Question

Many web sites describe how to add wires to your clothing to keep you warm while riding your motorcycle. The wires are added to the clothing; a current from the 12 V battery of the motorcycle passes through the wires, warming them. One recipe for a vest calls for 10 m of 0.25-mm-diameter copper wire. How much power will this vest provide to warm the wearer?

Answer 1

Answer:

$P=42.075W$

Explanation:

The power provided by a resistor (wire in this case) is given by:

$P=\frac{V^2}{R}$.

The resistance of a wire is given by:

$R=\frac{\rho L}{A}$

Where for the resistivity the one of the copper should be used: $\rho=1.68\times10^{-8}\Omega m$.

The area A is that of a circle, which written in terms of its diameter is:

$A=\pi r^2=\pi (d/2)^2=\frac{\pi d^2}{4}$

Putting all together:

$P=\frac{AV^2}{\rho L}=\frac{\pi d^2V^2}{4\rho L}$

Which for our values is:

$P=\frac{\pi (0.00025m)^2(12V)^2}{4(1.68\times10^{-8}\Omega m)(10m)}=42.075W$