Motion of a charge under two similar charges

Added: 2026-05-12

[00:00:06]So dear friends, in today's Shanka Samadhan, I have a question from a student, Nithya Tulasi.

[00:00:16]The question is on electrostatics. It's a long question. I will not read it here.

[00:00:22]And the four options are given, and the options are all in graphical form.

[00:00:29]And Nithya has solved this problem and gotten the right answer also, but still I congratulate Nithya.

[00:00:40]You have You are keen to understand deeper significance of the shape of this right option. So let me tell you Let me tell everyone the question.

[00:00:54]The question is that we have a two charges on the Y axis.

[00:01:02]Here is X axis. Plus Q here and plus Q here.

[00:01:08]The two equal charges, same sign, are placed symmetrically here and here. And a negative charge, say minus Q1.

[00:01:18]A negative charge comes from large distance from the origin in the negative side, and then it goes like this this this this, crosses the origin and then goes up to large up to large distance on the X axis.

[00:01:35]And the question is to plot the acceleration of this charge minus Q1. These two plus Q and plus Q are fixed. And this minus Q1, which is moving, what is the acceleration in influence of these two charges? For that, four options are given, and the right option is like this.

[00:01:58]The right option is is like from the large distance it goes, and then it goes up, and comes down, goes through the origin, to the negative side, and then it goes to large distances on the X axis.

[00:02:19]And acceleration is positive in the direction of the motion, that means positive X direction is also to be taken as the positive of acceleration.

[00:02:32]Okay?

[00:02:33]Now, the idea is quite simple in the sense that if you have minus Q1 at large distance from plus Q and minus Q on the negative side or positive side, the distance is very large. So each of these charges will exert a very small force, negligible force. So there will be a negligible acceleration. Acceleration is resultant force divided by the mass.

[00:03:03]So here it should be zero. Here also it should be zero. So here it is zero, and here also it is zero. Large distance on the positive side, large distance on the negative side. Don't look at the asymmetry of this figure.

[00:03:17]All right.

[00:03:19]And at the middle point, at this origin, the electric field due to our electric force due to this Q is attractive on the in the Y direction.

[00:03:31]And from this Q, the attraction, the force will be in the this negative Y direction. So there's no component in X direction. Y direction anyway cancels out. So you have zero resultant force here, and therefore acceleration here also should be zero, and therefore it is zero.

[00:03:52]Now, here as the negative charge comes towards the origin, the magnitude of force or the acceleration the the motion is in this direction, positive X direction. So the acceleration is positive. And here here, although the particle is going in this direction, the force because of this plus Q and this plus Q, if your minus Q1 is here, then the attraction force the attraction force will be in this direction.

[00:04:24]Attraction force will be in this direction, and so the resultant will be in the negative X direction. So the acceleration is also in the negative X direction. And therefore for this side, the acceleration is negative. This side it is positive. The magnitude the magnitude will increase first because it is accelerating in this direction. So that acceleration the distance is decreasing, force is increasing, and so on. But later on it has to become zero, and similarly on the other side.

[00:04:59]So this [clears throat] is okay. On the given in the given four options, this is the only option which satisfies these needs that the acceleration should be zero here, should be positive here, should be negative here, and then again zero here. So this is the correct option. No problem.

[00:05:22]Now, the question is why this kind of shape?

[00:05:27]And why this kind of shape? When does it turn? When does it become maximum? And so on. Why this kind of shape? On the basis of three points, large distance negative, origin, and large distance positive, one can choose the option, correct option out of the given four.

[00:05:45]But if you just don't you want to understand this whole figure, why it is like this, we can do that very simply.

[00:05:54]Okay, so also there is a constraint in the in the mail. She has written that don't base your arguments on a lot of mathematics. So [snorts] okay. Okay, I It was easy to do the mathematics and find all the nature of this curve, but some mathematics has to be done anyway.

[00:06:16]Now, if I look at this point, any point any point. Let's say here.

