AC voltage across pure inductor (derivation)

in this video we're going to take our alternating voltage generator and attach it to a coil of wire which is also called an inductor and find out what will be the expression for the current now just to be clear this is a circuit that only contains inductance it has no resistance or capacitance and immediately the question could be such circuits don't exist right i mean all circuits will have some resistance at least so then what's the point of studying this well the reason we study this is because if we do it this way then we can understand exactly how an inductor behaves when you have a voltage signal applied to it so it'll be easier for us to do understand that behavior and more importantly later on when we look at more real circuits this analysis will actually help us so pure inductive circuit attached to an ac voltage generator what's going to happen so let's see to figure out what the current is going to be i need to find an expression between the voltage and the current and let's say our current is flowing in this direction as of now so how do i figure out what that current is going to be well one relation that immediately comes to my mind when i think about voltage and current is ohm's law v equals ir but remember ohm's law only works for resistive circuits and there are no resistance over here so i can't use ohm's law so what do i do one of the things that we've learned about inductors or coils of wire is that they hate changes in current and we may have learned before that whenever there is a current changing through an inductor it induces an emf which we can treat as a voltage and we've seen in previous videos that the magnitude of that voltage which i'm just going to call as vl that will always equal l times di over dt di over dt and this expression is basically saying that you know inductors hate changes in current and the quicker the current tries to change the larger emf or the larger voltage it induces and of course if you need a refresher or more clarity on where this comes from we've talked a lot about that in our previous videos on inductors feel free to go back and check that out but now let's see if we can use this to build our equation so the next immediate question i have is what direction is this voltage compared to our source voltage of the generator so for that let's say our source voltage is continuously fluctuating at one point this will be higher at higher voltage and then this will be at a higher voltage and this will be a higher voltage that's why it's called alternating it's continuously changing its direction so at some point in time let's say this is at a higher voltage so i'll call this side as positive and let's say this is at negative now at this point because the current is flowing in this direction and our voltage is also pushing the current in the same direction in this situation the current is trying to increase or the current is increasing actually and therefore because the current is increasing our inductor says ah i hate changes in current i'm going to try and decrease the current okay and because it wants to decrease the current it's going to try and push back the current and to do that it puts a high voltage on this side and a low voltage on this side and so this is the polarity of that induced voltage and you can immediately see that the polarity of the source voltage and the induced voltage is exactly the same and because there are no circuit elements in between we know that this voltage in this point and this point is at the same potential this point and this point is at the same potential in other words this voltage should exactly equal this voltage both in magnitude and direction and so i can use that to figure out what the current is so let me go ahead and write that so the voltage across the inductor should exactly equal the source voltage or the generator voltage both in magnitude and direction and so if i substitute i get vl equals l d i over dt so d of dt over i and at this point one confusion i always had is if you go back and you know go back to inductors we said that the emf induced is negative ldi over dt so should i put a negative sign over here well remember the negative sign is only telling us the direction of the voltage it's saying it's in the opposite direction of the change in the current and over here we've already included the direction over here including the direction we said we realized that they both have the same polarity right so you've already taken care of the direction so you don't have to worry about the negative sign and that equals v naught sine omega t and now we see an equation which is not just an algebraic equation it's a differential equation which basically means there is a differentiation term over here and some differential equations can be very tricky this turns out to be one of the easier ones because to solve this all we have to do is separate all the i terms and all the t terms on one side separate them on two different sides and then we could just integrate now would be a great idea to pause the video and see if you can integrate this yourself and see what the expression for current is going to be all right so i'm going to rearrange just to have d i on one side so left side will only put d i and so on the right hand side i will get if i just rearrange i'll get v naught sine omega t i get a dt over here and i'll get divided by l and now because i've separated the items and the t terms i can integrate if i integrate integral of d i is just i and that equals v naught over l is a constant i can pull that outside the integral and what's the integral of sine omega t dt well that integral i'll directly write the integral over there that's going to be minus cos omega t divided by omega and whenever we integrate this is going to be a constant we need to put that constant and there we go so this is our expression for the current and you can immediately see that this is something that we don't know if you can figure out what the value of that constant is we are done so how do we figure this constant out first of all what does it even mean well remember integrals is basically like integration is like doing reverse of differentiation asking what function should i differentiate to get this as the answer so if you actually take this and differentiate and you can try that you will actually get this number differentiation of cos is negative sign so the negative cancels out and the omega pops out and that cancels with this however remember differentiation of constant is a zero which means i can also differentiate this plus 100 or i can differentiate this plus thousand and then all of those numbers will still give me this value and so the integral is saying i don't know what this constant is that's that's your job mahesh it's your job to figure the value of the constant out and so now quick question is how do we figure this out math is not going to answer that we need physics and one of the powerful ways of figuring out values of constants whenever we give an equation is we say hey it's a constant which means it does not depend on the values of voltage or v naught or does not depend on the values of omega or l so we can put whatever values of v naught you know omega whatever values we want and then see if we can figure out what the value of c is going to be it's a very powerful method let me show you what i like to do over here let's say in this particular example we put the v naught value to be zero if this is zero that means our source voltage is always going to be zero so i'm basically saying i'm basically saying i'm not going to put any generator in this circuit okay and i want to know what the current is going to be based on this equation so from this equation current is going to be well this part will be 0 because v naught is 0 but there is a constant so according to this equation the current is going to be a constant but i know in my heart in my bones in my stomach that if i don't put any generator over here the current in the circuit must be zero because if it was not zero we have problem with energy conservation where is that energy coming from there is no source so i know that in this situation my current should be zero so i know that in this situation that constant value should be zero and because it's a constant if it's zero for these values it should be zero for any values of v naught and therefore we can argue that this constant should be zero and ta-da we have found the expression for the current and so for the purposes of this video we are done but we still need to dig deeper and figure out what does it mean how does this explain the behavior of inductors in the presence of alternating voltages we'll explore all that fun stuff in the future videos