RC circuits is going to be the topic of this lesson in my new General Physics playlist which when complete will cover a full year of University algebra based physics now in this lesson we're going to talk about circuits that have both resistors and capacitors hence the name RC circuits uh we alluded to the fact in an earlier chapter that it takes time for these capacitors to actually reach their Max charge and it's the Dynamics of that process that we're going to study in this lesson my name is Chad and welcome to Chad's prep where my goal is to take the stress out of learning science now if you're new to the channel we've got comprehensive playlists for General chemistry organic chemistry General Physics and high school chemistry and on Chad prep.com you'll find premium Master courses for the same that include study guides and a ton of practice you'll also find comprehensive prep courses for the DAT the MCAT and the oat now in talking about the Dynamics of both the charging and the discharging of a a capacitor in an RC circuit I found it instructive to think about a defibrillator it's an example you're probably somewhat familiar with hopefully from just just the movies or a show or something uh and hopefully not in person but potentially but the idea is that the paddles that either the doctor or the first responder or the paramedic or something is holding uh are functioning as the opposite plates of a capacitor and it takes some time to charge them notice they got to wait for a little bit and then they say clear and then they touch it to the the patient's chest and then it discharges it's kind of completing the circuit so that charge can flow from one plate to the other so if that's not successful they've got to wait for a given period of time for them to charge up again as well so it's a great example of the fact that capacitors take both time to charge as well as discharge now if we take a look so the the real diagram of what's going on uh in an RC circuit for both charging and discharging capacitor is usually a little more complicated than we give it so but usually it involves at least a couple of different switches and and and here the defibrillator might be the exception because we don't need that second switch because we'll actually discharge it by just touching the patient's chest but in another capacitor it's usually not that way and the idea is this you've got a couple of switches and so so you can figure out which Loop of the of the circuit here you want to close and so the first one I'm going to close is the one involving the battery here so we're going to close that circuit and in this case now we've got a complete loop with the source of EMF the resistor and the capacitor all in series together so and current is going to flow from the battery through the resistor charging up the positive side of the capacitor leaving the opposite plate negative charge as current continues to flow through the other half of the circuit so what we'll find is that the current actually flowing through the circuit is intermittent it's transient so it turns out it starts at a Max but as as more and more charge builds up on the plates of that capacitor less and less current is going to flow until it reaches its Max charge and once it reaches its Max charge all current flow in the circuit stops completely and at that point you can actually open back up this switch and that capacitor will remain fully charged chared all right so let's take this current we used to charge it back off that was transient and is stopped now and now we're going to close that other switch and in closing out of the switch we now have a complete Loop involving just the resistor and the capacitor and this charge that's built up on the capacitor can now flow back around the other direction so until the plates have the same charge once again so and it flows right back through that resistor uh and and notice the EMF of the battery is not involved in this part of it at all and this happens until it's fully discharged now again initially we're going to get current flowing through the circuit until the plates of the capacitor once again have the same charge and at that point the current in the circuit completely stops and comes to an end so that's the charging and the discharging process and we want to look at a few graphs as well as some of the math involved in this process because you're probably going to in all likely to have to do some calculations with it so we'll start with the charging process and if we want to look at at the charge buildup uh on the plates of the capacitor over time they start out having no overall charge to having a positive plate and a negative plate so and in this case you start off with zero charge and you see it's actually exponentially approaching some Maximum value and if you recall the definition of capacitance was Cal Q over Delta V and if you rearrange that Q is equal to C Delta V now a couple chapter chapters ago when we introduced capacitors we just use this math so it turns out this is the final result when a capacitor is fully charged so but we alluded to the fact that it actually takes some time that we'd study it later well now in this lesson is the later that we alluded to so it turns out when we calculate this C Delta V that is the maximum charge once it's fully charged but it takes a little bit of time