 Let's solve a question on electric potential energy. The image below shows the electric field lines of a positive point charge. Consider the potential energy of the system UM or UN when a small negative charge is present at the point M or N. We need to choose the correct statement from the following. As always, hit pause, try this one on your own first. Alright, hopefully you thought about this. Now let's try to see what the question is trying to tell us. So we have a small negative charge, we have a small negative charge and this is present at the point M and N and the potential energy when the charge is at N that is UM denoted by UM and when the charge is at N that is denoted by UN. So which one is bigger, UM or UN? Now to answer this we should ask ourselves when there is a small negative charge in the presence of a fixed big positive charge where and why will the negative charge move? Now to answer that we can say that it will move towards the positive charge that is where it will move because of the electrostatic force of attraction opposite charges attract and it's moving because there is a force of attraction. The positive charge has an electric field which goes away, it radiates outwards and because the negative charge is present in that electric field it experiences a force not in the direction of the electric field because it's a negative charge in the opposite direction towards the positive charge. And finally we can ask ourselves when these two charges come closer to each other what does that do to the potential energy of the system? Now to think about that we can draw a parallel, we can think about a block or any ball which is dropped from height and it goes rushing towards the earth because of the force of gravity. In that case what do you think happens to the potential energy of the system? Well, turns out it decreases. The ball moves towards the earth and the earth really does move but it's just negligible because of the mass of the earth. So similarly there is an attractive force here as well between the negative charge and the positive charge. So when you let them be, when you let go of the negative charge it goes towards the positive charge and in doing so the potential energy of the system decreases. So if we have a negative charge which is kept at M or N we can ask ourselves in which direction should the negative charge move so that the potential energy of the system that is the positive and the negative charge the potential energy of the system decreases. Let's try and make a tunnel. Let's say we have a tunnel over here and the charge, the negative charge is only restricted to move between these two points. So if we leave a negative charge at point N due to the electrostatic force of attraction that negative charge will start moving towards point M and that tells us that the only reason that the charge is moving from N to M is because doing such movement going in that direction is reducing or decreasing the overall potential energy of the system. So just by that logic we can say that the potential energy at M that must be less than the potential energy at N. But let's try and also answer this mathematically. So we can use the relation of the relation of potential energy and we can take a positive charge. We can say this is plus Q plus Q and minus Q. So there are two charges there is plus Q and minus Q. So potential energy at M, potential energy at M we can write this as this is equal to, this is equal to minus capital Q into small Q divided by 4 pi epsilon naught into Rm Rm. We can say that is a distance and potential energy at N, potential energy at N this is equal to minus capital Q into small Q divided by 4 pi epsilon naught Rn. Now we can see that Rm really Rm is less than Rn. So in the denominator if we compare these two if Rm is less potential energy at M there should be more and if Rn is more Un should be less. But be cautious because these are negative numbers. So a higher negative number really is less than a lower negative numbers. Think about this like minus minus 50 is less than minus 30. Similarly a higher negative number that is Um is less than a lower negative number. Now we can also draw a graph and try and make sense of it further. So if you draw a potential energy distance graph the curve looks like this. You have U and R, U versus R. So what this really means is that the point M, let's say point M is over here, point N is over here, we know that Rm is less than Rn, R is greater. So the potential energy at M that really is this value. This is Um and the potential energy at N that is really, that is really this value. Even from the potential energy distance graph we can see that Um that is really less than Un. So turns out Um minus Un should be negative and the answer is option B. Let's look at one more question and that will be on work done. So here it is. Okay the image below shows the electric field lines of a positive point charge. Here it is. These are the electric field lines. The work turned by an external agent in slowly bringing a small negative charge. Okay this one's negative. And in brackets it's written zero acceleration from infinity to the point X is W. Choose the correct statement about this work. Is it positive, negative or is it zero? Alright let's think about it. So a charge is being brought from infinity to this point X and it is brought with zero acceleration. What is zero acceleration? Well we know from Newton's laws all the way back to Newton's laws that when the net force on any object when it is zero then that object is moving with zero acceleration. There is no, there is no acceleration with which the object is moving. So is the net force on this negative charge zero really? Let's think about the forces that are acting on this negative charge. So this is the negative charge, positive charge, there must be an electrostatic force of attraction. So that force is directed towards the center. Let me draw that over here itself. This force is directed towards the center. Let's say this is FE. And now if there is a force directed towards the center but someone is, someone is bringing a charge from infinity to this point with zero acceleration. So there must be a force that is balancing this electrostatic force of attraction and that must be in the opposite direction. Must be in the opposite direction like this. We can, we can call this F external and external agent is doing this work. Now in order to move the negative charge from infinity to a point X with zero acceleration, this F external must always be in the direction opposite to the electrostatic force of attraction because this will be always directed towards the center. And we also know that the charge is being brought from infinity to this point. So the displacement really is in this direction. This is the direction of displacement. So this force, force external and displacement, they are in the opposite directions. And from work done, we know that work done is F dot D. This is FD cos theta. Theta here would be 180 degrees because displacement and force external, they are in opposite directions. Cos 180, that is minus one. So work done by external agent, this really is less than zero. It's a negative, it's a negative value. All right, you can try more questions from this exercise in the lesson. And if you're watching on YouTube, do check out the exercise link which is added in the description.