Question

Two wire of same length is made of brass and sliver but wires are different diameter. Diameter of the brass wire is 4 times the diameter of the silver wire and the resistivity of brass is 5 times greater than the resistivity of silver. If $R_B$ denotes the resistance of the brass wire and $R_S$ denotes the resistance of the silver wire, which of the following is true?

• A. $R_B=\frac{5}{16}{R}_S$
• B. $R_B=\frac{4}{5}{R}_S$
• C. $R_B=\frac{5}{4}{R}_S$
• D. $R_B=\frac{5}{2}{R}_S$
• E. $R_B=\frac{16}{5}{R}_S$

Answer 1

The correct option is A $R_B=\frac{5}{16}{R}_S$

Resistance $R=\frac{\rho L}{A}$ where $\rho$ is the resistivity

Also $A=\frac{\pi D^2}{4}$ $⟹R=\frac{4\rho L}{\pi D^2}=k\frac{\rho L}{D^2}$

Given : $D_B=4D_S$ ${\rho }_B=5{\rho }_S$

For silver : $R_S=k\frac{{\rho }_{S}L}{D_S^2}$

For Brass : $R_B=k\frac{{\rho }_{B}L}{D_B^2}=k\frac{\left(5{\rho }_S\right)L}{(4D_S{)}^2}=\frac{5}{16}{R}_S$