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Current time:0:00Total duration:12:29

Video transcript

in a previous video we attached our generator to an inductor a pure inductive circuit and found the expression for the current in this video we'll dig deeper we'll take this expression figure out from this what's the relationship between the voltage and the current and from that we'll be able to draw plot a graph so this is the voltage graph that we have seen earlier we will be able to plot a current graph and see the relationship between them and eventually we'll be able to figure out exactly how inductors behave when you put an alternating voltage across them so where do we start well we can start by trying to find the relationship between the current and the voltage and for that let's try to put the current in the same fashion same format as the voltage so the first thing i immediately see is this part over here now represents the maximum current which we can call i naught just like how we call this as v naught and an interesting thing to see which will not explore here we'll do that in a future video is that even though there is no resistance in the circuit our current is limited there is a maximum value which is very interesting that means inductors also sort of kind of provide some kind of resistance to the current but how why what we'll talk about that all of that in future videos separately but now let's concentrate on this part okay because we want to compare these two and draw a graph so i need to get this part right so what i want to do is convert this into a very similar function like this so i want to convert this into sine so that i can compare them easily so that would be a great time to pause the video and use your trigonometry and see if you can convert negative cos into positive sign so that i can compare with this so pause the video and see if you can give this a shot yourself all right let's see so let me try that over here we have seen in trigonometry videos before technology lessons before that cos theta can be written as sine of pi by 2 minus theta so minus cos omega t can be written as minus i'm keeping the negative as it is with the bracket minus sine pi by 2 minus omega t does that make sense okay but i want to get rid of the negative sign as well i don't want the negative sign and another thing we've seen in trigonometry is that you can always put minus sine theta can be written as sine of minus theta which when i can put the negative sign inside these are all trigonometric relationships i'm not deriving any of that over here i'm just using the identities and so this now i can write it as sine omega t minus pi by 2. so when i put a negative sign inside the pi by 2 becomes negative omega t becomes positive and so we put it all together we can now write the current relation current expression we can write the current as i naught times sine omega t minus pi by 2. and there we go we now have an expression which we can compare with the voltage so what is this expression telling me it's telling me that the current will also be an oscillation sinusoidal oscillation no surprise over there alternating current but it's saying that it's not oscillating in sync with the voltage in fact you can see there's a negative pi by 2 over here this means it's oscillating 90 degrees behind the voltage what does that mean well remember 360 degrees or 2 pi represents one full cycle so 90 degrees one fourth of it represents a quarter cycle this means the current is oscillating a quarter cycle behind the voltage and in a minute we'll look at the animation and it'll be all clear but before we do that let's go ahead and draw the graph for the current and i want you to pause the video and see if you can plot the graph yourself you have the voltage graph so you have reference for that with respect to that try plotting the current graph okay so let's start at this point i know at this point we have omega t to be zero therefore the voltage is zero and so when omega t is zero notice you get a sine minus pi by two which is minus one so i becomes minus i naught so at this point we are at negative maximum let's say current negative maximum is somewhere over here this would be our minus i naught all right let's consider this point now i'm only considering the easy points this is the point where the voltage is maximum meaning omega t should be 90 degrees that's when the sign is maximum that's the point and then omega t is 90 degrees pi by 2 pi by 2 cancels oh that's when you get sine zero that's when the current is zero so at this point the current is zero the voltage is maximum but current is zero interesting then comes the point where voltage goes to 0 and this is the point where omega t is pi you get that right 0 pi by 2 pi sine pi 0 that's why the voltage is 0. and when you put pi pi minus pi by 2 is pi by 2 again this time you get maximum plus 1 so you get i equals plus i naught oh at this point voltage is 0 but the current goes to maximum so maybe this is the point where you have plus i naught and so you can kind of now see i know it's a sine graph and this is enough for me so i know that the graph is going to go like this and then maybe come down somewhere over here and sort of like that you know what i already have a better graph let me let me just draw that so here is our current graph this looks much better but one of the immediate doubts i get when i look at this graph is wait a second doesn't this say that the current is i mean doesn't it look like the current is ahead of the voltage i mean if this is when the current and voltage are in sync with each other shouldn't this be current being ahead not really because remember this is the future so think of it this way right now the voltage is at zero but the current comes to zero a little bit later by the time the voltage is already at maximum but the current comes to maximum little bit later so indeed the current is lagging behind the voltage but you know what a better way to visualize this is i have an animation for you here it is it's the same graph but here instead of going forward moving our time axis forward we'll move the graph backward and see what happens it's the same thing right so i'm going to move the graph backward and concentrate on the here so we'll dim everything and you can now see the oscillations and you can clearly