 Let's solve a couple of questions on AC circuits involving only a single component, either an inductor, a capacitor or a resistor. Now in this question we have this AC circuit where the current in a circuit is given by the function I0 sin omega t. And the question is to figure out which of these best represents the voltage of this source. So let us see the options. The options are V0 sin omega t, cos omega t and the option C and D. Alright, before I jump into this, why don't you pause the video and first attempt this one on your own. Alright, let me scroll this up. We will have a look at the options later. Okay, let's try and think about how would this inductor behave when there is a current in the circuit. As there is current passing through the circuit, there is current passing through the coils of the inductor as well. And we know if there is current passing through the coils or any wire, there will be a magnetic field generated due to that. Since this is an AC source, the current constantly oscillates. So that means that even the strength of the magnetic field that is produced due to the current in these coils, even that changes. And that implies that the flux of the magnetic field through these coils, even that changes as the current is changing. So let us write that there is current which changes. That leads to change in the magnetic field. And that leads to the change in flux. Now why don't you pause the video and think about how would the inductor respond to that? How would the inductor respond to change in magnetic flux through its coils? Alright, we know from Paradis Law that whenever there is a change in flux through a coil, the coil responds to that by opposing the change in flux. In this case, how would that really look like? So to oppose the change in flux, the inductor would have to oppose a change in current. And that implies that as the voltage is changing, the current is not changing immediately with it. But in fact it is changing with a lag because of the opposition. It takes some time for the current to build up. The current is in fact lagging the voltage or we can say that the voltage is leading the current. This can be expressed using the phase or the idea of phase difference. There is a phase difference between the current and the voltage. And using that phase difference, we can try and figure out the phase of the voltage source and maybe then the function of the voltage source. So let's do that. Now here we have the current function which is I0 sin omega t. And this argument omega t, this is what is called the phase. In this case omega t, the argument omega t is the phase of the current function. And now you might know that the phase difference between the voltage and the current in a purely inductive circuit, in a purely inductive circuit is 90 degrees and the voltage leads the current by 90 degrees because it took some time for the current to build up right. So the current is lagging the voltage or we can say that the voltage is leading the current by 90 degrees or pi by 2 radians. So we can express that in this manner, the phase of the voltage, it leads the current. It leads the current by 90 degrees. Now if we try and write the function of the voltage that would be V0 sin and now the phase of the voltage that would be this right here. And instead of this argument, we can write the phase of the current plus 90 degrees. So when we do that, that would be V0 sin omega t, I'm writing omega t, but this right here is just omega t. So that is omega t plus omega t plus 90 degrees. And now we can use one trigonometric identity over here. So let me write that, let me write that, let me write the identity right here. This is sin 90 plus theta, 90 plus theta that equals cos theta. And here theta is omega t and we already have 90. So sin omega t plus 90 would just become cos omega t. So if it becomes that then the final function of the voltage that is V0 cos omega t. Now let's check our options and in this case that is option number, option number b. You can try more questions from the exercise in this lesson and if you are watching on YouTube then the exercise link is added in the description. Let's move on to our second question now. Here the question says that the AC voltage and the current are given by these two functions. Current is given by I0 sin omega t and the voltage is given by minus V0 cos omega The question is to figure out the circuit element. The circuit element attached to the AC source. There are three options, there is resistor, inductor and capacitor. Before I get into this, why don't you pause the video and try to figure this out on your own. Alright, now the job is to figure out the circuit element. So if we know whether the current is leading or lagging or in phase with the voltage then we should be able to figure out what the circuit element is, right? Because I know that for resistor current and voltage are in phase and these two definitely do not look like that they are in phase. So but anyway, we will check that. I know that in an inductor the current lacks a voltage by 90 degrees or pi by 2. And in a capacitor the opposite is true. The current leads the voltage by a phase difference of pi by 2 or 90 degrees. So how can we figure out if the current is lagging or leading the voltage? For this, maybe the first step could be that we bring both of them to the same trigonometric function. Like both of them should say sine or both of them should say cos. After that we can then look at the phases. We can compare the phases and identify which one is leading, which one is lagging. Since current over here is given in sine, let's try and bring the voltage function to a sine function also. Voltage over here is given in cos omega t. And if I want to change cos into sine, I can use a trigonometric identity which looks like this. Cos theta that equals sine 90 minus theta. There is this one. There is also sine 90 plus theta. Now which one can I use? Maybe I can use both. Let's see. Let me use the second one first. Let me use this one first and see where we land. So cos theta is sine 90 plus theta. That means cos omega t. That means cos omega t. That will be sine 90 plus omega t. So this is minus v0 cos omega t, the voltage function. That becomes minus v0. That becomes sine 90. Let me write this in this manner. Let me write omega t plus 90. It's the same. Let me write omega t plus 90. But there is a negative sign outside. Over here we can use one more trigonometric identity. This is the first one. We can use one more which looks like this. When it says minus sine theta, we can write this as sine of minus theta. The argument can become minus theta. This is minus v0 sine omega t plus 90. Therefore we can write this as v0 sine minus omega t plus 90. This becomes v0 sine minus omega t minus 90. The current function is i0 sine omega t. But there is a problem. How can we compare this now? There is plus omega t over here and there is minus omega t. It's not a good way to compare and figure out the phase difference. This one didn't work. Let's use the first identity. Let's use this one. Let me make some space. The sine 90 plus theta 1 didn't work. Now we are writing v equals to minus v0 cos omega t as minus v0 cosine of 90 minus omega t. What is theta? Theta here is omega t. 90 minus omega t. If we use the second identity, minus sine theta is sine of minus theta. Minus goes into the argument and when it goes into the argument, I can write this as, in this manner, this is minus 90 minus omega t. Now when this minus goes inside the bracket, this becomes v0 sine minus of minus becomes plus. That's plus omega t and minus 90. Now I think we have something. Now we can compare these two functions. And what do we see when we compare them? We can see that the phase of the voltage function, the phase of the voltage function is omega t minus 90. And the phase of the current function is just, this is just omega t. Now definitely they are not equal to each other. So it cannot be a resistor. So it's all between these two, inductor or capacitor. Now let's try and express one phase, the voltage phase in terms of the current phase. So when we do that, we can write this, voltage phase is this is omega t, this is omega t minus 90. And the current phase, the current phase is just omega t. You might be seeing where this is going now. If we subtract them, this is phase of voltage minus phase of current that is equal to minus 90. That is equal to minus minus 90 because omega t just gets cancelled. And if we take the current, the phase current to the right hand side, that becomes, that finally becomes the phase of voltage that equals the phase of current minus 90 degrees. Now with this expression, we can see that voltage is in fact lagging the current by 90 or pi by 2 radians by a phase of 90 degrees. The current is leading the voltage. The voltage is lagging the current. And I know that in an inductor, the voltage leads the current. It does not lag the current. Whereas in a capacitor, to change the voltage across a capacitor, we need to change the amount of charge that the plates have, right? For this, we need to run a current first. And only after the current runs, does the voltage of the capacitor change. So the current leads the voltage in a capacitor or as we see with this expression, the voltage lags the current. And therefore, the right option for this one is option C. Again, try more questions from this exercise in this lesson. And if you're on YouTube, do find the link to the exercise that is added in the description.