All right, guys. In this example, we're upping the ante. Now we're going to figure out what the potential difference is, not just to one charge, but due to 2 charges. So we've got this 5 nano nanocoulomb charge and a negative 3 nano coulomb charge on the same line. So I'm just gonna draw a quick little sketch right here. I've got this 5 nano coulomb charge, negative 3 nano coulomb charge, and I know that they're on the same line and that distance is 6 millimeters. Now, in part a, what I'm asked for is I'm asked for the potential directly between them. So in other words, this is gonna be the potential at VA. So let me do a better job at sort of highlighting that. So that is gonna be the potential at VA. Alright? So let's go ahead and do work that out.

We've got the potential at VA is basically gonna be the potential due to both of these charges at this location. But one of the things we know is that if this is half of the distance between these two charges, then that means that this distance right here, this 3 millimeters, is actually gonna be the same for both of them. So in other words, we've got the potential at point a is gonna be the potential due to the 5 nanocoulomb charge plus the potential due to the negative 3 nanocoulomb charge. And we're just gonna add those things up together because they're scalars. So we have that the potential difference over here is just gonna be k⋅5ra1+k⋅3ra2. But one of the things we notice is that we have the same exact Ks and RAs for both of these things.

So in other words, since both of these are the same, we can actually use a little shortcut that we've used before in reducing these potentials down to, like, even simpler forms. We have the potentials basically just equal to kra⋅(5-3). And it's only because we have symmetry. It's only because these things are the same distances that we can actually use this rule. So in other words, that the potential at point a is gonna be 8.999×109, and then this is gonna be divided by the distance. We have to be careful because this is 3 millimeters, which means it's 0.003, and then we're just gonna add up the charges. We have 5 nanocoulombs minus the 3 nanocoulombs.

So that means that the result here so both of these things together are just gonna add up to 2 nanocoulomb charges. Right? So you're just gonna add in both of these charges, and you get 2 nanocoulombs, which by the way can be represented by 2×10-9. That's gonna be in coulombs. Right? So that means the potential at point a is just going to be Let's see. I got 6×103, that's in volts. Alright? So we can use that shortcut for this case right here.

So in part b now, in part b, we're supposed to figure out what is the potential at this point b, which is now halfway between the charges, but now it's some extra distance above that line. So in other words, point b is somewhere over here, and we're supposed to be figuring out what is the potential at this point. So basically what we need to do is now that we have a different point, we actually have to calculate what the distance is to both of these charges. Now fortunately, what we've got here, is we've got a 3, and we know that this line right here is 4 millimeters above, so we can recognize this as a 3, 4, 5 triangle. If you wanted to figure out this r distance, if you didn't know if it's a 3 4 5, you could always just use the Pythagorean theorem. But in any case, we've got 5 millimeters right here for this distance and this RB is actually gonna be the same for both of these things. Right? Because it's symmetrically placed around around this, this axis.

So you can kinda use the same shortcut that we did here, but now we're just gonna sort of like do it a little bit quicker. So this VB is just gonna be krb⋅(5-3). And if you write it all out if we write it all out, we get that the potential at point b is 8.999×109. And now we've got the distance which is 0.005, and now we've got 2×10-9. And if you work that out, you should get the potential at this point is equal to 3.6×103, and that's in volts. So that is part a and part b. And now the last part over here, which I wanna do, let's say over here, is I need to figure out what the work that's done on a 1 nano coulomb charge. And let's see. We've got from the first point to the second point. In other words, we're going from point a over to point b. And we know that a work that's done on a feeling charge through a potential difference is negative q, which sets the feeling charge times the potential difference at this point, and this is the potential from a to b. So in other words, what happens is the work that's done is equal to negative q times the difference in the final minus initial potentials, vb minus va. If it had set the second point to the first, then we would actually reverse that. So it's very important that you do the right step here.

Alright. So we've got the work that's done. This is gonna be negative. We've got 1×10-9. That's the feeling charge that we have. This negative one_nanocoulomb charge is this feeling charge over here. And then we've got the potential difference. So v b was equal to 3.6×103. V a, was 6×103. And if you work all that out with your calculator By the way, you could have actually just figured out what the potential difference is. Just like I actually gotten the subtraction, and then you would just plug that in here. But I'm just doing it sort of like the expanded way. And if you work this all out, you should get a work that's done, which is equal to 2.4×10-6, and that's in joules. So that's the work done."]