Mutual Inductance - Online Tutor, Practice Problems & Exam Prep

Mutual inductance describes how a changing current in one coil induces an electromotive force (emf) in a nearby coil. The mutual inductance (M) is defined by the equation M = NΦI, where N is the number of turns in the secondary coil, Φ is the magnetic flux, and I is the current in the primary coil. This relationship highlights that mutual inductance depends solely on the coils' properties, not the current, and is measured in henrys (H).

Mutual Induction

Hey, guys. So, we saw how a current-carrying wire can induce a current inside of another coil. That's just Faraday's law. When solving some problems, you're going to need to know what the mutual inductance is between two coils, and that's what we're going to cover in this video. Now, just in case you've seen the video on self-inductance, this is going to be very similar to that. If you haven't, that's perfectly okay because you're going to see it later. Let's check it out.

Basically, the mutual inductance is just that for two conducting coils that are close to each other, a current through one coil, a current that's changing through one coil, is going to induce an EMF on a current on the other. And that's what this is all about. Now, the coil with the changing current is often referred to as the primary coil. So, that's the one with the current that's changing, and the other one is going to be the secondary coil. Alright? So, let's take a look at this diagram here that I have. Imagine that we have a coil like this, and we have the currents, and I know this is sort of hard to visualize here, but the current is coming out towards you. So, if you use your right-hand rule, you should be able to figure out the magnetic field goes to the right. In any case, don't let the direction of the magnetic field really trip you up. This is more of a conceptual thing.

So, imagine this magnetic field points off in this direction. The fact that it goes through some area in Coil 2 means that there is some magnetic flux, and that magnetic flux is equal to B times A times the cosine of theta. Now what happens is I have two coils here, Coil 1 and Coil 2, so I want to start labeling things properly. This is the magnetic field that is produced by Coil 1 because it has a current going through it, and this is the area of Coil 2. Right? So what happens is that the total amount of flux through this second coil here depends on the number of turns, which I have right here, and also depends on the magnetic field. Now, this magnetic field depends on the current that is going through Coil 1. So what happens is this magnetic field right here is actually proportional to the current that's going through Coil 1. So what we say is that the total amount of flux, if it depends on the magnetic field and the magnetic field which depends on the currents, we just say that the total flux is proportional to current 1. So all we're saying here is that the total amount of flux through this second coil is actually proportional to the amount of current that is going through Coil 1. So whenever we have a proportionality, there could be multiple constants that go out here. Alright. It's just a proportionality constant.

And so what we say is that the total amount of turns times the total amount of flux through this second coil is actually equal to some proportionality constant called m times the current in that first coil, and that's what mutual inductance is. This mutual inductance m, this letter m here, is just a proportionality constant, which is called the mutual inductance. It sort of just describes the relationship between two coils in which you change a current in one, and it induces a current in the other. Now we can get from this equation here that if you just, if you just move over this current to the other side, we get that the equation for mutual inductance is nφ2÷I1. And the units for this mutual inductance are given as henrys, which is just this capital letter h here. In more fundamental units, it's just the flux which is given as a weber, and then we have the current on the bottom which is just an ampere.

The most important thing that you need to know about the mutual inductance is that it depends only on the number of turns and the shape of the coils. So what I'm saying is that it doesn't depend on the current. Now I know this sounds crazy because I clearly have a current right here in this formula, but we're going to see that the currents, this I one that's on the denominator, will always cancel out. So let's go ahead and check out an example in which we have two solenoids. We're given some letters and some information about those solenoids, and we need to find out what the mutual inductance is. Alright? So remember, if we're trying to find out the mutual inductance formula, that's always just going to be thisnφ2÷I1. So it's just going to be the number of turns times the total amount of flux divided by the total amount of current. So this is just a number right here, this n2 and that's just given to us right here.

