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# Resistivity and conductivity

## Video transcript

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 resistance the resistance of this 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 okay if I want a bigger resistance I'll just increase the voltage because that will give me a bigger number up top bigger resistance not doesn't work that way if you increase the voltage what you're going to increase the current and this ratio is going to stay the same the resistance is a constant at 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 will 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 going to look like well for simplicity 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 sin let's say and then here's the wire going out of the other ends is the blown-up version of this resistor one thing it 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 currents heading into this current is 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 semi conductor but it's a solid material right here 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 currents got to flow through a longer resistor it's got to 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 going to go up how about the area of this cross-sectional area let's say I increase this area and make it a wider larger diameter cylinder well it makes sense to me to think that that now that currents 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 because there's more room to flow so then over here in this formula my area has got 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 nonmetals typically offer more resistance so we need a way to quantify the natural resists material offers and that's called the resistivity and it's represented with a 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 like one point six eight times ten to the negative eighth we'll talk about the units in a second but the resistivity of something like rubber and insulator is huge it can be on the order of ten to the thirteenth so there's a huge range of possible values as you go from metal conductor to semi conductor 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 because 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 as area on top is length on bottom hopefully you could 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 this equal sign kind of looks like an e and the row kind of looks like a P and the L looks like an L and then the a looks like an a and it kind of 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 because 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 Khanda - 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 currents 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 just going to equal one over the resistivity because if something is a great resistor it's a bad conductor and if something is 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 had 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 going to 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 he to determine if your pipe is a certain length you're going to 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 a 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 effect 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 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 1.6 8 times 10 to the negative 8 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 got to use our formula remember replay so R equals Rho L over a the resistivity we have right here one point six eight times 10 to the negative eight notice how small this is hardly anything at all coppers a great conductor it's a terrible resistor it lets 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 is going to 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 this wire twelve meters and the diameter is 0.01 what do you 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 going to be pi times not 0.01 squared that's the diameter we need the radius we need to take half of this so a point zero zero five meters squared and if you calculate all this you get a resistance of zero point zero zero two six ohms hardly anything but there is some resistance and if this is going to have an effect on a very delicate experiment you've got to take that into account if you get this really long the longer it is the more resistance it has that could affect your system but typically it doesn't matter too much the copper wire electrons flow through that like water like it's not even there because the resistance is so small