Video: Inductance

In this video, we will learn how to calculate the self-inductance of a conducting loop and the mutual inductance of a pair of conducting coils.

15:27

Video Transcript

In this video, we’re talking about inductance. This term refers to the way that current-carrying conductors resist change. In particular, they resist a change in their own current. We’ll see how this all works in a bit. But to get started, let’s consider that inductance basically involves the interaction of two physical properties. It involves magnetic flux, symbolized Φ sub 𝑚, and it involves current, symbolized capital 𝐼. Now, we know that current is charge that flows through a closed circuit. And we can remind ourselves of what magnetic flux means.

If we have some area, and we can call that area capital 𝐴, and there’s a magnetic field 𝐵 passing uniformly through that area, then we would say the magnetic flux through this loop of area 𝐴 is equal to 𝐵 times 𝐴. This shows us that magnetic flux is a magnetic field spread over some area. Now, it’s worth saying a quick word about the units of magnetic flux because this will come up in example exercises. Considering how magnetic flux is defined, we know that the SI base units of magnetic field are teslas, capital 𝑇, and that the base unit of area is meter squared. So magnetic flux is measured in teslas meter squared. And the name for that simplified unit is the weber.

So magnetic flux is measured in webers, abbreviated this way. And each weber is equal to one tesla times a meter squared. Knowing this about magnetic flux, let’s get back to this idea that inductance is basically about these two physical quantities, magnetic flux and current. Say that we have a loop of wire, like this, that’s carrying a current 𝐼 clockwise as we see it. We can remember that a current-carrying wire produces a magnetic field around itself. So in the case of our wire, this would produce a magnetic field that points into the screen inside this loop of wire. And if we combine the effects of all of these magnetic field lines, what we have is some total magnetic field, we could call it 𝐵, within the area of the cross section of this wire. In other words, we have a magnetic flux through this loop.

Whatever this value is, if we take the ratio of that magnetic flux to the current 𝐼 that moves through the loop, then this is equal to what’s called the loop’s inductance, symbolized capital 𝐿. We mentioned that inductance is a measure of how much a current-carrying wire resists a change in its current. So the higher 𝐿 — the inductance of this circuit — is, the harder it is to change the current 𝐼 running through it. Now, it may seem obvious, but when we talk about the inductance of this current-carrying wire, we’re talking about how the current in this wire responds to the magnetic flux created by that current.

All that to say, when we talk about inductance symbolized this way, we’re really talking about the self-inductance of a wire, how its current responds to the flux that it creates. We bring all this up because there’s another kind of inductance, called mutual inductance. This has to do with more than one current-carrying wire, and we’ll talk about it soon. So far, though, we just need to know that the inductance of a current-carrying wire describes quantitatively how much it resists a change in current and that it’s equal to the ratio of the magnetic flux through a current-carrying loop divided by that current.

Now, talking about magnetic flux may remind us of the law known as Faraday’s law. This law tells us that if we have a conducting loop, even if that loop has no current in it, so long as the magnetic flux Φ sub 𝑚 through that loop is changing in time, then, believe it or not, an electromotive force, an emf, will be induced in the loop. Now, as far as the rest of this equation goes, this capital 𝑁 refers to the number of turns in the loop or loops in a coil. And then this minus sign here just refers to the direction of induced current movement. So Faraday’s law involves a magnetic flux, really a change in magnetic flux, and then an induced emf, which creates a current in a conducting loop.

So we can see some elements in Faraday’s law that also appear in this equation for inductance. And in fact, we can write an expression for the emf induced in a conducting loop in terms of the inductance of that loop. It looks like this. Given a conducting loop with inductance capital 𝐿, if we change the current in that loop over some amount of time, then that time rate of change times the loop’s inductance is equal to the emf induced in it.

Knowing this, we can actually combine this relationship with Faraday’s law. And as we do, let’s say that we’re talking about the magnitude of the emf induced. So that way, we can just neglect this minus sign here. Okay, so if emf is equal to 𝐿 times Δ𝐼 over Δ𝑡 and its magnitude is also equal to the number of turns in a coil multiplied by the change in magnetic flux through one of those loops divided by the change in time, then we can equate the right sides of those two equations. And notice that on both sides we have Δ𝑡 in the denominator, which means that if we multiply both sides by Δ𝑡, then that time interval cancels out.