And by the way, we're just working with letters here. There are no numbers. So what happens is we have the solenoid that's sort of wrapped up inside another one. So this short solenoid right here is actually Coil 2, and this long solenoid over here is going to be Coil 1. So let's say we had some sort of current that is going through the solenoid, and it's producing some magnetic field. Now again, the direction of this magnetic field doesn't really matter. That's not the important thing. So let's just imagine that the magnetic field goes this way through this solenoid, and that's our B field. Now, it's kind of hard to see. So we have some B field like this. Now if you vary the current in I 1, it's going to change and induce a current in Coil 2 because the flux is changing, right? So let's see. This flux, this given, this flux over here is just given as, let's see, we have n2, and now we have the B field produced by Coil 1, the area through Coil 2, and then the cosine of the angle.

Now what we can see here is that the magnetic field goes in this direction sort of down the solenoid, and the area vectors of each of these thing

An outer solenoid with 30 turns is wound tightly around an inner coil 25cm long with a diameter of 4cm and 300 turns. The current in the inner solenoid is 0.12 A and is increasing at a rate of 1.75×10³A/s. a) What is the average magnetic flux through each turn of the outer coil? b) If the resistance of the outer coil is 20mΩ, what is the magnitude of the induced current through the outer coil?

(Note:for the multiple choice, select the correct answer for part (b) only.)

• A. (b) i_ind = 0.17 A
• B. (b) i_ind = 5.0 A
• C. (b) i_ind = 20 A
• D. (b) i_ind = 150 A

Here’s what students ask on this topic:

What is mutual inductance and how is it defined?

Mutual inductance describes the phenomenon where a changing current in one coil induces an electromotive force (emf) in a nearby coil. It is defined by the equation:

$M=N\Phi I$

where $N$ is the number of turns in the secondary coil, $\Phi$ is the magnetic flux, and $I$ is the current in the primary coil. Mutual inductance depends solely on the properties of the coils, such as the number of turns and their shape, and is measured in henrys (H).

How does mutual inductance differ from self-inductance?

Mutual inductance involves two coils where a changing current in one coil induces an emf in the other coil. In contrast, self-inductance occurs within a single coil, where a changing current induces an emf in the same coil. The mutual inductance $M$ is defined by the relationship between the two coils, while self-inductance $L$ is defined by the coil's own properties. Both are measured in henrys (H), but mutual inductance depends on the interaction between two coils, whereas self-inductance depends on the coil's own characteristics.

What factors affect the value of mutual inductance between two coils?

The value of mutual inductance between two coils is affected by several factors:

• Number of Turns: The mutual inductance is directly proportional to the number of turns in both coils.
• Coil Geometry: The shape and size of the coils influence the mutual inductance.
• Distance Between Coils: The closer the coils are to each other, the higher the mutual inductance.
• Core Material: The presence of a magnetic core can significantly increase mutual inductance.

Mutual inductance does not depend on the current flowing through the coils but rather on their physical properties and relative positioning.

How is mutual inductance used in practical applications?

Mutual inductance is widely used in various practical applications, including:

• Transformers: Transformers rely on mutual inductance to transfer electrical energy between circuits through electromagnetic induction.
• Inductive Coupling: Wireless charging systems use mutual inductance to transfer power without physical connections.
• Induction Motors: These motors use mutual inductance to convert electrical energy into mechanical energy.
• Signal Transmission: Mutual inductance is used in communication systems to transmit signals between circuits.

These applications leverage the principle of mutual inductance to achieve efficient energy transfer and signal processing.

What is the relationship between mutual inductance and Faraday's Law?

Faraday's Law states that a changing magnetic flux through a coil induces an electromotive force (emf) in the coil. Mutual inductance extends this concept to two coils, where a changing current in the primary coil induces an emf in the secondary coil. The induced emf in the secondary coil can be expressed as:

$\epsilon =-M\frac{\mathrm{dI}}{\mathrm{dt}}$

where $M$ is the mutual inductance, and $\frac{\mathrm{dI}}{\mathrm{dt}}$ is the rate of change of current in the primary coil. This relationship shows how mutual inductance quantifies the induced emf due to a changing current in a nearby coil.