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

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{2 D_{ A }}{D_{ A }}\right)^{2}=\frac{4}{2}=\frac{2}{1} $