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F was down, The displacement D was also down, And so we know exactly what this becomes, It becomes mg for our force D going down for the displacement. The angle between those is zero degrees, Cosine of zero degrees is one. And so we get a work of M times G times D. Okay, where D is some positive number. This is the work that gravity did. It is a number that is positive. It's greater than 0 because the force and the displacement are in the same direction. Alright, but how do we put that in terms of Y? This variable Y that we said is the height above the Earth. Well, if I look at D I can see that that is actually going to be the negative of Delta Y. Right, Y F is down here That's going to be a small number Y I is up there That's going to be a big number so that Delta Y would be a negative number and so I have to add another negative sign right there. But that negative sign can come out in front and we can write it minus mg Delta Y where that Y, that Delta Y, is Y final minus Y initial. And that should be clear why we had to put that negative sign there because this stuff in parenthesis here that's going to be a negative number. Right, Y final is smaller than Y initial so that will be a negative number we have to add another negative to make sure it's a positive number, but now we can work that negative right on through. M G Y initial minus Y final And that is still a number that is positive. Which is good. Okay. This is the work due to gravity. Now, let's do the analogy between this gravitational problem and an electric field problem. So, let's ask the following question before we do that. Let's say I have a whole bunch of charge that is sitting here on this plate. And now I take another plate. I'm gonna put it right here. And I give it a whole bunch of negative charge. In the region in between those two, is there an electric field? What do you guys think? Is there an electric field in there? Yes. Why? Because electric fields leave positive charges and end up on negative charges. So there has to be some sort of field that is pointing in that direction. Okay, it's everywhere inside but we're just going to draw one for simplicity. If I put a particle in there, let's say I put a positive charge right there. Is that positive charge gonna feel a force on it? Absolutely. Right, we know what it's gonna do. Positive charges are repelled by positive charges and are attractive to negative charges. Positive charges follow lines of E. And so there's going to be a force on this positive charge, F, which is equal to Q times E. If it's positive charge, it goes with the field. If it's a negative charge, it goes opposite the field. Okay, but that positive charge that heads down in that direction, we can say, oh that just moved a distance D. And let's use the same picture that we just had with gravity, let's say that we're gonna measure Y going up from the bottom. This is like our surface of the Earth. And now we started up here at Y initial, we end down there at Y final. Okay, positive charge going to move downward. And let's calculate the work done by this electric field E. Before the work was done by the gravitational field but now it's going to be due to this electric field E. Alright, we can do that. We've got work equals F D cosine theta. In this case, the force is just Q times E. The distance is D. the theta is again zero degrees, so electric field is down, charge moves down, everything's in the same direction. so the angle between the is zero and cosine of zero is, of course, one. All right, so we get the work is just equal to QE times D. It looks very similar to what we had before, right, in gravity we had work was equal to mg times D. So everything that we just did we can take advantage of and write this out as the following: QED is what? Well, that is a positive number Q is a positive number. E D are in the same direction. These are magnitudes in this equation, and so this is a positive number, just like we had with the gravity. All right, but we know what D is, just like we had before, it's gonna be negative Delta Y, because Y F is going to be smaller than Y initial. So this is minus QE Delta Y Or minus QE times Y final minus Y initial. That number smaller than one, That- I'm sorry that number is smaller than Y I, and so that thing is negative the negative sign out in front. Make sure that this is still a positive number so we're happy. But now again we can distribute the negative sign and write it as QE Y initial minus Q E Y final. All right, Let's rewrite some of that over here. We've got W equals. And let's multiply this out. We've got Q E Y initial minus Q E Y final. And now let's pull out the Q from both of those and write it like this: Q times E Y initial minus E Y final And at this point we're going to make the key substitution, which is the following: this thing we're calling V sub I. This thing we are calling V sub F. And this V is our electric potential So this becomes Q times negative Delta V Or W equals negative Q Delta V This substitution that happened right here, we introduced the electric potential V. Okay, this is not speed, okay. This is a new term, "electric potential." And what we said was we must have VI is just equal to E times Y I And we must have VF is equal to E times YF. And so if we put these things together what can we say? Delta V is just VF minus VI but that is E Y F minus E Y I. The E is common and so this becomes E Y F minus Y I and YF minus YI is the same as negative D. And so there is a relationship between V the electric potential and E And it's just this. The change in the potential is negative E times D. Okay. Any questions up to this point? This was a little bit heavy on the derivation but I wanted to make sure you understood where this stuff comes from. It's very similar to what we did in gravity. We're just talking about a new potential. Now it's the electric potential.

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