Energy stored in inductor (1/2 Li^2)

current carrying coils or inductors store energy inside them and in this video we'll derive an expression for it but before we begin here's my question how do i convince myself that these coils really store energy i mean just because the textbook is saying that i can't believe it so for that let's take a circuit which i've stolen from our previous video with an inductor and a bulb in series current across the battery we saw that when you close the switch current slowly starts rising it doesn't instantly rise and that happens because inductors fight changes in the current and if this concept is not familiar be a great idea to go back and watch our previous video on self inductance highly recommended but anyways eventually once the current rises to its max value then it's no longer changing inductor no longer fights it and we have a steady situation now imagine we are in such a situation where the current is no longer changing it reaches max value my question now is what happens if i suddenly make the battery disappear and the way i do that is by short-circuiting it you might know when you create a short current tends to take the path of least resistance and now a shot has almost zero resistance so all the current would flow through this it's like effectively removing battery from our primary circuit so if i short circuit it it's like i've removed the battery so imagine i do it i remove the battery my question to you is what's going to happen to our bulb will it stop glowing immediately or we'll take some time and it'll keep going for some time or will it keep glowing forever what do you think will happen can you pause and ponder upon this okay my first thoughts are it is the battery that was maintaining the current so if i remove the battery the current should instantly die out and the bulb should instantly stop glowing but that's not what happens we find that the bulb will take some time before slowly turning off even if there is no battery why why is that happening because remember inductors hate changes in the current here the current is trying to decrease and go to zero and we've seen that whenever there is a change in the current the inductors will induce an emf or voltage now the moment we remove the battery a current immediately starts decreasing and therefore immediately there will be a voltage induced across the inductor another quick question do you think this induced voltage supports the current or do you think it opposes the current well again my first thoughts from this equation is well emf induced will oppose the current that's what the minus sign is saying right no no no remember inductors have no problems with the current they have a problem with changing current and so they oppose the change so if the current is increasing they'll try to decrease it but in our case the current is decreasing so the inductor will try to increase it so in this case to increase the current it will support the current so it will act like a battery and support the current and as a result the current will flow for some time but it can't keep maintaining that eventually di by dt will go to zero and so with time the emf decreases the current decreases and eventually the bulb runs out so this means even after removing the battery for a small time the bulb is still growing so during that time who is providing the energy to the bulb there's no longer a battery who is maintaining the current there's no longer a battery ah it's the inductor therefore the inductor must have had energy inside of it and that's how we can convince ourselves that inductors carrying current must have energy inside of them so the next question is what's the expression for that energy first i want to show you a shortcut derivation using a cool analogy inductors carrying current are very similar to masses having velocity at first you may like what what's the connection how can you even compare them but let's look at their equations and the equation will help us understand when you push on an object right you know from newton's third law it pushes back and so let's write the expression for that reaction force that reaction force f reaction is going to be equal and opposite negative to the force that you apply on it but what is the force that you apply on it from newton's second law we know that force you apply you apply on it is mass of that object multiplied by the acceleration of that object and that acceleration is dv over dt look at these two equations how similar they are so the induced emf is like the reaction force opposite so the inductance is like the inertia so just like how inertia of the mass resists changes in velocity electrical inertia inductance resists changes in the current and we know so if we now look at this comparison we know that moving masses have energy inside of them and the expression is the kinetic energy half mv square so from this can you guess what the expression for the energy stored in an inductor could be can you guess what it's going to be well if you use the comparison it's going to be half times inertia which is in our case electrical inertia inductance times instead of velocity square it's going to be current squared and that's exactly what you will end up getting if you derive it how cool is that in fact this is how i you this is how i remember the expression for energy stored inside inductor but this is kind of cheating because we thought deeply about physics and found this connection but that's not a derivation so let now let's go ahead and formally derive it using this circuit so let's go back to the time when we just removed our battery we know for the next few i don't know maybe milliseconds or microseconds there will still be the bulb will still be growing and during that time the energy would be transferred to the bulb by the inductor so since it's the inductor that's transferring energy to the bulb if i can figure out how much is the total energy transferred to the bulb that must be the total energy inside the inductor if the inductor has transferred let's say total of 100 joules then i know the inductor had 100 joules to begin with and so from this i can say that the total energy inside the inductor must equal the total energy that was transferred to the bulb but how do you calculate that well any time you want to talk about energies when it comes to circuits we fall back to one and only one equation the power transferred that is energy transferred per second always equals voltage times the current and in our case since the energy is transferred by the inductor the voltage will be that of the inductor voltage across the inductor and this will be the current through the inductor and so we know how much is the energy being transferred per second we know the expression for the voltage we also we know what we need to calculate the total energy transfer so although it's a little tricky to put this all together i highly encourage you to pause at this moment and see if you can try this yourself first it's completely okay if you make some mistakes or get stuck somewhere then you watch it then your learning would be much much better so i highly encourage you to pause and give it a shot yourself first okay so the first thing i do is i have an equation let me go ahead and substitute so i know the voltage across the inductor is minus l d i over dt times the current i okay what's next well one immediate thought is since this is energy transferred per second if i want to calculate the total energy transferred maybe just multiplied by the total time taken right total time for which the bulb was glowing so can i just multiply this number with the total time and would that give me the total energy very close to the final you know very close to it but that's not how you do it why that's important because these are some of the mistakes that we can make why can't i just do it that way because that would only work if the power was a constant for example if the inductor was transferring 100 joules per second and that was constant then in next 10 seconds i know the energy total energy would be 100 times 10 but here the power is not a constant right the current is continuously decreasing and so you can't just multiply it by time you have to use calculus and it's for that reason instead of directly saying it i can what i can do is i can say this is energy transfer per second and so i can write this as d u over dt that equals minus l d i over dt times i so you understand why you have to use calculus because sometimes i used to wonder why sometimes we use d or dt sometimes we don't because that's the thing because it's it's changing and now we can cancel out the dts and so what is this new equation saying it's saying when the current changes by a small amount what is the small amount of energy transfer that is this number okay so now if i want to calculate the total energy transferred when the current changes by a large amount from i to zero now we need to integrate this so again feel free to pause at any moment and integrate it okay so let's do that so now that we completely understand it's important to understand what's going on so if i come now because now i have to integrate this and i'm integrating with respect to the current and from where to where i'm integrating with from initial current i the moment when we just removed the battery all the way to when the current goes to zero because till then the energy is being transferred and i want to calculate what's the total energy transferred okay so minus l i'll keep the i first and times d i all right so over here l is a constant i can remove it outside so minus l comes out what's integral of i d i that comes from math it's going to be i square over 2 and so if you have troubles with that feel free to go back and check out our videos on integration so this will be i squared over 2 and we have to put the limits from i to 0. if you put the upper limit you get 0 minus i square over 2 and so if you put that together you'll get l i squared over 2. and indeed notice we get exactly the same expression so this is the energy stored inside the inductor