see the current is changing behind the voltage can you see that the pink is the current okay and so we can now draw an arrow mark to represent both the voltage and the current and that's how we like to visualize it so let me draw an arrow mark for the current as well and if you keep it over here you can clearly see they are not in sync with each other who current lags behind the voltage and current is quarter of a cycle or phase difference of pi by two behind the voltage okay now we could stop over here but if you're curious like me let's go one step deeper and ask the question why is this happening i mean this is the first time you're experiencing where current and voltage are not oscillating in sync with each other it's really weird why is the inductor doing that i mean think about think about certain points over here there are some points over here as you can see where voltage is zero in the circuit zero voltage but we have current to be maximum how does that make any sense and then you have these points over here where the voltage is maximum in the circuit and the current is at zero and what's going on and if you think also there are points where over here voltage is negative but current is positive i've never experienced anything like this in our previous circuits so i'm really really trying to figure out what's going on over here how to digest this luckily i found a mechanical analogy so here's what i mean here we have our current and voltage oscillating one more time but what we'll do now is we'll imagine the voltage to be similar to height of a plank above the ground which is pivoted to something that's connected to a wall so the plank is free to go up and down let's say and the voltage represents the height of that plank so an oscillating voltage that's all concentrated on that can be similar to can be imagined to be an oscillating plank going up and down okay all right and for the current what we'll do is you'll imagine there's a box kept over here and the box is going to slide and we'll assume there is no friction because it's a pure resistance less circuit right it's a pure capacitor only inductor is there so inductance is like the inertia the box's inertia represents the inductance and the box's speed represents the current what i want you to do before i show you the animation is i want you now to consider the speed and the height and look at their relationship and see if that follows the same current and the voltage relationship think about when the speed of the box would be zero when would it be maximum when it would be yeah when would be zero and maximum what what heights so imagine right now i've just kept the box and you imagine the thing goes back and forth so just visualize this yourself first all right so right now i the height is maximum and i just kept the box my question is what's the speed of the box well because of its inertia it's gonna be zero it'll take some time for it to pick up the speed right so we have a situation where speed is zero but the height is maximum but now let's wait and see what happens as the plank goes down notice the box will keep on picking up speed it pick up pickup pickup speed keeps on increasing until the plank becomes parallel to the ground that's when the height is zero but the box now has a lot of kinetic energy a lot of momentum it'll have maximum speed after this point the planck will go down but the box will not immediately come back down because of its inertia will continue to keep going up so even though the voltage is negative the speed or the velocity stays positive ooh does that make sense now let's look at the animation and as it goes up it starts losing its momentum and eventually when it comes to the negative height negative maximum that's when the speed of the box goes to zero very similar to what's happening to our circuit isn't it beautiful right i love this analogy it helps me digest what's going on and let's now look at the same thing with the arrow mark the arrow mark of this represents the speed or the velocity and that is equivalent to the current zero uh maximum voltage zero current and now notice zero voltage maximum current the current is positive even though the voltage has gone negative because of its inertia and now somewhere just you know if you just go just a second back now the current the current goes to zero but the voltage is negative maximum the thing keeps repeating beautiful isn't it beautiful okay okay so can you connect all the dots and see what's going on over here if you consider this point where the voltage is maximum and the current is zero it's like at that point over here the way i like to visualize this is imagine this is the point where i just turned on my voltage and the voltage is at maximum the current immediately wants to go to maximum but the inductor says hey wait a second i have inertia the current was zero i will not allow you to instantly go up and it's for that reason the current is still zero at this point okay but as time passes even though the voltage goes down it's still positive current is now starting to increase slowly and steadily just like how the speed of the box is increasing and eventually comes this point this is the point where the voltage has eventually gone to zero but because the current was continuously increasing this is the point where the current has reached the maximum value after this what's going to happen is the current the voltage says hey i'm going to zero current says fine i'll also come down to zero and the inductor says hold on i have inertia i can't let you go to zero i will continue to push the current and just just like how the blocks momentum keeps it going forward it's the you can kind of say the inertia of the inductor can make keeps the current going forward and that's why the current stays positive but of course now but as time passes by eventually the inductor loses all its juice just like how the box loses all its kinetic energy and finally comes a point where the inductor has ran out of all his juice current has gone to zero but that happens when the voltage has gone to negative maximum and then the whole story repeats in the opposite direction and the story goes on so the model of the story is like masses inductors have inertia they hate changes in current and because of that the current lags behind the voltage by an angle of pi by 2.