 Two charges A and B are fixed 25 centimeters apart as shown below Where should a third charge be placed such that it would be in equilibrium? So what does it mean for a charge or for anything to be in equilibrium? What does that mean? Well equilibrium in physics means that there is no net force acting on that object. So no net force So we need to find where should I keep a third charge in this universe Where the net force due to these two charges these two charges are gonna pull or push on it But the total force on that should become zero Now you may be wondering well Why don't A and B push on each other and they are doing that but it's given that it's fixed in place Right so that you can imagine their nail down But our third charge you need to figure out where should I keep such that the net force over there should be zero Now before we before we start another way to think about it is we could say look if the net force has to be zero That means the electric field at that point The net electric field at that point must be zero right if an electric field at a point is zero It means the net force over there is zero So our question now is slightly changed it our quick our goal now is to figure out where in this universe is The total electric field due to these two charges zero Now you may be wondering how do we do that? Right, I mean we know the formula to calculate electric field due to each charge I mean let's begin over there We know that electric field equals k q or r squared and we know if there are multiple charges Then we use the superposition principle and add them up But how do I know where are they gonna add up to become zero? I mean how do I even begin thinking about it? So here's how we can start For two because there are two electric fields for them to you know together cancel out They have to be equal and opposite to each other think about it If two vectors electric fields are vectors if two vectors have to cancel out There's only one way for that to happen. They have to be equal and opposite Based on just that we can narrow down we can narrow down where our where that point is going to be For example, I know for sure with 100% certainty that at this point electric field won't be zero You know, how do I know that because just think about it the electric field created by charge a would be away from it along the line Joining and from B would be again away from it along the line joining and forget about their magnitudes These two are not in the opposite direction. So they will not cancel out So I know electric field over here won't be zero similarly electric field here also won't be zero same reasoning here Also won't be zero the only place electric field can be zero is when the two Forces or two electric fields are along the same lines Along the same line and that can only happen on the line Joining a and B. So that's a good start. We now know it's not not anywhere in the universe It's got to be somewhere on this line Joining a and B only there there's a possibility the two field lines the two field vectors can be equal and opposite Does that make sense? Okay, but even here we can further narrow it down. See on this line. There are three options It can be somewhere towards the left of a it can be somewhere in between a and b and it can be somewhere towards the right Where do you think it's going to be and I here? I want you to pause the video and think a little bit about this use the same idea and see in which Which of the three regions can the electric fields be equal and opposite pause and try All right, let's start in this region if I pick some point over here Then what would happen when I add up the electric fields due to these two charges? Well, this charge is going to produce an electric field away This charge is also going to produce an electric field away and notice there in the same direction They're not going to cancel out. So there is no way electric field can be zero over here So it's definitely not in this region and Similarly, I hope you agree it cannot be in this region as well because again if you consider any point over here This is going to push away Electric field is going to be away. This is also going to put electric field away Both are positive charges. So anywhere you go on the right electric field is now going to become zero But what if you consider somewhere in between again pick some random point in between and Now if you consider the electric field due to this charge and let's be a little bit more careful now Due to this charge at this point. It's going to be away from there So it's going to be away and Electric field due to this charge over here is going to be away from that charge. So that's going to be away and look They are in the opposite direction So it can cancel out Somewhere over here Therefore now we know that the electric field can be zero only somewhere between a and p. It's incredible Right even without doing any maths just by logical deduction. We know in this entire universe Electric field can be only zero somewhere between a and b on the on the line joining them Which is I don't know super powerful, right? Super powerful. Okay, but now let's figure out Exactly where that's going to be. So let's do the math So let me get rid of all of that. So now I need to figure out. Let's say that point is P Let me call that point as P That is the point where the electric field is zero. The question is Where exactly that point is going to be how do I figure that out? Well, whenever like, you know always like how you do it in maths or physics whenever you don't know something We take that as X So we want to know where this point is that means we need to know its distance from a or its distance from B So let's call that distance as X. So let's take this. Let's let's call this distance from a as X and our goal now is to find what X is But how do I do that? Well, I know the condition For point P the electric field has to be zero, right? And that means the magnitude of the electric fields have to be equal to each other So if I call this is the electric field due to a because a is producing this and I call this as the electric field you to be EA in magnitude EA in magnitude should equal EB I'm only considering the magnitude. I'm not considering the direction. We've already taken care of the direction They're opposite So the magnitude has to be equal to each other and for that and how and what do we do next? Well, we know electric field formula so we can plug in and we can hope to calculate X So again, good idea to pause and see if you can try and arrive at the value yourself All right, let's do this. So electric field you to a is going to be