Cells with internal resistances: Worked example

[Laughter] let's solve some problems with cells connected to a simple circuit so the first one we have a cell with emf 10 volt it has an internal resistance of two ohms and it's connected to a resistor of three ohms our goal is to calculate this thing called terminal voltage something we've seen before and what the current in the circuit is going to be so there's going to be some current since this is the positive terminal the current will be this way to find out what that current is so how do we do this well i'm going to start by something that we've already seen before we've seen how to calculate terminal voltage before we saw that the terminal voltage i'm going to write that equation then first so the terminal voltage equals the emf of the cell e minus i times r and we don't have to remember this equation we can do this logically something we spoke about in a previous video but just to quickly recall what is this equation saying the terminal voltage is the voltage across the ends of the battery terminals of the battery all right and that represents the energy gained by a coulomb of charge when it when it moves from one end of the battery to another now what is that equal to that equals the emf which represents the energy transferred by the battery to that charge minus the heat lost due to the internal resistor so as the charge moves across the battery it gains some energy that is the emf and it loses some energy due to the internal resistor and that's why when you subtract them you get the net energy gained and if this concept is not familiar to you or you need more refresher on this then feel free to pause the video go back and watch the video on cells emf terminal voltage and then come back over here okay but if you're clear on this let's continue so let's see if we have everything we need we have the emf i want to calculate terminal voltage i need to find out what this vt is we have the emf that's 10 we have r that is 2 that's the internal resistor but we don't have the current hmm i don't have the current so i need to first find out the current and the question now is how do i calculate current over here all right can i do this here's my question and i want you to tell me where can i do this can i say hey the battery voltage is 10 volt so can i just say this this voltage is 10 volt and then i say a ohm's law so i is going to be equal to v over r v is given r is given so can i just say i is equal to 10 divided by 3 something like 3.33 amperes can i do this um pause the video and think about it if you've understood this concept you'll be able to answer whether i can do this or not and why so pause and think about it all right so you can't do this the reason why you can't do this is because the voltage across the resistor will not be 10 the voltage across the resistor is the terminal voltage it's going to be 10 minus something right and we don't know what that is we have to figure that out are we clear so all i'm saying is although it looks like the emf although it feels like the voltage across these two points is 10 volt it's not because there is internal resistance and some voltage is lost over there some energy is lost over there so that's why to make to make sure that we don't get confused like this we don't confuse emf and internal uh terminal voltage what we like to do is while drawing this we like to draw the internal resistance over here so let me show you what i mean so let me get rid of this and what we like to do is we like to draw that internal resistance here so this is our internal resistor and we like to imagine this whole thing now is our battery this is our battery so you can imagine like a normal battery it's the positive side the positive size has that button so this is our battery and now things become very clear so you can imagine your battery is made up of a perfect cell of 10 volt which has no internal resistor and then you can imagine there is a resistor connected in series with it of 2 ohms and now this voltage you can say is 10 volt but you can clearly now see the voltage across these two points will not be 10 volt it will be less than 10 volt it will be 10 minus whatever i get i times r is this clear and so whenever we have cells just to make things easier so we don't get confused all we have to do is attach an internal register in series with it and then imagine the whole thing as a battery and then solve the problem all right so with this let's see if we can calculate what the current is how would we do that well now what i can do is now i can say look i have a cell of 10 volt over here and i know the voltage between these two point is 10 volt right because i've removed the internal resistor and put it out over here so the voltage between these two point is 10 volt and i can say now that the resistance between these two points going from here to here i can go from here to here i can go from here to here the resistance between these two points if i go from here to here is 3 plus 2 they are in series so i know the voltage between these two points i know the resistance between these two points i can now use ohm's law so why don't you pause the video one more time and see if you can now solve the value of solve for current all right let's do this so ohm's law says v equals i r so i is going to be v over r so i is going to be here v is the voltage between these two points that is the emf see between these two points the voltage is just the 10. in these two points the voltage is this which i don't know okay so between these two points the voltage is just 10 that's the emf 10 volt divided by the resistance between these two points the total resistance and that is 3 plus 2 ohm that's 5 ohms and so the current is going to be i equals 2 amperes and there we have it we have the current as 2 amps and now that i have the current i can plug it in over here and calculate the terminal voltage again if you want you can pause and try so that you you you do it all right so over here vt is going to be emf which is 10 volt minus current i which is 2 amperes times 2 which is internal resistance so that's going