Magnitude of electric field created by a charge

- [Instructor] Okay, so we know that electric charges create electric fields. And we know the definition of the electric field is the amount of force per charge. What charge? Some charge that finds its way into this region, let's say this charge right here. If we took the force on this charge, and let's give this a name. Let's call this Q2, so we can keep these all straight, and I'll call it Q2 up here. If we took the force on this charge Q2 divided by Q2, that would be the electric field at that point in space, but something that would be useful to have is a formula that would let us figure out what's the electric field being created at that point in space, without even referring to Q2. It'd be useful to have a formula that would let us figure out what's the electric field that this Q1 is creating over here at this point in space, without even referring to Q2 at all. Is there a formula for that? There is, and it's not that hard to find, but the first thing I'll caution you about is that the formula we're about to find here is gonna be for the magnitude of the electric field. So I'm gonna erase these vector crowns on these variables. This formula we get will just be for the magnitude of the electric field, and I'll tell you why in a second. The way we'll find a formula for the magnitude of the electric field is simply by inserting what we already know is the formula for the electric force. Coulomb's Law gives us the force between two charges, and we're just gonna put that right in here. Coulomb's Law says that the electric force between two charges is gonna be k, the electric constant, which is always nine times 10 to the ninth, multiplied by Q1, the first charge that's interacting, and that'd be this Q1 over here, multiplied by Q2, the other charge interacting, divided by the center to center distance between them squared, and then because we're finding electric field in here, we're dividing by Q2. Notice what happens here. Q2 is canceling, and we get that the magnitude of the electric field is gonna be equal to k, this electric constant, and I'll write that down over here so we know what it is. K is nine times 10 to the ninth, and it's got kinda weird units, but it makes sure that all the units come out okay when you multiply. And then, what do we still have up here? We've still got a Q1 divided by the center to center distance between those two charges squared, and you might be like, well, the other charge went away. We canceled it out. Centered between which two charges? Well, this could be to any point in space, really. So you imagine your test charge at any point you want. I could put it here, I can move it over to here. The r would just be the distance from the first charge, Q1, to wherever I wanna figure out what the electric field would be. But since this Q2 always divides out, we don't even need to talk about that. We can just figure out the electric field that's created by Q1 at any point in space, so this r is just the distance from the center of the charge creating the field to the point in space where you wanna determine the electric field. And now we've got it. This is a formula for the electric field created by a charge Q1. Technically, though, this is only true if this is a point charge. In other words, if it's really, really small compared to the other dimensions in the problem. Or, if this is spherically symmetric, then it doesn't matter. If you're outside of this charge and you've got a spherically symmetrical charge distribution, where all the charges are lumped on one side of this sphere, or anything like that, if it's evenly distributed throughout, then this formula also works just as well when you're outside the sphere. And what's this formula saying? It's saying that the absolute value, or the magnitude of the electric field created at a point in space is equal to k, the electric constant, times the charge creating the field. This is important. This charge, Q1, is creating this electric field. And then you plug in the distance away from that charge that you wanna determine the electric field, r, you square it, and that'll tell you what the magnitude of the electric field is created by Q1 at any point in space around it. Now why are we being so careful, saying that this is just the magnitude? Here's why. Imagine we plugged in this charge as positive because the charge creating it is positive. You'd get a positive value for the electric field, and you might think, oh, that means positive. That means to the right. And in this case, it works out. It does go to the right at this point. But let's say you put those same calculations for a point over here, and you wanna determine what's the value of the electric field at this point? Well, if you plugged in k, it's a positive number, your Q is a positive number, r is gonna be a positive number, even if you tried to plus in r as a negative, it'd square away that negative anyway. This would all come out positive. So you might think, oh, over here points to the right as well because that's the positive direction, right? Well, no. This formula's not gonna tell you whether this electric field goes up, down, left, or right. If you really wanted to include a positive or negative sign for this charge, all that positive would be telling you is that the field's pointing radially away from the charge. But radially away could mean left, it could mean up, it could mean right, or it could mean down, and all of those count as a positive as far as this formula would be concerned. So that's why we say be careful. You're just finding the magnitude when you find this formula. The way you find the direction is just by knowing that the field created by a positive is always radially away from that positive. But even though this formula just gives you the magnitude, that's still really useful. So we're gonna use this. This gives you the magnitude of the electric field from a