24. Electric Force & Field; Gauss' Law
Coulomb's Law (Electric Force)
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Atoms are very interesting because the whole universe is basically made up of atoms, right? There was a very famous physicist named Richard Feynman and they asked him the following question. They said if you could type a letter to an alien species telling them something about what we understand about the universe but it was a Twitter type message where you could only send one sentence what would it be? Okay, and this is a Nobel Prize winning physicist that they're asking this, so it's sort of an interesting thought experiment, right? If you could send one message to an alien what would it be? And he said the following. Stuff is made of atoms. That would be his one sentence that he sent to this alien race which I think is pretty cool, right? I mean they might not know the word stuff but you could put some other word there, right? They probably don't know English at all but let's pretend that you had to say one thing and you could write it a binary code or whatever it was. Stuff is made of atoms. That is really powerful, okay, because it tells you oh wait a minute, this human that's right here, there's something more complex happening inside it and this solid right here this table there's something happening inside of it and there's all these little things in there that are bound together, they are bound together by electromagnetic forces. That's what holds material together. This glass that I'm looking at you through, right, it has atoms in it and those atoms are held together by electromagnetic forces. It turns out that it's transparent because visible light is not absorbed in those atoms but other things are not transparent right? My shirt is not transparent, it's made of atoms, but visible light is absorbed in that material in this class visible light is not absorbed. So stuff is made of atoms what are these atoms that we're talking about? Well we said it just a second ago, it's a positive charge and a negative charge, and the negative charge is whirling around the positive charge. This is a very simple picture of the atom but this is a great starting point for discussing the atom. If I think about the hydrogen atom, the hydrogen atom has one proton in it, that's why it's number one on the periodic table. The electron is whirling around that proton and in the ground state it's more or less whirling around in a circle, okay? Now when you get to quantum it gets a little bit more complicated, you really can't talk about the electron having a specific position and going around like a point particle. It gets a little fuzzier because you have to invoke quantum and talk about the wave functions and things like that, but this is a great starting point for thinking about the atom. Now if something is moving around in a circle there has to be a force, right? We know that, circular motion tells us that there must be a force that is keeping this thing moving in the circle. And when it was the earth going around the Sun, it was gravity that was keeping that earth in its orbit. But here it is of course a different force. It is the electromagnetic force. Likes repel opposites attract. The electron is bound to the proton because this force that's holding it there. All right, so let's talk a little bit about the force and let's introduce this idea of Coulomb's law. So Coulomb's law says the following the force is K q1 q2 over R squared. K q1 q over R squared, what are these different things? K is a constant, it's Coulomb's constant, it is 8.9 times 10 to the 9 and then it's got some funky units on it and you can sort of figure out what the units have to be, right? K is sitting up here, I got to end up with Newtons, so there's got to be a Newtons there. I got an R squared in the bottom, that's going to be a distance, so there must be a meters squared up in the top, and then I have q1 and q2 and we said the charge is in coulombs. So there must be coulombs squared in the bottom. Okay, and this is Coulomb's constant. All right, there are some other digits there, you can add other digits if you want. It's like eight point nine nine eight or something, so we often say it's just nine times 10 to the nine. What about q1. Q1 is the charge of the first particle. Q2 is the charge of the second particle. And then finally R is the distance between them and here we're talking about q1 and q2 so it's the distance between those two charges. Q1 and q2, okay? This is Coulomb's law this is a very powerful law and it's going to apply to a great number of cases, okay? We're gonna talk about a lot of those this term, later on we'll see when this starts to break down but for a great number of cases this is really it. This is all you need, okay? It starts to break down when we have to add magnetism but we'll get to that later. Let's write down Coulomb's law again. F equals K q1 q2 divided by R squared. Now F is a force. Is a force a scalar or a vector? It's a vector, right. And so we need to put an arrow on top of that and if there's an arrow on top of that we better put some direction over here on the right side, and I like to just write it like that. Let's put an N-hat right there. N-hat is along the line from q to q2. And so you can decide if it's attractive or repulsive but it's definitely going to be along the line between those two. All right. Let's see if we can figure out what that force is for a couple charges. Let's say we take the following. We had just talked about electrons and protons, let's put them together and see what we get. Let's put an electron on the page, we'll put a proton on the page, and we will separate those two by a distance R and let's figure out what the force is on the electron due to the proton. Okay, pretend the proton is nailed down to the page it's not going to move around the electron we put down there and now we're gonna see what's a force on the electron due to that proton. We know it's going to be attractive so it's towards the proton. The strength is right here, it's K q1 q2 over R squared. Okay, but we know exactly what K is, we have that number, and we know what the charge of the proton is, it is 1e, which we write just like that. We know what the charge on the electron is, it's also an e but it's negative, and then we have R squared in the bottom. Let's do this now and put in some real numbers. All right, so we have F equals negative K e squared divided by R squared and let's say that we're gonna let R equal 10 to the minus 10 meters. Okay, that's called an angstrom, 10 to the minus 10 meters. And let's put in some numbers and see what we get. So K we said was eight point nine eight eight times seven nine, we're gonna say that is nine times 10 to the nine in SI units. E is one point six times ten to the minus nineteen coulombs, we have to square that and we're going to divide by R squared, which is ten to the minus ten, and we're going to square that. So once you guys pull out your calculator or your phone and let's plug in some numbers, I will approximate it here. Nine times one point six is about probably 13 nah, a little bit more 14. 14 and then I've got--oh but we have to square that don't forget that. So one point six square, that's about three so we've got a nine times a three and then we've got a 10 to the nine and then we have a 10 to the minus 19 squared which is ten to the minus 38, and then we're gonna divide by 10 to the minus 10 squared, which is 10 to the minus and so what do we get? We get 18 times 38 and 9, we have to add so we get 10 to the minus 29 and we're going to divide by 10 to the minus 20 and so we get times 10 to the minus 9 which is pretty close to 1.8 or 2, times 10 to the minus 10 Newtons. That's my guess what did you guys get? Did you get a real number? 2.3 times 10 to the negative 10. Okay, so that's a little bit more exact our guess was a little bit off but that's okay. Two point three times ten to the minus ten is the force in SI units, right, Newtons, for the electron and the proton at this separation. So that sounds like a small number but electrons and protons are pretty small. Turns out this is the binding force of an electron to a proton in a hydrogen atom, roughly okay. This is about the right distance between the proton and the electron in a hydrogen atom. It is about one angstrom 10 to the minus 10 meters. So how hard is that electron held in its orbit? The force is that, okay? That's the force that is holding it in its orbit, two point three times ten to the minus ten Newtons.
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