Why current leads voltage in a capacitor (logic)

when you connect a capacitor across an alternating voltage generator we saw in a previous video that the current in the circuit leads the voltage by a phase angle of pi by 2 90 degrees this basically means that the current oscillations are a quarter of a cycle ahead of the voltage oscillations in this video we're going to dig deeper and figure out why does that happen logically we've seen it you know mathematically we've seen the previous video why it happens but logically why is that happening and i'll be able to answer questions like why is it that at you know there are points where voltage is zero but the currents are maximum and there are points where currents are zero but voltages are maximum does that make any sense all right so where do we begin well we can begin by reminding ourselves that the voltage across generator should always equal the voltage across the capacitor in fact this is how you start the derivation for the current and we saw that in the previous video to to derive the expression for the current and why is this true well that's because you know this put in the point the potential at this point should be the same as the potential at this point there are no circuit elements in between potential at this point should be the same as potential at this point therefore the voltage here should be the same as the voltage over there why does that matter why should i care about this that's because this means as the generator voltage changes the capacitor voltage should immediately adjust itself it should follow the generator voltage but how do you change the capacitor voltage well the capacitor voltage depends on the charge of the capacitor more charge more voltage less charge less voltage that means to change the voltage of the capacitor you need to change the charge but how do you change the charge by running a current therefore can you see to change the voltage you need to first run a current and therefore the current must lead the voltage does that make sense this kind of helps me uh digest this fact that in a capacitor the current must first you know current must lead the voltage because it's the current that charges the capacitor which in turn changes its voltage all right with this insight let's dig a deeper and see exactly why we get this graph so let's look at the voltage over here let's consider let's say right now i just closed the circuit and the voltage is zero and it's starting to rise now if the voltage is starting to rise the capacitor says hey i need my voltage to rise okay immediately that means the capacitor needs to get charged so current should start running immediately but here's the thing how much current should run that depends upon how quickly the voltage should rise if the voltage should rise very quickly the capacitor says charge me up very quickly if the voltage should rise very slowly the capacitor says charge me up slowly okay so at this point the question is how is the voltage rising is it rising very quickly or is it rising very slowly and for that you can look at the slope of this curve you can see it is very steep if you think of this as a mountain a very steep mountain and therefore the voltage is rising very quickly and so the capacitor says hey my voltage should rise very quickly charge me up very quickly and therefore at the at this point you get a very high current because you need to have the quickest charging over there and that's why this point represents the fastest charging of the capacitor because voltage is changing very quickly and therefore you have the maximum current so think about it even though the voltage is very low it's changing very quickly that's why you need a very high current does that make sense and notice the current has already reached maximum voltage is yet to reach maximum current leads the voltage all right now what happens as time passes by as time passes by notice the generator voltage keeps increasing becomes larger and larger and larger capacitor says hey keep keep charging me keep charging me okay but look at how the voltage is increasing the voltage is increasing very fast here but it slows down the increase slows down look at the slope it slows down slows down slows down slows down the slope becomes it becomes flatter and flatter and therefore the capacitor says hey keep charging me but slow that charging down slow that charging down and if you slow the charging down the current reduces that's why from here to here as the voltage increases the current reduces so this represents the part where the capacitors charging rate slows down eventually we reach a point where the voltage has reached the maximum and it's not going to increase anymore at that point the capacitors voltage also reaches maximum and the capacitor says hey i don't need any more charging i mean i reach the maximum value the voltage is not going to increase anymore and therefore the current at that point becomes zero so that makes sense and that's why at that point when the voltage is maximum acro across the capacitor the capacitor is max charge capacitor doesn't need any more charging the current stops and again notice the current has already reached zero the generator voltage is yet to reach the zero so current leads the voltage okay what happens after that after that notice the generator voltage starts reducing and it goes down and so capacitor says hey my voltage should also reduce how how can i reduce my voltage oh get rid of my charge get rid of oh discharge how do you discharge run the current in the opposite direction negative current that's why after this point you get a negative current even though the voltage is positive because the voltage is reducing the capacitor voltage needs to reduce you need to discharge that's why the current has become negative is that making sense now and also notice initially the voltage reduces very slowly very slowly and then the voltage starts reducing very quickly how do i see that look at the slope initially it's it's not that steep but it becomes steeper and steeper and steeper so the capacitor says hey discharge me slowly slowly but then as time passes by quickly discharge me quick discharge and so the discharge rate keeps increasing that's why the current is negative and keeps increasing so this is the point which represents this region represents the discharge rate becomes faster and faster and faster and as a result the current becomes higher and higher and higher and eventually as you keep discharging faster and faster and faster we reach a point where the capacitor finally loses all its voltage when the generator voltage has become zero capacitor at that same time loses all its voltage all the charge is gone and immediately as you can see the the generator now starts going in the negative the capacitor also says don't stop that current keep that current running because i want to now get charged in the opposite direction because the voltage is becoming negative the capacitor voltage should also become negative so immediately it says continue that current continue that current charge me up in the opposite direction and it's for that reason at this point you see voltage is zero but because the voltage will quickly rise in the opposite direction the current becomes maximum you now see same thing as over here quickest charging but in the opposite direction the last time passes by notice the voltage keeps on increasing in the opposite direction but it increases slower and slower and so the capacitor says keep charging me in the opposite direction keep doing that but slow it down slow it down so that my voltage also already increases slower and slower and that's why the current starts becoming lower and lower and eventually the current goes to zero and this represents where the capacitor has maximum charge in the opposite direction and then the story continues so this is how i really love to think about capacitors and what's going on inside and what i absolutely love about this is we can actually go one step further and ask more questions like for example what would happen if i were to increase the frequency of this generator without changing the maximum voltage so uh think of it as the voltage oscillation becomes faster okay but the maximum voltage stays the same what changes will we find in the current well of course one thing is that the current will also oscillate faster but i want you to think about the maximum current do you think that would change why would it change or why would it not change you can argue from the same you know you can use the same logic without having to look at the equations to try and answer this question so can you pause and think about this all right so let me get rid of these points first for a second and uh let me let's look at the what the new voltage graph is going to look like since the voltage is oscillating very quickly but within the same two points the voltage graph will be quicker like this and so it's kind of gonna like shrink like this so that makes sense so this is the new voltage graph same height as before but higher frequency okay so one immediate thing you can see is at this point the point where you get the maximum current because the capacitor needs to charge very quickly notice if you make your voltage oscillate quicker the slope increases it goes to it it increases the voltage increases even more rapidly at the beginning compared to before and as a result now when i turn on my generator at that point the capacitor says hey i need to get charge even faster than before because the voltage is increasing even quicker than before which means the current that runs in the circuit now has to be higher than before and as a result of that you will now find the maximum current in the circuit is much higher than before does that make sense mainly because the voltage is changing very quickly so the capacitor must charge and discharge very quickly that means the current in the circuit must be much higher so you increase the frequency you get higher max current i naught increases you decrease the frequency the the current i naught decreases and we can look at this purely based on logic and this agrees with the equation that we derived in the previous video we derived this expression and notice in this expression you see that if the omega increases which represents the frequency the maximum current increases so it's now starting to make sense and you can think about why if you increase the capacitance then also the maximum current increases i'll leave that to you but i just wanted to show you how these equations can start making sense if you deeply start thinking about what's going on inside so moral of the story why does the current lead the voltage in a capacitor because to change the voltage of the capacitor you need to first charge it or discharge it and to charge or discharge it you need to first run a current so current has to lead the voltage