Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

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Electric Fields

Jonathan Osbourne
Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.


An electric field is the ratio between the force felt by a small charge and that small charge. The small charge is called the test charge. This force can be found using Coulomb's law. Electric field diagrams are used to depict electric fields. When lines are close together, the field is strong.

Let's talk about electric fields. Electric fields are things that we introduce in order to make our lives easier. Now, we've seen that the electric force is similar to the gravitational force. We don't usually define something like electric fields for gravity, sometimes we do but we don't always do that. So why would we do that here?

Well, the issue is that atoms themselves consist of electric charge. So that means it's everywhere. And rather than having to keep track of the force on every single electric charge, all throughout whatever matter we're looking at, it's easier for us to define something that does it all at once for every single charge. And that's called the electric field.

So what we're going to do, is we're going to define the electric field as the force felt by a small charge. And this is usually called a test charge. So I've got some sort of configuration of electric charge all over the place and I'm going to take a little teeny tiny charge and I'm going to put it somewhere. And I'm going to go ahead and see what direction and how big the force that that little teeny tiny charge feels is in and then I'm going to divide that force by the small charge. And so then that allows us to determine the force on any charge just by taking the small charge and multiplying by the electric field. So it just kind of divorces any dependence that the force had on the charge that was filling the force. Alright.

So that's the electric field here, force divided by test charge. The electric field always points in the direction of the force on a positive test charge. Always in the direction of the positive force. Now we now from Coulomb's law, that the force will be equal to k times one charge times the other charge over r squared. And it will be attractive if the two charges are opposite and repulsive if the two charges are the same. So when we divide by one of those charges, we'll get ek charge over r squared. If we want to make it a vector, needs to be in the direction of the force on a positive charge. So if q is positive then it's going to be away because that will be a repulsive force. Now, notice if q was negative, then that's just going to change the sign of that. So it would be, the negative sign would make that toward. So it's very very very nice and useful. Alright.

So let's go ahead and look at some electric field diagrams. Alright. If I've got a positive charge sitting there all by himself, then the electric field is going to point away from the positive charge because any test charge that was positive, remember the electric field points in the direction of positive charge force, alright? So it's going to be pointing away because this positive charge is going to repel any other positive charges in the vicinity. Now it's going to be all nice and symmetric around this positive charge because if I've got a little test charge right there, well, repulsive, right? So very nice and symmetric and in fact in three dimensions but I've only drawn it in two. What about a negative charge?

Well, positive test charges are going to be drawn towards this negative charge. So that means that the electric field for a negative charge will point toward the negative charge. Alright. What if I want more complicated electric field diagrams? Well, there's five basic rules that we need to draw these diagrams. First rule, the electric field can only be created by positive charges and can only be destroyed by negative charges. So that means that if I've got an electric field and it starts somewhere. That place got to be a positive charge. So we got a bunch of electric field lines starting at that positive charge and we got a bunch of electric field lines ending at this negative charge. Alright. So that's the first rule.

Second rule. The electric field will always be symmetric near a point charge. It will always point away from a positive charge and toward a negative charge. Furthermore, the number of lines that I draw, the number of electric field lines that I draw near a point charge is proportional that's what that symbol means, proportional, proportional to the charge. So if I add plus plus up here, then I got to draw 8 field lines. So I'd have to add that one, that one, that one and that one, alright? Alright. So that's great.

Next one, number 3. Electric field lines can never be allowed to cross. Why not? Well, if they were to cross, let's say I got an electric field line here and an electric field line here. Well, what if I put a charge right there? What direction is the force on it? It's not schizophrenic. It doesn't get that split in half and part of it goes that way and the other part goes the other way. It's got a force on it. What direction is that force in? So it can't look like this. It cannot ever cross. Alright?

What about number 4? Number 4 says that the electric field lines will spread out whenever there is room for them to spread out. Alright? And we'll see that more clearly when we start drawing some of these.

And then the fifth one is more guidelines so that you can understand what the electric field line diagram means. It says that the electric field will be strong when the lines are close together. And it will be weak when the lines are far apart. So that's how we'll interpret this electric field line diagram.

Alright. So let's draw a simple one. I got a positive charge and a negative charge. So I'm going to go ahead and use the rules to draw this electric field line diagram. Alright. Near the positive charge got to have electric field lines that come out in a symmetric way. Near the negative charge I got to have electric field lines that go in, in a symmetric way. Now in order for us to complete this electric field line diagram, I need to go ahead and connect. So we'll have that. This guy here, oh, like that, will have this guy here going around like that and we'll have this guy here going around like that. And that's going to be my electric field line diagram for a positive and a negative charge. Alright?

What about 2 positive charges? Well, again, I start the same way. Near this positive charge, out, out, out, out. Near this positive charge, out, out, out, out. Now, these field lines can't connect like they did over here. Because they're not supposed to do that. They're going kind of in the same direction. So rather than doing that, they're going to come up and go out like that. Come up and go out like that. Come up, like that. So this is the field line diagram for 2 positive charges. Notice that we can kind of see the repulsion of these 2 positive charges just from the electric field line diagram. If I were to push them closer together, look these electric fields, they're not allowed to cross. they don't like that. So that's why these 2 positive charges are going to repel each other. so it's a real nice simple way to understand that fact, just directly from the electric field line diagram. Another nice thing about this diagram, what happens if I'm really far away? Well, if I'm really far away I don't see that the two charges are separate. Instead all I see are 1, 2, 3, 4, 5, 6, 7, 8 electric field line diagrams all coming out all coming out nice and symmetric from a double positive charge. And that's exactly what I would expect.

So all these stuff just makes sense. It's a wonderful wonderful wonderful way to understand the electric fields. Alright. Let's do one more. This is one of my favorites right here. So we got 2 positive charges together and then we got a negative charge right here. Alright. So, again, symmetric. Notice I drew 8. That's because this is a double positive charge. So I got 8 here, that means I need 4 here. In, in, in, in. And now I'm going to play my little connect game. Like that. This one, he got to go somewhere. Now this guy's got to connect with somebody because he's going in, that's a destroyed electric field. So he's going to come around and connect like that. This one, same thing. Notice that because there's room, I'm allowing them to spread out in accordance with the rules. Alright.

What about these 4 remaining guys? Well, again spread out, spread out, spread out, and spread out. And notice that if I look from really really really far away, what do I have? Well, I got 1, 2, 3, 4 field line diagrams. All these are just done. They're just inside, I don't see them from far away. So I got 4 field line diagrams. I wonder why. Well, the total charge is a single positive charge. + + and a negative gives me single +. So that means that I ought to have 4 electric field line diagrams coming out. and that's exactly what I got.

Alright. So that's electric fields.

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