitions, but the one that applies here is that it’s the potential difference between two points that will impart one joule (i.e. 1 N·m) of energy to a unit of charge (i.e. 1 C) that passes between them.] So we can measure the magnetic flux in volt-seconds, i.e. V·s. And then we take the derivative in regard to time, so we divide by s, and so we get… Volt! The emf is measured in volt!

Does that make sense? I guess so: the emf causes a current, just like a potential difference, i.e. a voltage, and, therefore, we can and should look at the emf as a voltage too!

But let’s think about it some more, though. In differential form, Faraday’s Law, is just that ×E = −∂B/∂t equation, so that’s just one of Maxwell’s four equations, and so we prefer to write it as the “flux rule”. Now, the “flux rule” says that the electromotive force (abbreviated as emf or EMF) on the electrons in a closed circuit is equal to the time rate of change of the magnetic flux it encloses. As mentioned above, we measure magnetic flux in volt-seconds (i.e. V·s), so its time rate of change is measured in volt (because the time rate of change is a quantity expressed per second), and so the emf is measured in volt, i.e. joule per coulomb, as 1 V = 1 N·m/C = 1 J/C. What does it mean?

The time rate of change of the magnetic flux can change because the surface covered by our loop changes, or because the field itself changes, or by both. Whatever the cause, it will change the emf, or the voltage, and so it will make the electrons move. So let’s suppose we have some generator generating some emf. The emf can be used to do some work. We can charge a capacitor, for example. So how would that work?

More charge on the capacitor will increase the voltage V of the capacitor, i.e. the potential difference V = Φ1 − Φ2 between the two plates. Now, we know that the increase of the voltage V will be proportional to the increase of the charge Q, and that the constant of proportionality is defined by the capacity C of the capacitor: C = Q/V. [How do we know that? Well… Have a look at my post on capacitors.] Now, if our capacitor has an enormous capacity, then its voltage won’t increase very rapidly. However, it’s clear that, no matter how large the capacity, its voltage will increase. It’s just a matter of time. Now, its voltage cannot be higher than the emf provided by our ‘generator’, because it will then want to discharge through the same circuit!

So we’re talking power and energy here, and so we need to put some load on our generator. Power is the rate of doing work, so it’s the time rate of change of energy, and it’s expressed in joule per second. The energy of our capacitor is U = (1/2)·Q2/C = (1/2)·C·V2. [How do we know that? Well… Have a look at my post on capacitors once again. :-)] So let’s take the time derivative of U assuming some constant voltage V. We get: dU/dt = d[(1/2)·Q2/C]/dt = (Q/C)·dQ/dt = V·dQ/dt. So that’s the power that the generator would need to supply to charge the generator. As I’ll show in a moment, the power supplied by a generator is, indeed, equal to the emf times the current, and the current is the time rate of change of the charge, so I = dQ/dt.

So, yes, it all works out: the power that’s being supplied by our generator will be used to charge our capacitor. Now, you may wonder: what about the current? Where is the current in Faraday’s Law? The answer is: Faraday’s Law doesn’t have the current. It’s just not there. The emf is expressed in volt, and so that’s energy per coulomb, so it’s per unit charge. How much power an generator can and will deliver depends on its design, and the circuit and load that we will be putting on it. So we can’t say how many coulomb we will have. It all depends. But you can imagine that, if the loop would be bigger, or if we’d have a coil with many loops, then our generator would be able to produce more power, i.e. it would be able to move more electrons, so the mentioned power = (emf)×(current) product would be larger. 🙂

Finally, to conclude, note Feynman’s definition of the emf: the tangential force per unit charge in the wire integrated over length around the complete circuit. So we’ve got force times distance here, but per unit charge. Now, force times distance is work, or energy, and so… Yes, emf is joule per coulomb, definitely! 🙂

[…] Don’t worry too much if you don’t quite ‘get’ this. I’ll come back to it when discussing electric circuits, which I’ll do in my next posts.

Self-inductance and Lenz’s rule

We talked about motors and generators above. We also have transformers, like the one below. What’s going on here is that an alternating current (AC) produces a continuously varying magnetic field, which generates an alternating emf in the second coil, which produces enough power to light an electric bulb.

transformer

Now, the total emf in coil (b) is the sum of the emf’s of the separate turns of coil, so if we wind (b) with many turns, we’ll get a larger emf, so we can ‘transform’ the voltage to some other voltage. From your high-school classes, you should know how that works.

