Potentiometer - calculating internal resistance of a cell

a common 9 volt battery provides a pretty high internal resistance to the flow of charges because of which the maximum current that you can ever draw from that is very tiny and so this is safe and you know we can let kids play with that on the other hand if you take a car battery which is just 12 volts not that much difference compared to this battery it has an extremely tiny internal resistance now because of that the amount of maximum current you can draw from the battery is lethal so forget kids even adults are not supposed to play with this so clearly internal resistance plays a major role in the working of a battery and so the goal of this video is to figure out how do we measure this internal resistance of a battery and from the title of the video you may have already guessed we're going to use a potentiometer but we will not start with the potentiometer circuit because you know that's always been confusing me instead we'll first logically understand how to measure the internal resistance and then we will build the circuit together so let's say my friend throws a random unknown battery at me and asks me to calculate its internal resistance how do i do that i don't know it's emf let's say i don't know anything and i'm not gonna read over here it's given but i'm not gonna read because i'm not gonna cheat so the question is how do i calculate how do i measure what experiment will i do to calculate the internal resistance so the first step would be what i would do is i would first model this battery i would say let's imagine that this particular battery internally is made up of an ideal cell which has no internal resistance of its own an ideal cell of emf e and along with that um all the internal resistance is connected in series with that cell that's how we can imagine it and that's what we always do in a circuit right we concentrate all the resistance at one point and we are concentrating all the emf of that battery at one point all right now how do i calculate r from here well the first thing would be to build an equation for that so to build an equation let's first attach it to you know let's let's make a circuit out of it and let's see if you can use ohm's law to build an equation so imagine i attach it to some resistance over here i will show it as a bulb so let's say i attach it to some resistance or some bulb which has a resistance capital r now comes the question can i build an equation with small r in it and the answer is yes all i have to do is figure out what's the voltage across the battery and that equation will have small r in it all right so can you try and give this a shot can you figure out you know just by using ohm's law um what is the voltage across these two points of the battery right now and i'll give you a clue the voltage across the battery is the same as the voltage across the resistor all right with that i want you to try give it a shot and remember uh the answer should be in terms of e capital r and small r all right so you prime pause the video and see if you can you know give this a shot all right let's see so the voltage across the battery i'm just going to write as v bat this is not a pad it's a battery you get it right so the voltage across the battery is going to be the voltage across the bulb and what's the voltage across the bulb from ohm's law it's going to be the current i through the bulb times the resistance of that bulb all right now i don't want to i don't want i in this i want to somehow bring e and r into this so how do i get rid of that i will again use ohm's law in this circuit to figure out what the current is now this time i will use ohm's law between these two points because i know what's the voltage across these two points that's e i also know what's the resistance across these two points that's r plus r so if i use ohm's law i equals e divided by the total resistance r plus r so what i can do is i can get rid of this i and i can say that's equal to the emf e divided by r plus r and i've built my equation my first step is done now let's see what i need to measure i need to measure small r so let's see what are the other things i know i know the value of capital r now you don't have to use a bulb right i used a bulb but you can use a standard resistance so if you use a standard resistance in a lab capital r value is something that you know so that's not a problem okay next how do i calculate the voltage across the battery well we have potentiometer with us it's an excellent measuring device so all i have to do is hook this up to a bulb and measure the voltage across the battery using that potentiometer and i can do that so to calculate this measure measure the voltage across the battery with the bulb i can directly measure it with potentiometer so measure with the bulb so this i can measure in my experiment with the potentiometer okay now the interesting question is what about e emf how do i measure that because the emf at least in this drawing this you know ideal drawing it's the voltage voltage between this point and this point that you can't directly measure so what do you do again i want you to pause the video and think a little bit about this how would you measure e because if we do that we are done okay have you given it a shot the secret is you get rid of that bulb and recalculate the voltage across the battery let's see if you know calculate this would this point would have the same voltage as this point there is no resistance in between but now the voltage at this point will be the same as the voltage at this point because i'm not drawing any current from this battery right now and so there is no potential difference over here in other words now if i calculate the voltage across the cell without any drawing any current from it now that voltage is going to be the emf so all we have to do is measure the voltage across the battery but without measure it without attaching it to a bulb and just to quickly remind you this is the battery equation which basically says when you connect to anything across a battery the voltage that you measure drops it becomes smaller than the emf because there's a small drop that comes across the internal resistance anyways now we can bring in our potentiometer and just