We defined the electric field, we've thought about the electric field, now let's find a formula for the electric field. Let's start with the electric field around just a small sphere of charge or a point charge. So to start, recall Coulomb's law. So remember we have Coulomb's law. Let's say we have a big charge plus Q here, and let's think about it like we did with the electric field. Let's draw a little charge further away, a little q. So big Q and a little q. Coulomb's law told us the force between big Q and little q. So F, and we had this notation, Q to q, big Q to little q, that's the force big Q will apply to little q equals Coulomb's constant times this charge big Q, this charge little q, and then we had to have the separation. So the separation between them, we just call r. We can call it r_Qq, big Q little q the separation between the two. So r_Qq squared and then we have to have the vector direction, and that came from that unit vector r-hat. Again, we keep the same notation, subscripts big Q little q, and that meant the unit vector. So its magnitude is one because it's a unit vector along the Qq axis and pointing in the direction from big Q to little q. If you get all that right, then you're going to get the force right. Now, let's think in terms of the electric field. The electric field generated by this big charge. So thinking from our last lecture, we know that the electric field, and we're going to get the electric field of big Q, it's defined to be the force that big Q would apply to this little charge divided by the little charge. Remember, it's the force per unit charge is our definition of the electric field. Now you'll notice I've changed our notation a little bit. I didn't put big Q, little q, and that really comes down to the whole philosophical point of the electric field. Is because we're not calculating really the electric field right here, we're calculating the electric field anywhere, anywhere little q goes, that's what we're calculating it. We're not calculating something dependent on little q, so we change the name to just the E field due to big Q. So this is just the E field created by Q independent of the little charge, it's usually called the test charge. So to find it, well, we just plug in for big F and you can see what's going to happen here, it's going to be K_e. Big Q the little qs cancel over r squared r-hat. Again, I've changed the subscripts, and not just for fun, I've changed them because it has a significant meaning. There are now no subscripts on r and r-hat, and the reason is, we're now talking about any position around big Q. Big Q basically, if that sits at an origin, were just looking at a spherical coordinate system. So really now we've calculated electric field anywhere. I guess if you wanted to be specific and say, where big Q is at the origin, you could call them r big Q and r-hat big Q. If you wanted to specify the origin because your problem out of the r is floating around. Basically now, we're calculating the electric field pretty much in all of space, so these two rs, we'll just write down, is the distance and direction from Q. That's how you'd calculate it. What's the unit? The unit is, well, you can see what the unit is right there. It's a Newton per Coulomb and what's funny is the unit for the electric field has no name, it wasn't named after anyone, is just the Newton per Coulomb. We can also visualize the field now. So now we have mathematically described it. Here's our field, and you can already start to see it. It's a field whose magnitude goes down as one over r squared, then it goes down as you get farther away just like the force, and always points away from big Q. So let's draw that. Let's see what it looks like. So here we have big Q and the E field vectors now, not the force vectors, the E field vectors are going to point away from big Q. If you get farther away, they get shorter because it goes as 1 over r squared. If you get farther away, they get a little bit shorter, a little bit shorter, a little bit shorter. Oh, I forgot, I always like to draw this one too. So I am going to draw all of them and you can't stop me because I am here and you are there in both time and space. There. Beautiful E field around a point charge. Is it beautiful? I don't know, it's kind of beautiful. But does it represent everything? No, it doesn't. So all of the vector fields we're going to talk about are three-dimensional. You know that in reality, if I could draw it, I would have vectors coming out this way and going off that way, they'd be going away spherically from the charge. But I can't draw that. So what we're going to do to help you visualize the fields is we're going to look at them in a visualization lab and really be able to move them around in 3D to see them better. We can't do that all the time. So when you do physics, you always have to remember all these fields I'm seeing in the paper of this book are actually three-dimensional fields. When you see this, you got to see it coming out in three-dimensions. But to get us started, we'll actually look at them in a three-dimensional visualization. But remember, that's what that really means. So let's go to visualization lab and see what it looks like. It's a long walk, sorry. Welcome to the Rice Visualization lab. This is this nice screen that will let us see or vector fields in three dimensions. So as I said, I draw them in two dimensions on the board, but to understand the physics, you often really have to think about them in three-dimensions. So here's a perfect example. Here is a positive charge in red and green is the vector fields pointing away. Just like I drew on the board, the farther you get away according to Coulomb's Law, the smaller the magnitude of the field, therefore, the smaller we draw the vectors, and they always point away. But they always point away in three-dimensions. So to really see it, you have to manipulate the field in three-dimensions. So here you see as I spin the charge around, the field always points away from the charge. So keep in mind, every drawing I do on the board is actually a three-dimensional vector field and you always need to be thinking in terms of all three-dimensions. Let's keep going.