Video transcript
- [Voiceover] So we've already started to familiarize ourselveswith the notion of charge. We've seen that if twothings have the same charge, so they're either both positive, or they are both negative, then they are going to repel each other. So in either of these cases these things are goingto repel each other. But if they have different charges, they are going to attract each other. So if I have a positiveand I have a negative they are going to attract each other. This charge is a property of matter that we've started to observe. We've started to observe ofhow these different charges, this framework that we've created, how these things start tointeract with each other. So these things are going to, these two things are goingto attract each other. But the question is, what causes, how can we predict how strong the force of attraction or repulsion is going to be between charged particles? And this was a questionpeople have noticed, I guess what you couldcall electrostatics, for a large swathe ofrecorded human history. But it wasn't until the 16 hundreds and especially the 17 hundreds, that people started to seriously view this as something that they could manipulate and even start to predictin a kind of serious, mathematical, scientific way. And it wasn't until1785, and there were many that came before Coulomb, but in 1785 Coulomb formally published what is known as Coulomb's law. And the purpose of Coulomb's law, Coulomb's law, is to predict what isgoing to be the force of the electrostatic forceof attraction or repulsion between two forces. And so in Coulomb's law, what it states is is if I have two charges, so let me, let's say thischarge right over here, and I'm gonna make it in white, because it could be positive or negative, but I'll just make it qone, it has some charge. And then I have in Coulombs. and then another chargeq two right over here. Another charge, q two. And then I have the distancebetween them being r. So the distance between these two charges is going to be r. Coulomb's law states that the force, that the magnitude of the force, so it could be a repulsive force or it could be an attractive force, which would tell us thedirection of the force between the two charges, but the magnitude of the force, which I'll just write it as F, the magnitude of the electrostatic force, I'll write this sub e here, this subscript e for electrostatic. Coulomb stated, well this is going to be, and he tested this, he didn'tjust kind of guess this. People actually were assumingthat it had something to do with the products of the magnitude of the charges and that as the particles got further and further away the electrostatic force dissipated. But he was able to actually measure this and feel really goodabout stating this law. Saying that the magnitudeof the electrostatic force is proportional, is proportional, to the product of themagnitudes of the charges. So I could write thisas q one times q two, and I could take theabsolute value of each, which is the same thing as just taking the absolute value of the product. Here's why I'm taking theabsolute value of the product, well, if they're different charges, this will be a negative number, but we just want the overallmagnitude of the force. So we could take, it's proportional to the absolute value of theproduct of the charges and it's inversely proportional to not just the distance between them, not just to r, but to thesquare of the distance. The square of the distance between them. And what's pretty neat about this is how close it mirrorsNewton's law of gravitation. Newton's law of gravitation,we know that the force, due to gravity between two masses, remember mass is justanother property of matter, that we sometimes feel isa little bit more tangible because it feels like we cankind of see weight and volume, but that's not quite the same, or we feel like we can feel or internalize things like weight and volume which are related to mass, but in some ways it isjust another property, another property, especiallyas you get into more of a kind of fancy physics. Our everyday notion of even mass starts to become a lot more interesting. But Newton's law of gravitation says, look the magnitude of the force of gravity between two masses is goingto be proportional to, by Newton's, by the gravitational concept, proportional to theproduct of the two masses. Actually, let me do itin those same colors so you can see the relationship. It's going to be proportional to the product of the twomasses, m one m two. And it's going to beinversely proportional to the square of the distance. The square of the distancebetween two masses. Now these proportionalpersonality constants are very different. Gravitational force, we kind of perceive thisis as acting, being strong, it's a weaker force in close range. But we kind of imagine itas kind of what dictates what happens in the, amongst the stars andthe planets and moons. While the electrostaticforce at close range is a much stronger force. It can overcome thegravitational force very easily. But it's what we consider happening at either an atomic levelor kind of at a scale that we are more familiar to operating at. But needless to say,it is very interesting to see how this parallelbetween these two things, it's kind of thesepatterns in the universe. But with that said, let's actually apply let's actually apply Coulomb's law, just to make sure we feelcomfortable with the mathematics. So let's say that I have a charge here. Let's say that I have a charge here, and it has a positivecharge of, I don't know, let's say it is positive five times 10 to the negative three Coulombs. So that's this one right over here. That's its charge. And let's say I have