Electric Circuits Legacy Problem #4 Guided Solution

Charge flowing through wires encounters resistance or hindrance to its flow, which in turn reduces the amount of current and causes the wires to heat up. Here we have a problem about an electric toaster and we wish to determine the overall resistance of the nichrome wire or heating element within this toaster. In order to do so, we need to understand that resistance depends upon three different factors. It depends upon the material with which the wire is made out of. We typically differentiate between the different materials in terms of the so-called resistivity value. Here for nichrome wire, it's 110 times 10 to the negative 8 ohms times a meter. The other variables which affect the amount of resistance is how long the wire is with longer wires, offering more resistance to the overall flow of charge. And then finally, the cross-sectional area of the wire, which is typically viewed as a circular cross-section with a given diameter or radius. And here we're told the diameter is 0.56 millimeters. Now the crux of this problem will center around using the equation resistance is equal to resistivity multiplied by length and divided by cross-sectional area and giving significant attention to units. So what I have done is I've written down three bits of information. I've written down that L equals 220 centimeters. I've written down that diameter or D equals 0.56 millimeters. And I've written down that resistivity or the symbol is usually a Greek symbol, rho, 110 times 10 to the negative 8 ohms times a meter. And I've not only written the number down but also the unit. And what I'm going to have to do is calculate resistance in units of ohms. And I'll have to make sure that I get the meters in the numerator to cancel meters in the denominator. Now actually what you're going to have in the numerator of that equation is a meter coming from the resistivity unit and a meter coming from the length unit. And in the denominator what you're going to have is a meter squared coming from the area of a circular cross-section. So my first step is to take the 220 centimeters and convert that to 2.20 meters. That's just a movement of the decimal place two places since there's 100 centimeters in a meter. And then in the denominator I've got a more complicated situation. I need to calculate an area. And so what I'll need to do is remember that the area of a circular cross-section is simply pi r squared. So I'm going to need to use that equation. And that means that I'm going to have to first take the 0.56 millimeters and I'm going to have to divide it by 2 to turn it into a radius from a diameter. That would give me 0.28 millimeters. Now I'm going to need to move the decimal place three places to the left since there's 1,000 millimeters per one meter. That would give me 0.00028 meters. And now I'm going to go area equal pi r squared. Now I have my denominator. It's a really small number on the order of 2.4630 times 10 to the negative 7. And I plug that into the equation, resistivity times length divided by area. And I end up getting 9.8254 ohms. And I can round that to two significant digits.