Question

Two conductors are made of the same material and have the same length. Conductor $A$ is a solid wire of diameter $1.0mm$ and conductor $B$ is a hollow tube of outside diameter $2.0mm$ and inside diameter $1.0mm$ . Find the resistance ratio $\frac{R_A}{R_B}$ measured between their ends

• A. 3
• B. 2
• C. 1
• D. 0.5

Answer 1

Answer: (A)

Resistance of wire, $R_A=\frac{\rho l}{A}=\frac{\rho l}{\pi {\left[\frac{D}{2}\right]}^2}=\frac{\rho l}{\frac{\pi D^2}{4}}$
$=\frac{\rho l}{\pi \frac{{\left(1\right)}^2}{4}}=\frac{4\rho l}{\pi }$
Similarly, for hollow conductor $B$
$R_B=\frac{\rho l}{A}=\frac{\rho l}{\pi \left[\frac{D_1^2}{4}-\frac{D_2^2}{4}\right]}=\frac{\rho l}{\pi \left[\frac{2^2}{4}-\frac{1^2}{4}\right]}$
$=\frac{4}{3}\frac{\rho l}{\pi }$
$\therefore \frac{R_A}{R_B}=\frac{4\rho l}{\pi }\times \frac{3\pi }{4\rho l}=3$