Question

An aluminum wire of diameter D and length L has a resistance of 1.5 Ω. Find the length of another aluminum wire at the same temperature if its diameter is 2D and its resistance is 6 Ω.

Answer 1

Wire of the first case , Aluminum has the following information .

Length = L

Resistance = 1.5 $\Omega$

Diameter = D

Area A= $\pi r^2= \pi (\frac{D}{2})^2= \pi \frac{D^2}{4}$

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

$\rho$ = $\frac{\pi \times 0.25D^2\times 1.5}{L} = \frac{0.375D^2}{L} \pi$

$\rho$ = Resistivity is constant for all size of one type of material at a particular temperature .

Therefore the second aluminum also has the same resistivity as=$\frac{0.375D^2}{L} \pi$

the second aluminum piece has the following information,

R= 6 $\Omega$ Diameter = 2D ,

Area = $\pi r^2= \pi (\frac{2D}{2})^2 = \pi D^2$

Length of the wire = $\frac{AR}{\rho}$ = $\pi(\frac{D^2 \times 6\times L}{\pi \times 0.375\times D^2})= 16 L$