Question

two conducting wire so same length are made from the same material. one wire is solid with diameter 1 millimetre while the other one is Hollow with outer diameter 2 mm and inner diameter 1 mm. the ratio of the resistances will be:
a) 1:2
b) 1:3
c) 3:1
d) 2:1

Answer 1

$$\begin{gathered} Resis\tan ceofthesolidwire=R_{solid}=\rho \frac{L}{A} \\ A=\pi \frac{d^2}{4}=\pi \frac{1^2}{4}=\frac{\pi }{4} \\ {R}_{solid}=4\rho \frac{L}{\pi } \\ {R}_{hollow}=\rho \frac{L}{A\text{'}} \\ A\text{'}=\pi \left(\frac{{d_{outer}}^2}{4}-\frac{{d_{inner}}^2}{4}\right) \\ A\text{'}=\pi \left(\frac{2^2}{4}-\frac{1^2}{4}\right) \\ A\text{'}=\frac{3}{4}\pi \\ {R}_{hollow}=\rho \frac{L}{\frac{3}{4}\pi }=\frac{4}{3}\rho \frac{L}{\pi } \\ \frac{R_{solid}}{R_{hollow}}=\frac{4\rho \frac{L}{\pi }}{\frac{4}{3}\rho \frac{L}{\pi }}=\frac{3}{1} \\ {R}_{solid}:R_{hollow}=3:1 \end{gathered}$$