The thing I want to talk about here is something else, though. There is an induction effect in coil (a) itself. Indeed, the varying current in coil (a) produces a varying magnetic field inside itself, and the flux of this field is continually changing, so there is a self-induced emf in coil (a). The effect is called self-inductance, and so it’s the emf acting on a current itself when it is building up a magnetic field or, in general, when its field is changing in any way. It’s a most remarkable phenomenon, and so let me paraphrase Feynman as he describes it:

“When we gave “the flux rule” that the emf is equal to the rate of change of the flux linkage, we didn’t specify the direction of the emf. There is a simple rule, called Lenz’s rule, for figuring out which way the emf goes: the emf tries to oppose any flux change. That is, the direction of an induced emf is always such that if a current were to flow in the direction of the emf, it would produce a flux of B that opposes the change in B that produces the emf. In particular, if there is a changing current in a single coil (or in any wire), there is a “back” emf in the circuit. This emf acts on the charges flowing in the coil to oppose the change in magnetic field, and so in the direction to oppose the change in current. It tries to keep the current constant; it is opposite to the current when the current is increasing, and it is in the direction of the current when it is decreasing. A current in a self-inductance has “inertia,” because the inductive effects try to keep the flow constant, just as mechanical inertia tries to keep the velocity of an object constant.”

Hmm… That’s something you need to read a couple of times to fully digest it. There’s a nice demo on YouTube, showing an MIT physics video demonstrating this effect with a metal ring placed on the end of an electromagnet. You’ve probably seen it before: the electromagnet is connected to a current, and the ring flies into the air. The explanation is that the induced currents in the ring create a magnetic field opposing the change of field through it. So the ring and the coil repel just like two magnets with opposite poles. The effect is no longer there when a thin radial cut is made in the ring, because then there can be no current. The nice thing about the video is that it shows how the effect gets much more dramatic when an alternating current is applied, rather than a DC current. And it also shows what happens when you first cool the ring in liquid nitrogen. 🙂

You may also notice the sparks when the electromagnet is being turned on. Believe it or not, that’s also related to a “back emf”. Indeed, when we disconnect a large electromagnet by opening a switch, the current is supposed to immediately go to zero but, in trying to do so, it generates a large “back emf”: large enough to develop an arc across the opening contacts of the switch. The high voltage is also not good for the insulation of the coil, as it might damage it. So that’s why large electromagnets usually include some extra circuit, which allows the “back current” to discharge less dramatically. But I’ll refer you to Feynman for more details, as any illustration here would clutter the exposé.

Eddy currents

I like educational videos, and so I should give you a few references here, but there’s so many of this that I’ll let you google a few yourself. The most spectacular demonstration of eddy currents is those that appear in a superconductor: even back in the 1970s, when Feynman wrote his Lectures, the effect of magnetic levitation was well known. Feynman illustrates the effect with the simple diagram below: when bringing a magnet near to a perfect conductor, such as tin below 3.8°K, eddy currents will create opposing fields, so that no magnetic flux enters the superconducting material. The effect is also referred to as the Meisner effect, after the German physicist Walther Meisner, although it was discovered much earlier (in 1911) by a Dutch physicist in Leiden, Heike Kamerlingh Onnes, who got a Nobel Prize for it.

superconductor

Of course, we have eddy currents in less dramatic situations as well. The phenomenon of eddy currents is usually demonstrated by the braking of a sheet of metal as it swings back and forth between the poles of an electromagnet, as illustrated below (left). The illustration on the right shows how eddy-current effect can be drastically reduced by cutting slots in the plate, so that’s like making a radial cut in our jumping ring. 🙂

eddy currentseddy currents 2

The Faraday disc

The Faraday disc is interesting, not only from a historical point of view – the illustration below is a 19th century model, so Michael Faraday may have used himself – but also because it seems to contradict the “flux of rule”: as the disc rotates through a steady magnetic field, it will produce some emf, but so there’s no change in the flux. How is that possible?

Faraday_disk_generatorFaraday disk

The answer, of course, is that we are ‘cheating’ here: the material is moving, so we’re actually moving the ‘wire’, or the circuit if you want, so here we need to combine two equations:

two laws

If we do that, you’ll see it all makes sense. 🙂 Oh… That Faraday disc is referred to as a homopolar generator, and it’s quite interesting. You should check out what happened to the concept in the Wikipedia article on it. The Faraday disc was apparently used as a source for power pulses in the 1950s. The thing below could store 500 mega-joules and deliver currents up to 2 mega-ampère, i.e. 2 million amps! Fascinating, isn’t it? 🙂800px-Homopolar_anu-MJC