So this is another way of expressing inductance based on these two relationships here. Now, earlier, we said that self-inductance isn’t the only kind of inductance there is. Mutual inductance, and we saw an example of this on our opening screen, is another way for induction to happen. It works like this. Say that we have a current in a conducting loop such that it creates a magnetic field through that loop that looks this way. Now, say we put a second conducting loop in a position so that this magnetic field went through it. In that case, there will be a magnetic field moving through the cross-sectional area of this loop. And that would mean a magnetic flux in the loop.

Faraday’s law says that a change in magnetic flux happening over some amount of time induces an emf. And that means that charge would start to flow in this loop. Now, the change in magnetic flux we’re talking about could be the initial movement of this conducting loop into the magnetic field. But of course, once the loop is there, assuming there’s no change in the magnetic field through it, there’d also be no change in magnetic flux.

But what if we did this? What if we gradually increased the magnitude of the current in this loop over here? That would create an increasing magnetic field, which would lead to an increase in magnetic flux through this smaller loop. So long as current is changing then, in this primary loop we could call it, then it will drive the creation of current in this secondary loop. That induced current direction will be such that it creates a magnetic field which opposes the change in magnetic flux experienced by this current loop.

This overall process is referred to as mutual induction. And rather than symbolizing this kind of induction with a capital 𝐿, it’s typically a capital 𝑀 that is used. So, for example, in this equation over here, instead of an 𝐿, we might see an 𝑀. But it would still mean essentially the same thing, that some inductance, whether self-inductance or mutual, multiplied by a change in current over a change in time is equal to the induced emf in a current-carrying loop. The same thing is true for this equation here, or sometimes instead of an 𝐿, we would see the 𝑀 of mutual inductance.

It’s worth pointing out that rather than being a rare occurrence, mutual inductance is the operating principle for electrical transformers. Say we had a transformer here consisting of a core and a primary and secondary circuit. If the current in the primary circuit changes in time, then that will create a change in magnetic field transmitted by the core through the coils of the secondary circuit. Through mutual induction between these circuits, emf and then current will be induced in the secondary circuit. Now, whether we’re talking about mutual or self-inductance, the unit for inductance is the same. It’s the henry, named after Joseph Henry, and it’s abbreviated capital H. To understand this topic better, let’s now try out an example exercise.

A transformer with an iron core has a primary coil with 25 turns and a secondary coil that also has 25 turns. The current in the primary coil increases the magnetic flux through the core by 0.15 webers per second. The current in the secondary coil increases at 0.075 amperes per second. What is the mutual inductance of the coils?

To get started here, let’s draw a sketch of this transformer with the primary and secondary coils. Okay, so here’s our transformer core. And this is the primary coil; we’ll call this the secondary. And even though we can see that neither of these has 25 turns to it, like our problem statement tells us, we can pretend they do. The idea with the transformer is that as current passes through the primary coil and specifically as it changes and passes through this coil, then this creates a change in magnetic field that the core directs through the loops of the secondary coil. When the secondary coil experiences this change in magnetic flux, current is induced in it.

In our scenario, the current in the primary coil is increasing. We don’t know the rate it’s increasing by, but we do know that it’s affecting the magnetic flux in the core and that that flux is increasing at a steady rate, given as 0.15 webers per second, where a weber is the unit of magnetic flux. Now, just as a reminder, if we have some area 𝐴 and there’s a magnetic field 𝐵 passing through that area, then we can say there’s a magnetic flux Φ sub 𝑚 through that loop. And the units of magnetic flux, as we’ve seen, are webers.

So going back to our transformer, in the core, this iron material that connects our primary and secondary coils, the magnetic flux is increasing at this given rate. And that, our problems statement tells us, drives an increasing current in the secondary coil. We have then a change in current in one coil, creating a change in magnetic flux, which induces a change in current in another coil. This means we have mutual inductance going on. And we want to calculate the value of that inductance.

