Two resistors are made of the same material, one twice as long as the other. If the current through the shorter resistor is 5 A, what is the current through the longer resistor if they both have the same potential difference and cross-sectional area?

Hey, guys. So we're going to work this one out together. Hopefully, you guys took a shot at it on your own. So we've got these 2 resistors that are made of the same material, but one of them is twice as long as the other. So we have the currents and the resistances. We're supposed to be figuring out what the current is of the longer resistor. So we have the relationship between currents. So we have the relationship between VI and R. That's Ohm's law. So v equals I times R. So in other words, for both of these wires, the short one and the long one sorry, the problem of these resistors, the voltage is going to be the current times the resistance. And that holds true for the long and the short one. So I have v equals ilrl and v equals isrs because I'm told that these things both have the same potential differences. Okay? So we have the current. So in other words, IS is equal to V divided by RS, and we know that that is equal to 5 amps. We don't know what any of those actual numbers are. But what we do know is that one of these things is twice as long as the other. So if I need to figure out what IL is equal to, so it's v over r l. Well, we know that the potential differences are gonna be the same. So what's different about them? Well, what's different is that the resistances are changed because we know that the resistance is equal to row times l over a. So row for or sorry. The r for the short one is gonna be row times l over a, And then r for the long one, rl, is gonna be rho times 2 times l because it's twice as long divided by the same exact area. So in other words, this long resistor actually has twice the resistance of the short resistor because the only thing that's different about this equation is that we have a 2 on the outside. So we can basically pull it out, and this just becomes twice the resistance of the shorter one. So basically, what ends up happening is that the current of the short one is just V over 2 RS. In other words, it's one half of what this expression is. And we know that that's equal to 5 amps. So in other words, it's just 1 half of 5. In other words, that the current across the longer one has to be 2.5 amps. Alright. So one of the ways we can also see that is just sort of from this equation right here. V equals I times r. So if the shorter one is 5 amps and the resistance of the longer one is gonna be double, then that means that the current has to be cut in half in order to maintain the same exact voltage. So it has to be half of what 5 amps is. Okay? Anyway, that's the answer to this one. Let me know if you guys have any questions, and let's keep going.

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Two resistors are made of the same material, one twice as long as the other. If the current through the shorter resistor is 5 A, what is the current through the longer resistor if they both have the same potential difference and cross-sectional area?

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