[00:06:22]Let's say here. So the force due to this plus Q is in this direction.

[00:06:27]Is in this direction. Force by this is also in this direction.

[00:06:32]Now, if I'm going from here to here, let's see what happens.

[00:06:36]Let's see what happens if I'm going from here to here. What happens to the force?

[00:06:42]The [clears throat] resultant force is in X direction. So this angle theta, if this angle is theta, and this is F, say F1, and this is F2.

[00:06:55]Then when the particle is here, the component which is driving X component will be F1 into cos theta, and also from the negative this side. So it will be two times F1 into cos theta.

[00:07:10]So it is F and cos theta. F cos theta. That is deciding the resultant force. That is deciding the acceleration.

[00:07:21]All right. When you move from here to here, what happened to the force F, magnitude F? Since it has gone closer, the magnitude has increased.

[00:07:31]Okay?

[00:07:32]The F2 is greater than F1. Here F2 is greater than F1. So the force has increased. What happens to cos theta?

[00:07:41]Theta has increased. Theta has also increased. This theta is larger than this theta. And if theta increases, cos theta decreases.

[00:07:52]So [snorts] F is increasing, but cos theta to one, if you say, decreases.

[00:08:00]So that is the interplay. That is the competition. Once you are moving from the negative side, the force is increasing, cos theta is decreasing. The force is trying to increase the resultant resultant, but the [laughter] cos theta is trying to decrease.

[00:08:21]So when you are large at a very large distance, suppose I'm here, and I'm somewhere here.

[00:08:30]So it's a very large distance, and this angle is very small. Already it is very small.

[00:08:36]Now, if I move a little bit here, little bit here, the angle is hardly changing anything.

[00:08:44]The force is increasing. The distance is decreasing.

[00:08:48]Okay?

[00:08:49]>> [snorts] >> If I move delta X distance here, originally I was here, and now I'm here.

[00:08:56]So if I'm moving delta X, this distance between this and this is also almost almost delta X. Almost because very small. If it is theta equal to zero, then yes, if I'm moving delta X, the distance is decreasing by delta X.

[00:09:13]And since this theta is very very small, this decrease is also also delta X type.

[00:09:20]Okay?

[00:09:21]But the change in angle is very very nominal, very very nominal.

[00:09:29]So cos theta is not effective here much, and the force is effective. Therefore the force is increasing. So that has to be understood. At any point at any point, here also here also here also here also. As you go on the positive side, from this negative side to origin towards origin, your [snorts] When you are here, when you are here, then your F1, this is dominating.

[00:09:59]Theta 2 is almost same as theta 1.

[00:10:03]Because I have moved delta X distance, so the force has increased, but the angle has the cos theta is almost the same. But as you come closer, this theta change is now taking over.

[00:10:17]This is becoming stronger.

[00:10:19]Right? And when you are here, the distance change is very nominal. Suppose this is that point and I am just below that.

[00:10:29]>> [clears throat] >> And then I move delta X. I move delta X.

[00:10:32]I reach here.

[00:10:33]And then what is the what is the what What is the separation? What happens to the separation? This separation and this separation. If this is my, let's say, distance is S1, this is S2.

[00:10:48]How much has this changed?

[00:10:51]Very very nominal.

[00:10:53]Very very nominal. If I drop a perpendicular here or a a circular arc here of of this radius S1, it will just meet here. So the distance change is not much effective here.

[00:11:08]The distance change is not much effective here, and and the theta of course here theta is is dominating and this they two are balancing and that is why your total is is zero.

[00:11:25]Okay? But if you are at a larger distance, let's say you are here, and then from here you reach here, this point.

[00:11:35]Then from here to here if if I see, if it this this is delta X, even the separation is also the change in the distance is also delta X. And then again the theta will be less dominating.

[00:11:49]Huh? So you have on this side, of course this is just a mirror image on this.

[00:11:55]What happens to this side, that same thing will happen to that side. If I understand this, that is enough.