Question

Two cylindrical wires $A$ and $B$ have the same resistance. The ratio of their specific resistances and diameters are $1:2$ each, then what is the ratio of the length of $B$ to the length of $A$ ?

Answer 1

Answer: 2

Resistance of a wire $R=\frac{\rho l}{\pi r^2}=\frac{\rho l\times 4}{\pi D^2}$
$\because$ $R_A=R_B$
$\therefore$ $\frac{4{\rho }_A{L}_A}{\pi D_A^2}=\frac{4{\rho }_B{L}_B}{\pi D_B^2}$
or $\frac{L_B}{L_A}=\left(\frac{{\rho }_A}{{\rho }_B}\right){\left(\frac{D_B}{D_A}\right)}^2$
$=\left(\frac{{\rho }_A}{2{\rho }_A}\right){\left(\frac{2D_A}{D_A}\right)}^2=\frac{4}{2}=\frac{2}{1}$