to be 10 minus 4 and that's going to be 6 volts 6 volts so what we are saying is that the voltage across these two points that's the terminal voltage that we are seeing is 6 volts and again think about it why is that 6 volts why is it not 10 because out of 10 4 is lost as heat when the current goes through this resistor and that's why it's 10 minus 4 is equal to 6. now another way just to show you there are multiple ways of calculating this now another way i could have calculated the terminal voltage is i could have said hey the voltage across the ends of the battery is the same as the voltage across this resistor external resistor right because this point is same as this point this point is same as this point so i could have said hey the voltage across this resistor is the terminal voltage and i can apply ohm's law here is that making sense so think about it what would be the voltage across this resistor if i used ohm's law i would get that voltage across this resistor is going to be just ir i just found out i that's 2 multiplied by r which is 3 and that is what i get i get 6 volts i get the same answer as i did over here you have to because voltage here is the same as voltage here does that make sense i know at first this might be all confusing but you know as you practice and you can practice more with our exercises this will make a lot of sense so just for practice let's do one more problem over here we are given a battery again with emf 5 volt and some internal resistance 1 ohm but this time current is given to us and we are asked to calculate again what the terminal voltage is and what this resistance is so again can you pause the video and see if you can try this yourself first again there are multiple ways to approach this so feel free to try your own approach all right the first thing i like to do is just to make things easier for me i'm going to add this internal resistance over here add this internal resistance so this is our one and imagine this is our battery this is my battery all right and so i need to first calculate the terminal voltage i'm going to say well i know now how to calculate terminal voltage terminal voltage vt equals the emf minus ir so emf minus i times r and i know all of them the current is given to me i know emf i know internal resistance oh i can directly calculate so emf is 5 minus i is 2 and r is 1 so 2 times 1 is 2 and so the terminal voltage becomes immediately 3 volt and so what does that mean that means the voltage across these two points of our cell is 3 volt why is it three why is it less than five because it's charge gains five volt but then it loses two volts to heat and as a result you get only three volts so i got my terminal voltage now comes the question how do i calculate the external resistance and again i want you to pause the video if you haven't tried this pause the video and see if you can try this now we know the voltage across the battery which is the same as the voltage across the external resistor so feel free to try this now all right let's do this so i know that this voltage is three so this voltage must also be three volt right i know that and i know the current is also two two amperes i can use ohm's law so ohm's law says v equals ir so let me just write ohm's law v equals ir and so r is equal to v divided by i i know what v is that's 3 volts i know what i is that's 2 amps and so my resistance has to be 1.5 ohms so that is my internal oh sorry that's my external resistor all right if you've come to this fight i have a bonus question for you before we wind up the question for you is in what is the maximum current i can ever draw from this cell imagine you could change this resistor to whatever value you want my question to you is what's the maximum current i can ever draw from this cell can you pause the video and think a little bit about this again this is like a just to get practice of this whole logic all right let's see if i want to get maximum current the way i'm thinking is i need to try and draw as much current as possible so i need more current the question i ask myself is to get more current what should i do with the resistor well i should decrease the resistance now i don't have a choice with this this is fixed this is inside the battery but this i can decrease it i can make this resistance smaller and smaller and if i make the resistance smaller i'll draw more and more current what is the smallest value i can make well theoretically i can make it zero right so let's see what happens if i make this resistance zero so let me draw another circuit for that or you know let me just do that over here if this let me draw another circuit so if this was zero so it's sort of like a short circuit i'm not attaching any resistor over here i'm short circuiting it and here's my battery and here's my internal resistor what will be the current in this situation that should be the maximum current whatever current i get in this situation because i have the minimum resistance so can you pause now again and see if you haven't tried it before if you didn't get it before can you try now what would be the current in this situation well let's see again to calculate the current what i'm going to do now is i'm going to say look i'm going to consider between these two points because i don't have an external resistor i'm going to control between these two points what is the voltage between these two points it's 5 volt what is the resistance between these two points it's 1 ohm right from here to here there's only 1 ohm and therefore now the current which is the maximum current it's going to be the voltage phi divided by the internal resistance 1 and that's going to be 5 amps so in this from this cell the maximum you can ever derive current is 5 amps you cannot get more than 5 amps and to do that you have to short-circuit the battery and that's bad practically that's bad because all of that energy is dissipated over here because there is no external resistance all that energy is dumped into the battery and so basically the battery gets heated up so this is a bad idea to do that but this is how you calculate the maximum current