Resistivity in PipeJan 13, 2023
Resistance is one of the three fundamental components of electricity—and believe it or not, it’s in everything. Like, everything. From precious metals to rubber boots, electricity is flowing… albeit not always very well.
Controlling electricity is something that we as humans have gotten pretty good at. I mean, look around! We’re able to control so much of the world around us. This wasn’t always possible—luckily for us, a lot of really smart scientists were able to figure out how to control electricity. Nowadays, we’re used to seeing cords powering electronic devices all around. These cords, despite carrying electrical charge, are safe for us to touch. Why is that? Anything that is connected to a source of electrical charge will conduct electricity—but not always well. That’s why electrical cords, like this, are coated in an insulating material. Inside, there’s a material that’s really good at carrying electrical charge—a conductor. What makes these two things different from one another is their resistivity. The resistivity of a material is a number given to it, measured in ohms per meter. Every material has a different resistivity, or constant of proportionality. The total resistance of a material is given by the equation R=𝜌lA, where R is the total resistance, 𝜌 is the resistivity, l is the length, and A is the cross sectional area. This makes sense; we want a resistor to give us a certain resistance measured in ohms. By multiplying the resistivity by length in meters, and then dividing by area in meters squared, we are left simply with ohms. Let’s take a look at the resistivities of some common materials. Here, we can see that some materials, like silver or gold, have very low resistivities. This means that they are really good at conducting electricity. If we look at other materials, like rubber or glass, we can see that they’re not nearly as good. It’s also important to know that resistivity changes with temperature. These measurements are for materials at 20ºC. Looking back at our electrical wire, we now know why they’re safe for us to touch. The insides are conductors, while the outsides are insulators. Taking a piece of material, like this copper pipe, we can see just how resistance works. First, let’s do some calculations. We need to know the resistivity of copper, the length of our pipe, and the cross sectional area of it. Resistivity is a measured value that we can easily find on Google. Then, we can take a few quick measurements and we’ll see that the pipe is 6 inches long and has a 0.75 inch diameter. I live in America, so I’ll have to convert these measurements to meters soooooo… the pipe is 0.152 meters long and has a 0.0191 meter diameter. Let’s just plug this in and we’ll see that we have a resistance of *siren* And this pipe is hollow… who would've thought. I could just do a really quick calculation using the equation for the area of a circle, r2, twice by taking the outer area, and subtracting it from the inner area. Divide the diameter by two to find the radius and plug it into the equation for area and find that it’s 0.086 square meters. But that’s too easy. I’ll take the double integral, and still measure the inner and outer diameters. I won’t forget to divide by 2, since I want the radius. And then finally, I’ll integrate from 0.0405 to 0.4375 and from 0 to 2π to find that the pipe has a cross sectional area of 0.086 square meters. Wow, exactly what I got before. Don’t you just love calculus? Anyways, we’ll multiply our resistivity by length, divide by our cross sectional area, and we have a total resistance of 30.1 nanoohms. Unfortunately, theory isn’t always the same thing as reality. So, let’s measure exactly the resistance in this bad boy. We’ll hook up our handy ohmmeter and we get a reading of 0.2 ohms. That’s not actually too bad—accounting for the fact that copper oxidizes, leading to increased resistance. Regardless, this copper would be a great conductor.