point charge at any point away from that point charge. Let's solve some examples here. Let's use this thing. Let's say you had a positive two nanoCoulomb charge, and you wanted to determine the size and direction of the electric field at a point three meters below that charge. We wanna know, what's the size and direction of the electric field right there. To get the size, we could use the new formula we've got, which says that the electric field created by a charge, Q, is gonna be equal to k times that Q over r squared. We'll use that down here. We get k, which is always nine times 10 to the ninth, and then we multiply by the charge creating the field, which in this case, it's this positive two nanoCoulomb charge. Nano stands for 10 to the negative ninth, and then we divide by the distance from our charge to the point where we wanna find the field. That's three meters. You can't forget to square this. People forget to square this all the time. It doesn't come out right, so you gotta remember to square the r, and if we solve this, 10 to the ninth times 10 to the negative ninth goes away. That's pretty nice. And then nine divided by three squared is just nine divided by nine, so all of that's gonna go away. And all we really get for the electric field is that it's gonna be two Newtons per Coulomb at this point here. So that is the magnitude. This gives us the magnitude of the electric field at this point in space, that's how you get the size of it. How do we get the direction? We just ask, what was creating this field? It was a positive charge. Positive charges always create fields that point radially away from them, and at this point, radially away from this positive is gonna point straight down. So we've got an electric field from this two nanoCoulomb charge that points straight down, and has a value of two Newtons per Coulomb. What did this number mean? It means if we put another charge at that point in space, some little charge q, then there would be two Newtons for every Coulomb of charge that you put there, since we know that electric field is the amount of force per charge. Notice that even though this electric field came out to be positive, it was pointing down because all we're getting out of this calculation is the magnitude of the electric field. Let's do one more. Let's try this one out. Let's say you had a negative four microCoulomb charge, and you wanted to determine the size and direction of the electric field at a point six meters away, to the left of that charge. We use the same formula. We'll say that the electric field created by this negative charge is gonna equal k, which is always nine times 10 to the ninth. And then multiplied by the charge creating that field, which in this case is negative four microCoulombs, but I am not gonna plug in this negative sign because I know all this formula's giving me is the magnitude of the electric field. I'm not gonna get tricked into thinking that this negative sign would tell me the direction of the electric field. I mean, it does tell you the direction. It tells you that it points radially inward, but it's safest, gotta take my word on this, safest to just leave that negative sign out and know that this is just the magnitude of the electric field. So we have four microCoulombs. Micro stands for 10 to the negative sixth. And then we divide by the distance from the center of that charge to the point we wanna determine the electric field at, which is right here, and we square it. That's six meters, and we cannot forget to square. If we solve this for the electric field, we're gonna get, well, six squared is 36, and nine over 36 is 1/4. 1/4 of four is just one, so all we're left with is 10 to the ninth times 10 to the negative sixth, but that's just 10 to the third, which is 1000. So this electric field's gonna be 1000 Newtons per Coulomb at that point in space. That's the magnitude of the electric field. Or, in other words, that's the size of the electric field at that point. How do we get the direction? We're gonna decide this by thinking carefully about it. In other words, we don't include this negative, not because direction isn't important. We don't include this negative because direction is so important, we're gonna make sure we get this right. What I say is that I've got a negative charge. I know that negative charges create fields that point radially into them. And that means over here at this point to the left, the electric field is pointing radially toward that negative charge, it would point to the right. So we'd have 1000 Newtons per Coulomb of electric field, and it would point to the right. Note, if I would've just naively plugged this negative sign in over to here, I would've come out with a negative value for my electric field, and I might've thought, well, negative, that means leftward, right? So that means it points to the left. And I would've got the wrong direction. So that's why we don't do that. All this negative sign is representing, if you were to plug it in, is that it's pointing radially inward, but radially in could mean right if you're over here to the left. It could mean left if you're over here to the right. It could mean up if you're underneath the charge, and it could mean down if you're above the charge. In other words, it doesn't mean anything for a given particular problem. It often just screws you up. Leave that outta there. Don't put the negative signs in. Just use this formula to get the magnitude, and once you have that magnitude, just know which direction negative charges create their fields, and that'll tell you which direction the field points. So recapping, this is the formula for the electric field created by a charge, Q. And it tells you that the magnitude of the electric field is gonna be equal to k, the electric constant, times the charge creating that field, divided by the distance from the center of that charge to the point where you wanna find the field, squared.