# The Relationship of Power and Heat to Length of a Resistor

I have two sections of nichrome wire. They are identical in every way (gauge, etc.) except one is twice as long as the other. If I made it so the short nichrome wire received about 8.485 V and the long nichrome wire received 12 V, they would have equal power (both would release 3.6 W).

I know that the heat that both of these wires release would be the same. Does this mean that the heat released over the entire length of the wires are the same, or does it mean that the heat released at a given point along the length of one of the wires is the same as the heat released at a given point along the length of the other wire?

I use nichrome wires as heaters, and I want the heat at points along both of these wires to be the same; I don't care if the total amount of heats they release are the same.

Thank you!

The watt density is the power per unit surface area, determined over a vanishingly small area.

Since the wires are the same diameter, and the same resistivity, to get a constant power per unit area, the voltage per unit length must be the same.

Think about two exactly similar lengths of wire, powered, say, from 8.485V. Now connect them in series, and power them from a single supply.

• "I know that the heat that both of these wires release would be the same."

So when you enclose either wire in a box, the heat generated inside the box would be the same.

• "Does this mean that the heat released over the entire length of the wires are the same, or does it mean that the heat released at a given point along the length of one of the wires is the same as the heat released at a given point along the length of the other wire?"

We know that the heat generated in the box is the same for both wires. And we know that one wire is twice the length of the other. Hence the heat per unit of length for the longer wire is half of that for the shorter wire.

• "I use nichrome wires as heaters, and I want the heat at points along both of these wires to be the same; I don't care if the total amount of heats they release are the same."

If you want that, the voltage per unit of length must be the same.

• P = (V^2)/R, and therefore power and voltage are not directly proportional. If I increase voltage by a factor of 2, power will not increase by a factor of 2. It makes sense that the voltage per unit length must be the same, but I also know that heat equals power times time, and therefore I assumed that maybe I should double the power for the longer wire instead of the voltage. Am I thinking about this the wrong way? Thanks for your help. Commented Nov 27, 2018 at 18:58
• I think I figured it out - it has to do with the resistivity equation (resistance equals resistivity times length divided by cross-sectional area). When you combine this equation with the power equation (P = (V^2)/R) and set power per unit length of the short wire equal to power per unit length of the long wire, you can confirm that voltage per unit length of the short wire must equal voltage per unit length of the long wire ((V^2)/(L^2) for the short wire equals (V^2)/(L^2) for the long wire, and therefore V/L for the short wire equals V/L for the long wire). Commented Nov 27, 2018 at 19:14