for that to be possible and it turns out the resistance of the resistor and the capacitance of the capacitor affect how long it takes the greater their combination of values the longer it's going to take to fully charge this capacitor all right so the max valum that we're going to a Asm totically approach here is simply equal to C Delta V but we're going to ASM totically approach it and we've got a rather complex looking equation uh that gives us that kind of the value over time and you take that qax value and multiply it by one minus some number here so in this number is an exponential but it's an exponential to a negative power with time there and so as time goes up this becomes e to the negative let's say time goes to Infinity this becomes e to the negative Infinity well e to the Nega infinity is0 and you end up with 1 minus 0 and that term goes away and you just end up with qal Q Max at time equals infinity so the idea that this term is uh the limit of this term is going to be zero as T approaches Infinity so ultimately Q is going to approach Q Max at greater and greater amounts of time now here this - t over to it turns out that to is what we call the time constant for an RC circuit and it's simply equal to the product of the resistance times the capacitance and the greater this is the longer it's going to take to reach full charge and things of the sort and the way this works if you look so it's a ratio of time over that time constant RC so in the uh exponential here so and ultimately it works like this it takes about five times the time constant so to reach Max charge turns out that's when you're going to kind of get over 99% fully charged so if this time constant was equal to 10 seconds well then it would take 50 seconds to about reach maximum charge so and you know ASM totically get pretty close to the maximum charge up there somewhere that's kind of how it works all right so if we look uh going back over here if we rearrange this one more time we can see that Delta V is equal to Q over C well the capacitance is a constant so but the Delta V is directly proportional to the Q the charge and so we shouldn't be surprised that the graph looks exactly the same so if Delta V is proportional to Q then its graph is going to look nearly identical in this case we're going to approach Delta V Max which is just in this case the EMF of whatever the the voltage source or EMF source is in this case so and again you see that same term 1 minus E the- T over to and once again it takes about five time constants to reach the max potential difference across that Capac fter and then finally we see a graph that looks exactly the opposite this is actually the current flowing so and if you recall with ohms law Delta V equals I well I equals Delta V / R but it turns out that's just the maximum value and the maximum value is when you first start charging it but again once you start building up charges on the positive plate and negative plate that makes it harder and harder for more charge to flow throughout that circuit to lead to an even greater buildup of charge and so it ASM totically approaches zero instead so you start at the max and drop down to zero current once it's fully charged all current stops in that circuit and so in this case instead of 1 minus E T over to it's just times e t over to the entire term is Asin ASM totically approaching zero in this case so that's the charging process discharging is similar so in discharging it's now fully charged state so we're going to close that other switch and have the current kind of revers in that second Loop going back through the resistor in the opposite direction so we're starting at a maximum charge and so in both the case of maximum charge as well as maximum potential difference across that capacitor they both start at a Max and now ASM totically approach zero and we've seen for something ASM totically approaching Zer instead of 1 minus E to the T over to you're just going to take the maximum times e to the T over to and again this term right here is approaching zero which is why the entire side of the equation here is going to approach zero as time approaches Infinity and once again it's going to take about five time constants or more to kind of get full discharge or something that we say approaches full discharge greater than 99% so very similar equations and instead of the 1 minus E to the T of tower here it's just the qax or the EMF of the battery uh which is the Delta Vmax time e to the T over to and then notice though that the current one looks exactly the same so we saw a difference for charge and Delta V here the charge and Delta V were going up here they're going down but in both cases both during charging and discharging the current is going down when we first closed this switch over here when this one was open so we had current flowing to charge up those plates and as more charge built up on those plates less current flowed and so it made sense that it was going to go down well here now we're actually discharging it and we're going to have the charge going off those plates and we have the greatest impetus for that charge to want to flow when it's at a Max charge but eventually this is going to be overall back to neutral no overall charge on those plates and there's definitely no propensity for current to flow at that point and so with this one you