Let's complicate it a little bit. Let's say we do the following. Let's say I take a positive charge and I take a negative charge and I put them reasonably close to each other. What should the net electric field look like for those two charges? Okay, this is a popular term called a dipole, two poles positive and negative, and now let's see if we can figure out what the electric field should look like. All right, you might think well we can draw this field and we can draw the field for the negative charge and we should be done. But that's not quite right, and why is it not quite right? Because these field lines really extend forever. Okay, we can draw another one and another one and another one, and it gets smaller as it goes out of course but they go on forever. The positive one goes on forever and it just gets shorter as you go. So you look at these two and you say, well that's got to be the field lines but when we draw electric fields, we need to draw the net electric field. And electric fields are vectors and so we have to add them up according to our vector rules. So at this point right here, we have two vectors. One going down the left, one going down to the right but we need to figure out what the net electric field is. What's the net vector sum of those two? And so this picture isn't quite right. And so let's draw it again but let's draw the correct picture. Okay there's my plus and minus charge. I obviously I'm going to have one electric field line that goes straight down from the top one to the bottom one. There's gonna be another one that comes out of the positive but that's gonna go into the negative and somehow I have to join these two lines. You might think it just goes straight out and then makes a kink and then it goes down, but physics never behaves that way, right. Physics is smooth and so the curve has to be smooth, and so it looks like that. And there's one on the other side that looks like that, and then there's another one here that looks like this, and another one there that looks like that and another one there that looks like that and so forth. Okay, and they keep coming out of the top and eventually going into the bottom. And now we've drawn vector fields that look a little different than we had before, right? We had a whole bunch of arrows before but now I have a bunch of lines. This is an equivalent way of drawing electric fields. The strength of the electric field has to do with the density of the lines, if the lines are close together the field is strong, if the lines are far apart from each other they're very weak. So this is another representation of the dipole field. Okay. Dipole fields are really important because atoms are made up of positive charges and negative charges. Molecules, polar molecules like water will have a negative end and a positive end, and so the fields that surround them are dipole fields. And that explains how those things, those atoms and molecules, are going to interact with other atoms and molecules that are nearby. How are those different molecules going to combine to form new compounds? It's because of the interaction of the dipole field between the two. So understanding this picture it's really important. Now when you think about positive and negative charges and you talk about field lines, what you can say is field lines originate on positive charges and they terminate on negative charges. And this is often referred to as the source and the sink of the electric field, positive charges are the sources negative charges are the sinks. Let's talk about a solid sphere of charge and let's think about the electric field from this solid sphere of charge. So I'm gonna take, let's say a whole bunch of positive charges and put them everywhere here in the sphere. Okay, and it's solid. What is the electric field outside the sphere? What direction do you think it is? Sam what do you think? Probably doing this? Yeah. I'd say that's fine. Okay. It looks like it was a point charge, right? Everything concentrated at the center, stuff is going uniformly outward. Okay, that's the electric field outside the sphere. What about the electric field inside the sphere? If it's uniformly charged everywhere, then inside the sphere we're gonna have the following. Right at the center e equals zero, but as soon as you go out from the center it starts to go radial again and it increases its strength as you go out. Okay, this is a solid sphere of charged charge everywhere distributed throughout the sphere. Now let's take a look at a different example which is a hollow shell of charge. Okay, if I think about a hollow shell, here it is, I'm just gonna put charge all the way around the outside. And if I put charge all the way around the outside, what do you think the electric field looks like on the outside? Catherine what do you think? Which way should I draw my electric field on the outside of this hollow shell? Yeah, she said, shouldn't still be going up here and away from the charges there, right. Radially outward, it should look very similar to that, and that is exactly right. It's always pointing radially out from the sphere, but here's the $64,000 question. Elena, what is the electric field inside the hollow shell of charge. Okay, why do you think it's going in towards the center? Because all these are positive and so it's got to be going away from those positives, is that why? Okay, so her hypothesis is the sources are all around the edge, and so the electric field has to be going away from those sources, right? I think that's a great hypothesis except there is something very unique about a sphere. A sphere of uniform charge in fact has electric field equal to zero, not just in the center but anywhere inside. Okay, e is equal to zero inside. Where is over here, out other than the exact center, e was not equal to zero. Okay, and this is sort of a weird thing but it's this idea that, all right I might be closer to this charge over here but there's a lot more charge on the other side of me. Yes I'm closer to this one, but there's more charge over here and it just exactly balances out such that E is everywhere of zero, inside the hollow sphere. Okay, this is a very subtle point and we can prove it using something called Gauss's law, which is a slightly tricky concept but one that we will get to.