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Motional EMF | 9 mins | 0 completed | Learn |

Transformers | 10 mins | 0 completed | Learn |

Mutual Inductance | 18 mins | 0 completed | Learn |

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Concept #1: Mutual Inductance

**Transcript**

Hey guys, in this video we're going to be talking about the phenomenon known as mutual inductance. Alright, let's get to it. Now, if I have two coils, one on my left and one on my right and I bring them together, and the current in the coil on my left is changing, that will actually cause an EMF to be induced on the coil on my right, this phenomenon is known as mutual inductance, okay? It's mutual because it happens between the two of them, the same thing would happen, if instead of the current changing in the left coil, the current change in the right coil, then it would induce an EMF on the left coil, okay? Mutual inductance, it happens mutually between two coils that are close by, alright? Now, the coil with the changing current is typically referred to as the primary or the primary coil, and the coil feeling the EMF is typically referred to as the secondary or the secondary coil, okay? Let's look at this image right above me. I have two coils, coil 1 and coil 2, coil 1 has N 1 turns and a current i1 within it, coil 2 has N 2 turns and no current in it. Because there's a current in coil 1, there's going to be a magnetic field produced by coil 1, using the right hand rule, this current is coming out of the page towards you, so I'm going to curl my fingers out of the page, we can see that, that magnetic field goes to the right, you guys can all confirm this on your own, use your right hand, your fingers are going to curl out of the page towards you, your thumb is going to point to the right in the direction of the produced magnetic field, okay? Now, that magnetic field is going to change, if the current changes, because that magnetic field depends on the current, if that magnetic field is changing, then the flux through coil 2 is going to change and we know by Faraday's law, that means that EMF has to be induced on our secondary coil, okay? So, that is mutual inductance. Now, as I said, the magnetic field B produced by our primary coil depends upon current 1, which we call i1, just the current in our primary coil, just as a reminder, you guys have seen this equation a bunch of times, for a circular wire, the magnetic field is Mu nought times the current divided by 2 times the radius of that circle, okay? So, clearly the magnetic field depends upon the current, the magnetic field produced by the primary coil, depends upon the current through the primary coil. This means that the flux through our secondary coil right here, also depends upon the current in the primary coil okay, right? Magnetic field depends upon the current in the primary coil, the flux through the secondary coil depends upon the magnetic field, so the flux through the secondary coil depends upon the current in the primary coil. Now, taking this a step further, the flux doesn't only depend upon the current, the flux is actually proportional to the current, you can see here, that the magnetic field is proportional to the current, if the current doubles, the magnetic field doubles, if the current triples, the magnetic field triples.

The magnetic flux through our secondary coil, through each loop of our secondary coil, I should say, is the magnetic field produced by the primary coil times the area of the secondary coil, okay? Since the magnetic field is proportional to the current, and we can see here that the flux is proportional to the current, sorry, to the magnetic field, then the flux has to be proportional to the current through coil 1, which is our primary coil, okay? Now, we can write this out mathematically. The total flux, remember, is not just B A, B A is the flux of each individual coil, the total flux is the sum of the flux through all of the coils, so it's the number of coils in our secondary times the flux through each coil, that's proportional to current 1, okay? Now, an important thing about proportionalities, I can write just an arbitrary one, y proportional to x, is that you can always change this from a proportionality to an end equality, using what is called a proportionality constant, okay? In this case, little m is our proportionality constant between y and x, and this can be done with any proportion-, sorry, with any proportionality. In our case, the proportionality constant between the total flux or coil 2 and the current through coil 1, is referred to as the mutual inductance given by capital M. Something important here to realize is that mutual inductance has been said twice, has been said at the top of this page and right here, the one at the top of the page is referring to the phenomenon of mutual inductance, this is referring to a number, so this is sometimes referred to as the coefficient of mutual inductance, but it's more commonly referred to as just the mutual inductance. All this number represents is how strongly the mutual inductance or the effect of the mutual inductance between two coils is, okay? Now, the mutual inductance mathematically between two coils is the number of turns in the secondary coil times the flux through each turn of the secondary coil divided by the current in the primary coil, okay? So, this is flux over current and the units are called Henrys, given by capital H, says it's flux over current, this is equivalent to the units of flux Webers divided by the units of currents, which is amps, okay? Let's do a quick example. What is the mutual inductance of two solenoids of length L and area A, one with N1 turns and one with N 2 terms, okay? Now, the mutual inductance is going to be the same regardless of which is the primary coil and which is the secondary

Practice: A solenoid of 25 turns, with an area of 0.005 m^{2} is wound around a 10.0 cm solenoid with 50 turns, as shown in the figure below. If, at some instant in time, the current through the 10.0 cm solenoid is 0.5 A and changing at 50 mA/s, what’s the induced EMF on the 25 turn solenoid?

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Concept #1: Mutual Inductance

Practice #1: Mutual Induction Between Two Solenoids

A circular coil with 20 loops and a radius of 50 cm and a second circular coil with 40 loops and a radius of 10 cm are aligned such that they share the same center axis. What is the mutual inductance of the two coils?

Two coils have a mutual inductance of 3.4 mH. If the current in one coil is changing at a rate of 1 mA/s, and the resistance of the second coil is 10 Ω, what is the current induced in the second coil?

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