Mutual induction & inductance

check out this cool youtube video i found here's a coil that is connected to some power supply so there'll be current running over here but there's a second coil which is not connected to any power supply or any batteries it is just connected to a bulb now look at what happens when it brings this closer to the first coil it lights up even though there is no connection between these two look at that it lights up wow how is that happening now although it doesn't look so dramatic if you think about it we're having a wireless power transfer electricity transfer from one coil to another there is no connection between them and the idea is the same in wireless charging phones there's a wireless power transfer happening so how is the electricity able to jump from one wire and enter into another wire when there is no contact well let's find out so here's the same setup there is a current that's running through one of the coils let's call that as coil one and let's call that current i1 and everything related to this coil let's keep it blue so that we can differentiate between the two coils and so because some and because of this somehow the bulb is lighting up in the second coil how is that happening so let's think about this we know now that the current in the first coil generates a magnetic field in the first coil but what's important is that this current is changing it's continuously changing its direction its strength is changing which means the magnetic field that it's producing is also changing it's fluctuating and we know from faraday's law that a changing magnetic flux induces an emf and so what's going to happen there are a couple of things that are going to happen one thing that's going to happen is that this coil itself will oppose the change in the flux and therefore we say that the coil itself opposes the change in its own current this idea is what we call self-induction because the opposition is happening to the same coil so there's an amf induced in this coil due to the change in its own current self-induction however this time what's happening in this experiment is that this field generated by the first coil is also seen by the second coil is also linked with the second coil as you can see and therefore if the field is changing the flux through the second coil is also changing oh this means there will be an emf induced in the second coil as well so let's write that there will be an emf induced in the second coil due to what due to changing current in the first coil this idea where you have an emf induced in one coil due to the changing current in the another coil that's what we call mutual induction and we'll get to know exactly why it's called mutual in a second but that's the idea behind mutual induction hopefully you can differentiate between the two now and because there's an emf induced over here there's a current that starts running over here and that induced current causes the bulb to glow and that's what we saw in our experiment pretty cool huh that's how wireless power transfer works but we also saw that when you take that when you took that secondary coil far away notice the bulb stopped glowing it's only when you come closer it glows why did it stop glowing if you took it far away can you now answer this question well when you take it far away the magnetic the strength of the field reaching this coil becomes very weak in our diagram you can see that if you now take this far away the magnetic field linked with this coil becomes very weak and so the flux changes are very weak induced emf is very weak and as a result the current induced will be smaller and the bulb won't glow okay now let's go ahead and write an equation for mutual induction and i want you to take a shot at it it's going to be very similar to this equation but this is the self-reduction happening in the same coil so why don't you pause the video and take a shot at it okay so here we just have to be careful about which coil we are talking about which currents and which emfs we are talking about so we're talking about the emf in the second coil so emf e2 equals minus here we had self-inductance as the constant here we'll introduce a new constant called mutual inductance and we'll talk a little bit about it but who's mutual inductance is it the second coils or the first coils the second coils right because induction is happening in the second coil so minus m2 i'll write times di over dt but which di over dt is it the rate of change of the current of the second coil or of the first coil hey it's of the first coil that's the whole idea behind mutual induction emf is induced in the second coil due to the current changing in the first coil so we'll write d i one over dt and this mutual inductance just like self-inductance has units of henry and if the value is very high it means that a lot of emf is induced when the current changes in the other coil and what does its value depend on well just like self-inductance it does not depend on the voltages or the currents it's a matter of geometry it depends upon how many turns this coil has what's the radius of this coil it also depends upon how far it is from this coil the farther you go the mutual inductance drops the closer you are the stronger the mutual inductance the orientation how they are kept together it also depends upon the number of turns of the first coil that's right it depends on that as well and we will calculate mutual inductances for some cases in future videos but since the mutual inductance of the secondary also depends upon the properties of the primary we like to often say that this is mutual inductance of the second coil with respect to the first coil and this will make sense if you think there are many other coils involved over here then there will be a mutual inductance between two one mutual inductance between two three which will inductance between two four so for every pair of coil there will be a value for mutual inductance and that's why we like to write two one over here okay now here's where things get really interesting so because of the emf generator in the secondary coil there is a current induced in the secondary coil right there's a current induced and that's that's why the bulb is glowing but now think about this this current also generates its own magnetic field so here is the magnetic field generated by this current and because this current is also changing this magnetic field is also changing and as a result of that it induces an emf in the primary and therefore because of the changing current in the secondary there will be an emf induced in the primary as well and we'll have a very similar formula and i want you to you know take a shot at writing this down but now hopefully you understand why it's called mutual induction because they both induce emf in each other that's why it's called mutual induction so can you now pause the video and write down what would be the expression for the emf induced in the primary coil all right it's gonna be very similar it's gonna be emf induced in the primary coil is going to be negative mutual inductance of the primary coil times the rate of change of current of the secondary coil remember it's always the change in current of the other coil and again for similar reasons we will write this as mutual inductance of the primary with respect to secondary and the same thing holds over here this is also henry this also depends upon the configuration geometry and all of that and finally there is a very nice surprising connection between the two mutual inductances see if when i was learning this if you were to come to me and ask hey what is the relationship between m12 and m21 i would say i have no idea what the relationship would be right i mean how would i know what the relationship would be why should there even be a connection between the two right well however it turns out that those two values will always be equal to each other and that's right you have to box it with the exclamation mark because this is by no means obvious but that's great it makes our lives easy so what this means is if you have a pair of coils then there is one single value for mutual inductance whether you consider its mutual inductance of this coil with respect to this or you consider this coil with respect to this turns out to be the same and this is super useful when you're calculating mutual inductances because you can decide which one you want to calculate and sometimes in fact in most times one of them is always easier to calculate another one is almost impossible and we'll explore all of that in future videos and one final thing for folks who are really curious these are not complete equations because this emf is only due to mutual induction remember there will be an emf induced due to its own changes in the current so the complete equation would be a mutual induction along with that there will also be self-induction which is caused due to changes in its own current and similarly a complete equation for this one would be along with that self-induction which is happening to do changes in its own current and so these now represent complete equations for induction but unless you're working on advanced engineering concepts we really don't have to worry about these big equations