KQ by R square So it's gonna be K. I'm not going to substitute. You'll see why K into Q Q is the charge due to a so that's gonna be two nano coulomb again I'm not gonna substitute for nano and you will see why Divided by R squared R need to be careful. There are too many distances over here Let's not get confused R is the distance from the charge to that point we're considering a so from a to that point That's X. That's what we want. So divided by X So that equals What is eb eb is going to be K Q Q is the charge at B. Okay, I'm not gonna Can you see why I'm not substituting? Can you can you see now? Divide by R square. What's R for for this charge? Well, again, R is a distance from this charge to that point and That distance is well, the whole thing is 25. So this is X. So this is gonna be 25 minus X So that's going to be 25 Minus X or the whole square. I forgot to square it. There's a square over there KQ by R square. Okay, and again, notice that this is in centimeters We need to be always careful about the units But I'm not gonna worry about it and you'll see why now when I'll do it I'll tell you why now because when you're equating you can cancel stuff out. So K cancels Nano cancels the centimeters square and the centimeter square over here also cancels So I don't have to worry about anything the coulomb cancels all the units cancel out. Okay, so that's great So all of those things cancel out. So what remains now is just pure algebra So I have let me read that over here. So I have two over X squared equals 18 divide by 25 minus X the whole squared and so I can do goes nine times and I can take square root on both sides because there is a square. So I get one over X Equals three over 25 minus X And just to save space, I'm gonna do it over here now. So just algebra 25 minus X equals 3x so 25 equals 4x So X equals 25 divide by 4 and that is 4 6 are 24 or 0.10 for two zero eight six point two five six point two five what Well, we cancelled everything or it has to be centimeters, right because because the distances were in centimeters So we know this is six point two five centimeters and there we go We have our answer X equals six point two five centimeters so that point P is two point six point two five centimeters from a and that's the place where if you keep a Third charge, it's gonna be in equilibrium. It's gonna stay in equilibrium. And if you notice something you can see that that point Like we drove we drew over here. We didn't draw it properly. It's not at the midpoint It's very close to point a that point actually somewhere should be somewhere over here And then if you think about it that kind of makes sense Because for the magnitudes to cancel out because B has a higher charge You need to be farther away from it So hopefully it kind of makes sense that our point that we were looking for is actually closer to a compared to B All right, how about we try one more just to check our understanding and this is fun also, right? So almost almost the exact same question except let's say I make this charge negative Okay, what do you think is gonna happen? Let's do the same process again But a little faster we now know you can guess that has to be the point of equilibrium where the electric field is Zero that point has to be again somewhere on the line Joining these two charges, but again, where is it gonna be? Do you think it's gonna be the same as before or maybe somewhere else again? Can you try and pause and think a little bit about this? All right, let's do the same exercise again. Let's start with this point somewhere over here If I take some point over here now this charge Well, it's gonna electric put the electric field away from it and This charge because it's negative. It's gonna put the electric field towards it. Hey, so it's possible It's possible that they can cancel out over here, right? So this is one possibility What about in between? Let's let's check in between if I check somewhere over here from a the electric field is going to be away from it and Due to be the electric field is gonna be towards it So over here now in between they're gonna add up so electric field cannot be zero now things have changed dramatically Interesting so this is not gonna be the place. What about over here somewhere? Let's see due to a electric field is Gonna be away and Due to be electric field is gonna be two words. So yeah, even here they may cancel out So now comes an interesting question. Where do you think in which region do you think the electric field would be zero? Maybe it might be zero in both No, if you if you see carefully, we can use something that we learned earlier. See in order for the electric field to be zero Not only is it enough that they are They're in opposite direction their magnitudes have to be equal as well, right? So if I consider This point over here Notice that this point will always be closer to point B the bigger charge Compared to the smaller charge since the bigger charge since this point is always bigger to the big closer to the bigger charge This electric field will always be Always be bigger than the electric field produced by this chart. Does that make sense? Think about it Wherever you go, I will always be closer to the bigger charge. These two will never cancel out. So this is impossible This can't happen But over here it's possible I can I can have some point where it is so close to that this small charge even though the charge is small It's close to it so that the electric field due to this one and this one can cancel out Okay. Now again, the next question is where exactly is it going to be and we're gonna do the same exercise And we're gonna do it a little quicker now So let's say that point is P and again, we don't know where that point is We're gonna call that distance as X X and now again the condition over here is electric field at P to be 0 We need electric field due to a in magnitude should equal the electric field due to B And in the spirit of time I already done my calculation. I'm just gonna show you what we get and again feel free to do this yourself first So just one thing you might notice over here is I haven't put the negative sign because I'm only considering the magnitude The negative sign is only to check the direction So for magnitude, I don't have to consider the signs and over here It's gonna be 25 plus X because it's so far away from B And then if I simps if I simplify which you can just pause and see you will get the answer to be 12.5 centimeters. So that means X in this case is 12.5 centimeters to the left of a that's where the electric field is zero So if I keep any third charge over here, it's going to be in equilibrium