 Let's solve a couple of questions on mutual inductance. Now over here we have two coils 1 and 2 which have 50 and 75 turns respectively and they're placed near each other as shown below. On passing a current of three amperes in the blue coil that is the second coil, a flux of 2.4 Weber passes through each turn of coil 1. Now the question is to figure out the value of mutual inductance of coil 1 with respect to coil 2. Why don't you first attempt this on your own and see if you're able to figure out the value of mutual inductance. Alright, now how do we even get to something like a mutual inductance? Well, let's start from what we already know. We know that if there is a current moving in a coil, it will produce a magnetic field and in this case the magnetic field could look like this. And this current of three amperes must have changed from zero to this value. It must have taken some amount of time, very small amount of time. Which means its magnetic field must also have changed. That means that the flux must have increased from a value of zero when there was no flux and there was no current to a value of 2.4 Weber. Now we can straight away write an equation linking the flux with the mutual inductance, but I like to start with the idea of self-inductance and then write an equation linking the EMF and the mutual inductance, an equation which looks very similar to the equation linking EMF and self-inductance and then from there onwards try to build an equation which links the flux to the mutual inductance. So, let's do that. Now, over here two things are going to happen. This coil, the blue coil, it will oppose the change in flux. The coil itself opposes the change in its own current and this idea is called, this idea is called self-inductance. This idea is called self-inductance. Because the opposition happens in the same coil, so there is an EMF induced in the same coil due to a change in the current. But we can see here that the field generated by the second coil, that is the blue coil, it's also passing through the first coil. It is also linked with the first coil and we can see these field lines passing through. So, therefore if the field is changing, the flux through the first coil must also be changing. Which means there must be an EMF induced in the first coil as well due to the changing current in the second coil. This idea where we have EMF induced in one coil due to the changing current in another coil is called mutual induction. And because there is an EMF induced, there is a current that starts running in the first coil. So, let's try and write an equation for mutual induction. It will be very similar to this one. Now over here we are writing EMF induced in the first coil. This is equal to a constant that is negative m. But whose mutual inductance, the first coil, so we write m1 times di over dt. But which di over dt, that will be for the second coil. That's the idea behind mutual inductance, right? EMF induced in the first coil is due to the current changing in the second coil. So, this is di2 divided by dt. And because we are interested in figuring out the mutual inductance of coil 1 with respect to coil 2, so we add a subscript of 2 over here. This shows that we are calculating mutual inductance of first coil with respect to coil 2 due to the changing current in coil 2. But over here we do not really know the amount of EMF that is induced in the first coil. So, maybe we can use some knowledge that we already know. Whenever there is a changing flux through a coil, there must be an EMF induced according to Faraday's law. And that EMF is given by the rate of change of flux. So, this is minus d5 minus d5 by dt. And this equation over here has a constant of mutual inductance. So, I can take this constant inside the differential. That won't change anything because this is a constant after all. If we do that, this will become d by dt of mutual inductance of coil 1 with respect to 2 into i2, the current that is flowing in the blue coil. Now, we can try and compare these two equations. When we do that, we will realize that flux, that flux has to be equal to this quantity right here. Mutual inductance multiplied by the current has to be equal to flux because both of them are after all equal to the EMF induced in the first coil. So, when we equate these two, this becomes flux passing through the first coil, the total flux passing through the first coil. This is equal to the mutual inductance of coil 1 with respect to 2 into the current that is flowing in the second coil. Why don't you pause the video here and think about what will be the total flux that is passing through coil 1. Now, we know the flux passing through one turn of the coil, that is 2.4. And in total, there are 50 turns. So, 2.4 Weber's will pass through 50 turns. So, the total flux passing through the first coil, the red coil, this will be n that is 50, 50 multiplied by 2.4 Weber's. That is the total flux passing through the first coil. And this is equal to again the mutual inductance of coil 1 with respect to 2 multiplied by the current flowing in the second coil, that is 3 amperes. And now, when we work this out, when we try and work out the mutual inductance, this will come out to be equal to 50 into 2.4 divided by 3. And finally, this comes out to be equal to 40 Henry. 40 Henry. It has the same unit as that of self-inductance. So, let's write 40 Henry over here. And when we do that, when we check our answer, yes, that is right. This is the correct answer. Now, there are different types of questions that you can try. And the link to the exercise is in the description. We will now move on to a different question. Okay. Now, over here we have two coils that are placed in two different orientations with respect to each other. So, we have one case and we have the second case. In which case is the mutual inductance between the coils greater? There's one note which says that a distance between the centers of the coils remains unchanged. All right. And the options say that case two has a greater mutual inductance. Mutual inductance is the same. And the last one is that case one has a higher mutual inductance. So, why don't you pause the video and first try this on your own. All right. In order to figure out the mutual inductance between the coils, we can start off by letting current flow in one of the coils. Now, over here we have the red coil, which is in the same orientation in both the cases. So, let's say there is a current flowing through red coil in this direction, in the anticlockwise direction in both the cases. And as a result of this current flow, there will be some magnetic field lines set up produced and they will be passing through the blue coil as well. And mutual inductance is a constant which tells us how big the effect of this current in the red coil would be. If the effect due to this changing current is high, if the effect on this blue coil is high, then that means the mutual inductance would be high as well. If the effect on the blue coil due to the changing current in red coil is low, then the mutual inductance would also be low. It's a constant which tells us how big the effect is. Now pause the video and think about how would you even know whether changing current has more effect on the blue coil or less effect. All right. So, due to this current flow, there will be some magnetic field lines produced and they will look like this. Now, we can see that in the second case, more number of magnetic field lines will pass through the blue coil compared to the first case. So, the flux that is linked with the blue coil will be more when we compare it to the first case. And from Faraday's law, from Faraday's law, we know that the EMF induced that is equal to the rate of change of magnetic flux. This is delta phi by delta t. So, if there is more flux changing across the blue coil, then that means that the EMF induced in the blue coil in the second case is more than the EMF induced in the blue coil in the first case. We can also extend this and say that the current induced current in the blue coil in the second case will be more than the current induced in the blue coil in the first case. So, the effect in the second case on the blue coil is more. We can see that the EMF induced is more and even the current induced is more. If we had a bulb attached to the blue coils in both the cases, the bulb would glow brighter in the second case compared to the first case. So, due to the changing current in the red coil, the effect of this current is more on the blue coil in the second case. And therefore, the mutual inductance between the coils in the second case is more than the mutual inductance between the coils in the first case. So, over here, we will choose case 2. Now, let's check this and when we check this, that is correct. This is the right answer. You can again go check out some more questions from this exercise and the link to this exercise is also in the description.