Ohm's law and circuits with resistors
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Resistivity and conductivity
- [Voiceover] You probably know that if you hook up a battery of voltage V, to a resistor of resistance R, then you'll get a certain amount of current, and you can determine how much current flows here by using Ohm's law. And remember, Ohm's law says that the voltage across a resistor equals the current through that resistor times the resistance of that resistor. So, this pretty much gives you a way to define resistance. The resistance of the resistor is defined to be the amount of voltage applied across it divided by the amount of current through it. And this is good, we like definitions, because we want to be sure that we know what we're talking about. That's the definition of resistance. Remember, it has units of ohms. But be careful, don't fall into the trap of thinking about this the way some people do. Some people think, "Oh okay, if I want a bigger resistance, "I'll just increase the voltage, 'cause that'll give me "a bigger number up top." Bigger resistance doesn't work that way. If you increase the voltage, you're gonna increase the current. And this ratio is gonna stay the same. The resistance is a constant. And this resistor, if you're not changing the material makeup or size or dimensions of this resistor, this number that is the resistance is a constant, if it's truly an Ohmic material. So, Ohmic materials maintain a constant resistance, regardless of what voltage or current you throw at them. It'll just be constant. Yeah, if you throw too much current or voltage, the thing'll burn up. I don't suggest you do that. So, there's an operating range here, but if you're within that range, this resistance, this number, this number of ohms is a constant. It stays the same no matter what voltage or current you put through it. So, we define it by talking about voltage and current, but it doesn't even really depend on that. If you really want to change this ratio, this number that comes out here for the resistance, you need to change something about the resistor itself. Its size, what it's made out of, its length, its shape. So let's figure out how to do that if we take this resistor. Imagine taking this resistor, bringing it into the shop. What is it gonna look like? Well, for simplicity's sake, let's say we just have a perfectly cylindrical resistor. So, this is the wire going into one end. This is your resistor. It's a cylinder, let's say. And this is the wire going out of the other end, so this is the blown up version of this resistor. One thing you could depend on is the length. So, the length of this resistor could affect the resistance of this resistor. Another thing it could depend on is the area of this front part here, this cross sectional area. It's called the cross sectional area, because that's the direction that the current's heading into. This current's heading into that area there, like a tunnel, and it comes out over here. Now this is full, this isn't hollow. This is made up of some material. Maybe it's a metal or some sort of carbon compound or a semiconductor, but it's a solid material right here that the current flows into and then flows out of. So, what would happen if we made this resistor longer? Let's say we start changing some of these variables, and we increase the length of this resistor. Well, now this current's gotta flow through a longer resistor. It's gotta flow through this resistor for more of this path. And it makes sense to me to think that the resistance is going to increase. If I increase the length of this resistor, then the resistance is gonna go up. How about the area, this cross sectional area? Let's say I increase this area, I make it a wider, larger diameter cylinder. Well, it makes sense to me to think that now that current's got more room to flow through, essentially. There's a bigger area through which this current can flow. It's not as restricted. That means the resistance should go down. And if we try to put this in a mathematical formula. What that means is, if I increase the length, R should depend on the length. It turns out it's directly proportional to the length. If I double the length of a resistor, I get twice the resistance. But area, if I increase the area, I should get less resistance. 'Cause there's more room to flow. So, over here in this formula, my area has go to go on the bottom. The resistance of the resistor is inversely proportional to this cross sectional area. But there's one more quantity that this resistance could depend on, and that's what the material is actually made of. So, the geometry determines the resistance as well as what the material is made of. Some materials just naturally offer more resistance than others. Metals offer very little resistance, and non-metals typically offer more resistance. So, we need a way to quantify the natural resistance a material offers, and that's called the resistivity. And it's represented with the greek letter rho. And the bigger the resistivity of a material, the more it naturally resists the flow of current through it. To give you an idea of the numbers here, the resistivity of copper. Well, that's a metal, it's going to be small. It's 1.68 times 10 to the negative eighth. We'll talk more about the units in a second. But the resistivity of something like rubber, an insulator, is huge. It can be on the order of 10 to the 13th. So, there's a huge range of possible values as you go from metal conductor to semiconductor to insulator, huge range of possible resistivities. And this is the last key here. This is the last element in this equation. The resistivity goes right here. So, the bigger the resistivity, the bigger the resistance. That makes sense. And then it also depends on these geometrical factors of length and area. So, here's a formula to determine what factors actually change the resistance of a resistor. The resistivity, the length, and the area. So, what are the units of resistivity? Well, I can rearrange this formula, and I can get that the resistivity equals the resistance times the area of the resistor divided by the length. And so that gives me units of ohms times meters squared, 'cause that's area, divided by meters. And so I end up getting ohms. One of these meters cancels out. Ohms times meters. Those are the units of these resistivities. Ohm meters. But how do you remember this formula? It's kind of complicated. I mean, is area on top, is length on bottom? Hopefully you can remember why those factors affected it. But sometimes students have a hard time remembering this formula. One of my previous students from a few years ago figured out a way to remember it. He thought this looked like "Replay". So, this R is like R, and the equal sign kind of looks like an E. And the rho kinda looks like a P. And the L looks like an L, and the A looks like an A. And it kinda looks like "Replay". There's a missing Y here, but every time I think of this formula, I think of it as the "Replay formula". 