 Let's derive the expression for self-inductance of a long solenoid. So where do we begin? Well, let's begin with one expression that we saw in our previous video for self-inductance. We saw that the induced emf by any coil or any inductor is given by minus L L being the self-inductance times Di over dt. The whole idea was coils hate changes in current and the quicker you try to change the current the more emf they induce opposing that change and the self-inductance basically tells you how much emf you get for a given change in rate of change of current More self-inductance means more capacity to resist that change meaning more emf induced Now if you're not familiar with this get an idea to go back and watch our previous video on self-inductance where we explore this in great detail Okay, but our goal is to figure out what the self-inductance of the solenoid is So how do I do that? Well, I can use this equation But what I like to do is go back to Faraday's law which says that any coil in general any loop in general Whenever there's a flux change There's an induced emf and that induced emf is given by minus n d phi or dt where this is the Magnetic flux Faraday's law says that you take any loop whether it's a solenoid or coil or anything if you change the magnetic flux through it induces an emf and and The rate of and the quicker you change that flux the more emf gets induced and just some clarity This is the flux through one loop if there are multiple loops, then you just multiply that by n So to calculate inductance what we can do is we can look at these two and we can we can say they are equal They have to be equal and which means that L times I Should equal n times phi B. So let's write that so n times phi B should equal L Times I and before we go forward. Let's pause here for a moment because this is a cool relationship What is it saying? This represents the total flux through our solenoid So it's n times flux through each coil. So this is the total flux a equals It's it's saying that the total flux is proportional to the current. Does that make sense of course? It makes sense right when you pass some current through it some current I It's that current that's generating the magnetic field It's that magnetic field that's causing the flux so more current gives you more magnetic field giving you more flux so it makes perfect sense and The constant is the self inductance and why I like this is because you can do something very cool with it You can you can compare this with something. We've learned in mechanics flux is like momentum Think about this for a second. Okay Why we like why I like to think of flux as momentum is because in Mechanics we've seen that if you want to change momentum of an object then you have to put a force on it But when you put a force on it you from Newton's third law it pushes back on you So that means whenever you try to change momentum of an object that object will push back on you It's very similar to what we're seeing over here We can say objects hate changes in momentum and flux is very similar coils hate changes in flux You try to change the flux. They will push back on you by inducing an emf and We've seen the momentum equals mass times velocity where mass is their inertia and Similarly self inductance is the electrical inertia Velocity is the speed at which objects move and you can think of current Is proportional to the speed at which charges are moving. So do you see a neat relationship exists between the two? Beautiful isn't it? In fact, you can use the relationship here also and you get something very similar force equals mass times dv over dt And you can explore that yourself But anyways now we can calculate what the self inductance is so you can say from this expression Self-inductance of our solenoid equals the total flux If you pass some current through it you calculate the total flux and you divide it by the current divide by the current All right, so how do I calculate the total flux through a solenoid? We know how to calculate the flux through any any loop. We know the flux equals b times a times cos theta the theta is the angle between the area vector and the b vector here notice the area vector So if I take one such loop The area vector would be this way and the magnetic field is also in the same direction if I assume the field to be uniform And so the the flux just becomes b times a And so if I plug that in I will get self inductance equals n times The flux which is just be the magnetic field over here times the area Divide by a divide by I sorry And now comes the question. What is the magnetic field inside a solenoid now? That's something that we have derived before and we can just directly substitute but my problem is I don't remember that Okay, my principle was I don't remember a lot of formula Because physics is not about formula So what do I do over here what I like to do is go back and think about how we derived it and quickly make That derivation and so if you're like me then you would also want to do something like that if you don't remember the formula But if you do remember the formula feel free to pause or skip ahead to directly Substitute and getting the expression But if you're like me then this is what I like to do so I go back and ask myself Hey, where did the magnetic field? How did I derive the expression for the magnetic field? So there are two ways to derive it you use be yourself are or you use Ampere's law and I remember I use Ampere's law In some special cases so I go back and I use Ampere's law It says the closed loop integral of b.dl equals mu naught times I am closed and so now I have to choose a loop and the clever loop I choose because I have a straight field over here I remember our Amperian loop is going to be some kind of a rectangular square and When I go around it to calculate b.dl notice b.dl here is zero Because there's no field outside here and here b and b and DLR in perpendicular So b.dl becomes 90 beer it becomes zero cos 90 is zero b.dl over here is zero if you can see that same reason outside is zero So the only b.dl you get is over here and when you do that you just get magnetic field b Times the length which is x. So I'm doing a quick derivation not a complete derivation if you've done this before So b times x. So this is how I'm doing my derivation. I don't remember the field value at all So that equals mu naught times I am closed. So that's a total current enclosed by the loop What is I am close? That will be the total number of loops enclosed and enclosed Multiplied by the current through each loop. That is just I and so now I ask a how many loops are enclosed And I can do a simple ratio for that So I know that if I take the total length of the solenoid to be say L And let's say the total number of loops is n that I know L has n number of loops So the link so the length x has how many loops? So it's gonna be nx by L. So I can just substitute this as nx over L. So this is nx over L and There we go X cancels out and that's the expression for the magnetic field. So you don't have to remember things So what I like to do is remember basic Fundamental rules like Ampere's law, B. Osawa law, Faraday's law and Deeply understand the derivation and if you do that you can pretty much derive all of these very quickly So you don't have to remember stuff All right, so that's how I do it. So now I can just plug this in. So if I plug in I get n times the expression for the magnetic field is mu naught n I by L times a divided by I and The I cancels out and our final expression for L turns out to be a pull that mu naught out So it's gonna be mu naught n squared Times a where a represents the area of that loop divided by L The length of the solenoid and this is the expression It will only work for long solenoids if you have very tiny solenoids and the magnetic field is not gonna be uniform So what is this saying well first of all it's saying that it does the self-inductance does not depend upon voltages or currents But only on the geometry and on the material just like capacitances or resistances So that makes a lot of sense But how does it depend on the geometry of the material? Well, let's look at geometry first See it says that if you have more tons you get more self-inductance How why is that happening because for more tons you tend to get more field for a given current You also tend to get more flux for You know more flux for a given current. So that's why more tons gives you more self-inductance It says that more area also gives you more self-inductance. Why is that happening because with more area you tend to get more flux For a given current. So that's how self-inductance tends to increase and it says that if you increase the length You tend to decrease the self-inductance. Why is that? Well, that comes from here So if you tend to increase the length and then for a given current the field won't be much If the coils go far away the fields do not add up that much that nicely and so the fields will tend to reduce So you tend to get less self-inductance, which means to have a higher self-inductance Not only it's important to have more tons, but you tightly coil them tightly make them short I mean make the length shorter you get more self-inductance But how does it depend upon material? So it doesn't depend upon the material of the wire, but depends upon what is wrap around See if it's wrapped around air then you use mu naught permeability of air or vacuum But say you wrap it around some kind of a ferromagnet Then the permeability of a ferromagnet is much much higher than that of the air So then the value will not be mu naught It'll be mu naught multiplied by some number and that number is gonna be in thousands or tens of thousands if you use something like ion or soft ion and so material also matters and so to increase permeability tremendously we use wrap it around some kind of a ferromagnet and to indicate that we use some multiplier over here and We often call that multiplier mu R. It's called relative permeability Its value is one for air or vacuum but for any other material the value would be different for ferromagnets and stuff It can be in tens of thousands and so that's a great way of increasing self-inductance