actually start at a Max and drop down to zero as well and so that's the one thing that's kind of unique here is that both during charging and discharging current flow starts at a maximum and goes to zero so but we got kind of opposite looking graphs for for both charge and the potential difference across the capacitor uh and for from here we're ready to do some plugin and chug in so the question at hand has this lovely circuit here we've got a 12vt battery a 100 ohm resistor and a 6.0 mad capacitor in a circuit here with what's starting out as an open switch and the question says for the circuit shown to the right what is the time constant first question what will be the charge stored by the capacitor when fully charge second question and what will be the charge on the capacitor initially uncharged and the potential difference across the capacitor 1.8 seconds after the switch is closed all right so a few questions answer and the first is the easy one just what is the time constant time constant to again is just equal to R * C just easy enough in this question so R * C which is the 100 ohms times in this case 6 * 10-3 farads so as long as I use SI units ohms and farads so I had to convert the mil farads to farads uh the time constant here is going to come out with SI units of seconds so in this case 100 * 6 * 7 - 3 is going to be 0.6 and in this case it should have two6 so 0.60 seconds and that's our time constant and if you recall again we said the time constant it takes about five time constants to kind of reach full charge or full discharge in this case well five times that would be about 3 seconds so this capacitor to get over 99% charged or discharged depending on where you're starting from takes about 3 seconds that's something we can just kind of ballpark right off the bat so the first question is answered what is the time constant the second one is what will be the the charge stored by the capacitor when fully charged well again that's just going to come right off the definition of capacitance where Q is just equal to C time Delta V and our capacitor again is the 6 M farads here then Delta V is the 12 volts so and as long as I'm using milars you guys might recall that this is going to come out in mom so in this case 6 * 12 is 72 and so we could write it as 72 mum or we could go back and make it like 072 kums or something of that sort uh really is up to if it's multiple choice you know how the answer choices look and stuff like that all right the next question says what will be the charge on the capacitor initially uncharged and the potential difference across the capacitor 1.8 seconds after the switch is closed we're told it's initially uncharged so we're doing the charging process and so here's our two lovely equations here here and so in this case Q at time T here is going to equal qax and we just solve for qax so I'm just going to plug that right back in but had we not solve for it I would just plug in C * Delta V again right into the equation that's 72 molom * 1us e- T over to so in this case we're at 1.8 seconds all over 0.6 seconds so and in this case we can see that I chose a time that was exactly three time constants so and if you look on your on the study guide here and I'll put it up on the screen but uh three time constants is about 95% fully charged or discharged depending on which way you're going uh in this case and so it's going to come out to be 95% of that 72 micum number uh if you want to check it when we're done so but in this case we'll let the calculator definitely do the heavy lifting here um we've got 72 * parentheses 1 minus so you look at your natural log button typically where you get the E button e to the 1.85 /65 oh I need another set of parentheses there in all likelihood so e to the negative I'll insert another parentheses 1.85 divided 65 and then I've got two sets of parentheses to end right there all right we're going to get 67.8 one9 which rounded with two sigfigs would be 68 so 68 mum here after 1.8 seconds and that looks right around 95% of 72 so that makes sense so now we're also asked for the potential difference across the capacitor and there's a couple different approaches you can take had we not already answered this question right here we could then go proceed to this equation right here and we' use 12 volts for the EMF into again 1 minus E the T over to and life would be good but again so from our definition of capacitance here we know that Delta V is equal to Q over C and so the Delta V at any point in time is going to equal the Q at that time over the C which is a constant for a given capacitor and so in this case because we know the Q at 1.8 seconds finding the Delta V at 1.8 seconds we're just going to take the Q value at that same time point and divide it by the Capac make our life a lot easier again we totally could use this equation right here we just don't have to in this case and so in this case Q is going to equal I'm sorry let's go Delta V is going to equal that's 68 mums all over a capacitance of in this case 6.0 mads and as long as they're both in millas they'll cancel and the Delta V will come out in volts here and 68 divided 6 comes out to 11.3 which will round down to 11 volts and again that should be effectively 95% of the 12vt maximum uh the EMF on that battery if you found this lesson helpful consider giving it a like happy studing