'Cause my former student Mike figured out this mnemonic. And it's handy, I like it, so thank you, Mike. And since we're talking about resistivity, it makes sense for us to talk about conductivity. Electrical conductivity. Now the resistivity gives you an idea of how much something naturally resists current. And the conductivity tells you how much something naturally allows current. So, they're inversely proportional. And if you're thinking it might be this easy, it is. The resistivity is just equal to one over the electrical conductivity. And the symbol we use for electrical conductivity is sigma. So, this Greek letter sigma is the electrical conductivity. And rho, the resistivity, is just one over sigma, the electrical conductivity. And vice versa, sigma is gonna equal one over the resistivity, because if something's a great resistor, it's a bad conductor. And if something's a great conductor, it's a bad resistor. So, these things are inversely proportional. They're like two peas in a pod. If you know one of these, you know the other. All right if this all seems a little bit too abstract still, there's a nice analogy you can make to water. We saw that a resistor depended on a few things, like the resistivity. The bigger the resistivity, the bigger the resistance. And we saw that the bigger the length of the resistor, the larger the resistance. And if you divide by the area of the resistor, it shows that the resistance is inversely proportional to the area of the resistor. So, let's make an analogy to water. Let's say you have, instead of electrons, flowing through a wire. Instead of the wire, let's say you had a tube, a pipe that water could flow through. So, instead of electrons, you've got water flowing through a pipe. Different pipes are gonna offer different amounts of resistance to the water flowing through that pipe. What would affect it? Well, imagine you had a constriction in this pipe. If this pipe got constricted, it'd be harder for the water to flow. You'd find that it resists the flow of water more because of this constriction. And what would it depend on? Well, the smaller this area of the constriction, the larger the resistance. And that agrees with what we have up here. If you have a really small area, you're dividing by a small number, and when you divide by a small number, you get a big number. That'd be a big resistance. So, that makes sense. Also the length, if you increase the length of this constriction, the water will have a harder time flowing. There's manuals for plumbers, and you can look it up. There's a key to determine if your pipe is a certain length, you're gonna need more pressure over here. So, the smaller the constriction in terms of its area, and the longer it is, the more pressure you need back here. The pressure is like the source of the battery. So instead of a battery providing the voltage to this circuit, you'd have something offering pressure to get the water flowing. And just like a battery, what matters is the difference in electric potential. What matters for the pressure here is the change in pressure between one point in the system and another point in the system. So that makes sense. A longer constriction means more resistance. A smaller area means more resistance. What would this resistivity be analogous to? Well, it would be kind of like what the pipe is made out of. If this pipe has a rough inner surface, the water wouldn't flow as smoothly. You would get a greater resistance regardless of how long it is or what the area is. Just the natural built-in affect of the pipe itself is what the resistivity would depend on, just like up here. The resistivity depends on what the material is made out of. The resistivity of this pipe depends on what this pipe is made out of, at least the inner wall. So hopefully this analogy makes this formula seem a little more intuitive. But just in case, let's do an example. Let's get rid of all this. And let's say you got this question. "How much resistance would be offered by "12 meters of copper wire with a diameter "of 0.01 meters?" If copper has a resistivity of 1.68 times 10 to the negative 8th. Now, what units does resistivity have? Turns out resistivity has units of ohm meters. So, ohms times meters. Well, let's try this out. We've gotta use our formula. Remember "Replay", R equals rho, L over A. The resistivity we have right here. 1.69 times 10 to the negative 8th. Notice how small this is. This is hardly anything at all. Copper's a great conductor. It's a terrible resistor. It let's electrons flow through it like a charm. All right, so the length, that's pretty easy. The length is 12 meters. Notice, we're asking, what's the resistance of the wire itself? Now there's not really a quote-unquote resistor in here. But every piece of wire's gonna offer some resistance. And this formula applies just as well to a piece of wire as it does to a resistor. So, the length of the wire's 12 meters, and the diameter is 0.01. What do we do with that? Well, we need the area. Remember the cross sectional area. And the area of a circle is pi r squared, so the area down here is gonna be pi times not 0.01 squared. That's the diameter. We need the radius, we need to take half of this. So 0.005 meters squared. And if you calculate all this, you get a resistance of 0.0026 ohms. Hardly anything, but there is some resistance. And if this is going to have an affect on a very delicate experiment, you've gotta take that into account. If you get this really long, the longer it is, the more resistance is has, that could affect your system. But typically, it doesn't matter too much. The copper wire, electron's flow through that like water, like it's not even there, because the resistance is so small.