To begin to do this, we can recall Faraday’s law. The reason we’re thinking of this law is because we have a magnetic flux that changes in time. Faraday’s law tells us that this change in flux, ΔΦ sub 𝑚, over some change in time, Δ𝑡, multiplied by the number of turns in whatever coil we’re considering is equal to the magnitude of the emf induced in that coil. Now, in terms of mutual inductance, induced emf is equal to something else as well. We can recall that induced emf is equal to the mutual inductance between two conducting loops multiplied by the time rate of change of current and what we can call the secondary loop or, in the example of our transformer, the secondary coil.

So it’s 𝑀, the mutual inductance, that we want to solve for. And we see that that value multiplied by Δ𝐼 divided by Δ𝑡 is equal to the emf induced in a secondary coil, which by Faraday’s law is also equal to negative the number of turns or loops in the coil multiplied by the change in magnetic flux divided by the change in time. Since we’re given this time rate of change of magnetic flux, let’s combine these two equations for induced emf by equating the right sides with one another.

When we do this though, we’ll leave off this minus sign here because in our case we’re only concerned with the magnitude of the emf induced. So then we get this. 𝑀, the mutual inductance of the coils, times Δ𝐼 divided by Δ𝑡, the time rate of change of current in the secondary coil, is equal to the number of loops in that coil multiplied by the time rate of change of magnetic flux that each loop in the coil experiences.

What we want to do is isolate 𝑀 since we’re solving for the mutual inductance. And to do that, we’ll divide both sides of the equation by Δ𝐼 divided by Δ𝑡. When we do, on the right-hand side of the resulting equation, we have 𝑁 — the number of turns in the secondary coil, that’s 25 — multiplied by the time rate of change of magnetic flux in the core, that’s given as 0.15 webers per second, divided by the time rate of change of current in the secondary coil. This is the induced current, and its rate of change is 0.075 amperes per second. When we calculate this fraction and keep two significant figures, we find a result of 50 henrys. That’s the mutual inductance of the two coils.

Let’s look now at a second example exercise.

A loop of wire carries a current of 180 milliamperes. The magnetic flux produced by the current is 0.77 webers. What is the self-inductance of the loop?

So then, in this example, we have a loop of wire. Let’s say this is it. And it’s carrying a current of 180 milliamperes. Because of this, a magnetic field is produced that passes through the loop. Let’s just say the magnetic field points like this. And because we have this field passing through the interior area of the loop, that means we have a magnetic flux through the loop. And that’s the value we have given to us here in our problem statement, 0.77 webers. We can refer to that symbolically as Φ sub 𝑚, magnetic flux.

Knowing all this, we want to solve for the self-inductance of the loop. And we can recall that the self-inductance of a loop, capital 𝐿, is equal to the ratio of the magnetic flux Φ sub 𝑚 produced by a current 𝐼 in the loop to that current 𝐼. For our particular loop then, we have exactly that information about it. Our current 𝐼 is a current in the loop. And the magnetic flux is produced thanks to a magnetic field generated by that current. So to solve for the self-inductance of our loop, we only need to divide Φ sub 𝑚 by 𝐼.

With these values plugged in, the one change we’ll want to make before dividing is to convert the units of our current from milliamperes to amperes. Now, since 1000 milliamperes is equal to one amp, we can say that 180 milliamperes is equal to 0.180 amps. And now we’re ready to divide. And when we do, to two significant figures, we find a result of 4.3 henrys. That’s the self-inductance of this loop.

Let’s summarize now what we’ve learned in this lesson on inductance. Earlier on, we saw that inductance is the tendency of a current-carrying conductor to resist a change in current. The inductance of a circuit, sometimes called its self-inductance, is equal to the ratio of the magnetic flux through the current loop divided by that current. We also learned that Faraday’s law led us to another relationship describing inductance, that the inductance in a circuit multiplied by the time rate of change of current in that circuit is equal to the number of loops or turns in the circuit times the time rate of change of magnetic flux.

We learned further about mutual inductance, symbolized using a capital 𝑀, which occurs when a change in current, in what we could call a primary coil, creates a change in magnetic field, which is then experienced by a secondary coil where that change in flux through the secondary coil induces a current. This, we saw, is the basic principle behind electrical transformers and can be described by saying that the mutual inductance between two coils or two loops of wire multiplied by the time rate of change of current in the secondary wire is equal to the emf induced there. This is a summary of inductance.

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