thisother charge right over here and this has a negative charge. And it is going to be, it is going to be, let'ssay it's negative one... Negative one times 10 to the negative one Coulombs. And let's say that thedistance between the two, let's that this distance right here is 0.5 meters. So given that, let's figure out what the what the electrostatic force between these two are going to be. And we can already predict that it's going to be anattractive force because they have different signs. And that was actuallypart of Coulomb's law. This is the magnitude of the force, if these have differentsigns, it's attractive, if they have the same sign then they are going to repel each other. And I know what you're saying, "Well in order to actually calculate it, "I need to know what K is." What is this electrostatic constant? What is this electrostaticconstant going to actually be? And so you can measure thatwith a lot of precision, and we have kind of modern numbers on it, but the electrostatic constant, especially for the sake of this problem, I mean if we were to getreally precise it's 8.987551, we could keep gone on andon times 10 to the ninth. But for the sake of ourlittle example here, where we really only have one significant digit for each of these. Let's just get an approximation, it'll make the math a little bit easier, I won't have to get a calculator out, let's just say it's approximately nine times 10 to the ninth. Nine times 10 to the ninth. Nine times, actually let memake sure it says approximately, because I am approximating here, nine times 10 to the ninth. And what are the units going to be? Well in the numerator here, where I multiply Coulombs times Coulombs, I'm going to get Coulombs squared. This right over here is going to give me, that's gonna give me Coulombs squared. And this down over here is going to give me meters squared. This is going to give me meters squared. And what I want is to get rid of the Coulombs and the meters and end up with just the Newtons. And so the units here are actually, the units here are Newtons. Newton and then meters squared, and that cancels outwith the meters squared in the denominator. Newton meter squared over Coulomb squared. Over, over Coulomb squared. Let me do that in white. Over, over Coulomb squared. So, these meter squared will cancel those. Those Coulomb squared in the denomin... over here will cancel with those, and you'll be just left with Newtons. But let's actually do that. Let's apply it to this example. I encourage you to pause the video and apply this informationto Coulomb's law and figure out whatthe electrostatic force between these twoparticles is going to be. So I'm assuming you've had your go at it. So it is going to be, and this is really just applying the formula. It's going to be ninetimes 10 to the ninth, nine times 10 to the ninth, and I'll write the units here, Newtons meter squaredover Coulomb squared. And then q one times q two,so this is going to be, let's see, this is going to be, actually let me just writeit all out for this first this first time. So it's going to be times five times ten to the negative three Coulombs. Times, times negative one. Time ten to the negative one Coulombs and we're going to takethe absolute value of this so that negative is going to go away. All of that over, all of that over and we're in kind of thehome stretch right over here, 0.5 meters squared. 0.5 meters squared. And so, let's just do alittle bit of the math here. So first of all, let's look at the units. So we have Coulomb squared here, then we're going to haveCoulombs times Coulombs there that's Coulombs squareddivided by Coulombs squared that's going to cancel with that and that. You have meters squared here, and actually let me just write it out, so the numerator, in the numerator, we are going to have so if we just say nine times five times, when we take the absolute value, it's just going to be one. So nine times five is going to be, nine times five times negative... five times negative one is negative five, but the absolute value there, so it's just going to be five times nine. So it's going to be 45 times 10 to the nine, minus three, minus one. So six five, so that's going to be 10 to the fifth, 10 to the fifth, the Coulombsalready cancelled out, and we're going to haveNewton meter squared over, over 0.25 meters squared. These cancel. And so we are left with, well if you divide by 0.25, that's the same thing as dividing by 1/4, which is the same thingas multiplying by four. So if you multiply this times four, 45 times four is 160 plus 20 is equal to 180 times 10 to the fifth Newtons. And if we wanted to writeit in scientific notation, well we could divide this by, we could divide this by 100and then multiply this by 100 and so you could write this as 1.80 times one point... and actually I don'twanna make it look like I have more significantdigits than I really have. 1.8 times 10 to the seventh, times 10 to the seventh units, I just divided this by 100and I multiplied this by 100. And we're done. This is the magnitude ofthe electrostatic force between those two particles. And it looks like it's fairly significant, and this is actually a good amount, and that's because this isactually a good amount of charge, a lot of charge. Especially at thisdistance right over here. And the next thing we have to think about, well if we want not just the magnitude, we also want the direction, well, they're different charges. So this is going to bean attractive force. This is going to be anattractive force on each of them acting at 1.8 times tento the seventh Newtons. If they were the same charge,it would be a repulsive force, or they would repel eachother with this force. But we're done.