to quickly recap a potentiometer is basically a battery connected to a slide wire maybe with some resistors in between and the whole idea is if you want to calculate you know voltage between any two points then you attach it with respect to a galvanometer and the whole idea is the voltage when and as you slide this and as you slide this particular slider when the galvanometer deflection shows zero say at around this point the voltage across these two points must be exactly equal to the voltage of this part of the wire so to measure this all i have to do is calculate the voltage of this part of the wire and how do you calculate the voltage of this part of the wire well you calculate what that length is and if we know what the voltage per meter of this wire is which we call the potential gradient then we just multiply it by l so the voltage required would be the potential gradient which is the you know voltage per meter of this wire into the length of you know the balancing length and if you're not familiar with this concept or you know you need a refresher feel free to go back and watch our video on potentiometers we have explained this intuitively logically where all of this comes from all right we have recalled our potentiometer we have our theory i want you to think about how you connect these two and what experiment will you do what procedure will you carry out to figure out the internal resistance remember it's okay to be wrong but wondering and exploring is super important so please please give this a try think about what circuit will you build now all right if you've given this a shot let's see let me first bring back the bulb okay so how do we attach our potentiometer so here's our potentiometer slide wire since i want to calculate in both these cases i want to calculate the voltage across the battery i'm going to connect my galvanometer and the other circuit also other wire also across these two points so here's how i'm going to make the connection okay remember i want to do two measurements first measurement what i will do is without the bulb to calculate the emf i need to find the potential difference here without the bulb and the way to do that i mean this is a practical circuit i don't want to detach again so what i'll do is i'll just introduce a switch in between okay think of it as a key if i remove the key the bulb got detached if the if i add the key the bulb got attached so i first detach the bulb and calculate what the potential difference across the battery is and how do i do that well i go back to my galvanometer i notice there is a deflection right now which says that the voltage here is not exactly the voltage of the battery so let's start sliding as i slide to the right notice the deflection reduces reduces reduces reduces reduces reduces and here we go zero because the deflection is zero i know the voltage here must be exactly equal to the voltage of the battery which is the emf so now the question is what is the voltage here to do that i'm going to calculate the length of that wire so let's say that length is l1 and just like what we saw before now length is n1 l1 that means the voltage over there the voltage across the battery that's going to be phi which is the voltage per meter times l1 and you may be wondering well do i know the value of phi well usually when a potentiometer is given to you the value of phi is usually given but over here you will see that's not even required as you will understand okay now the next step is i want to calculate what the voltage of the battery is with the bulb so i'm going to plug in the key now to connect the bulb over here and my question to you is when i plug in the key do you think the galvanometer deflection will still be zero will it still be balanced or not take a moment to think about this let's see it's not the reason it's not is because as we discussed earlier now the voltage across the battery has reduced so we are not at the balancing length so i have to slide again should i slide to the right or should i slide to the left let's see if i slide to the right oh the reflection increases of course because the voltage is reduced here right if i slide to the right i'm increasing the voltage difference even more so i need to slide to the left the voltage has reduced i need to reduce the voltage here as well so i'm going to slide to the left slide slide slide and here we go so now i'm getting the new balancing length this balancing length refers to the voltage across the battery with the bulb and so if i call this length as l2 then this voltage which is the voltage across the battery with the bulb is going to be phi times l2 and guess what we are done because notice in this equation phi cancels out and you can do the algebra right again feel free to pause the video and do the algebra and see what r equals to in terms of l1 and l2 and if you do that i'm pretty sure you can you'll end up with this equation but again i want to stress you don't have to remember the equations how many equations will you remember you know how to do the algebra right so you don't have to remember the equations stop remembering equations concept is all that matters so to quickly summarize the first thing we did to measure the internal resistance is built the battery equation which can be done using ohm's law and the insight over here is that the voltage across the battery reduces when you attach anything across it and when you don't have anything across the voltage of the battery is the emf once we got that knowledge then we went ahead and used the potentiometer first we use the potentiometer to calculate the voltage of the battery without the bulb that gives us the emf then we went ahead and hooked up the bulb and recalculated the voltage across the battery since that voltage has reduced a little bit we got a little smaller value of the balancing length and once we got that we plug in we simplify and that's how you measure the value of r and of course in your lab you'll also have a switch over here you don't want to keep this primary circuit on all the time and you might have another internal resistance over here which i've ignored and of course you would be you know doing this ex doing this experiment two or three times and calculating the average value that you would get