Question

A nichrome wire of length \(l\) and area of cross-section \(a / 4\) has a resistance R. Another nichrome wire of length \(3 l\) and area of cross-section \(a / 2\) has a resistance of \(\text{R}_{1}\). Find the ratio of \(\text{R}_{1}: \text{R}\).

Answer 1

Case 1: \[R;\ \text{length}=l;\ \text{area}=\frac{a}{4}\]

∴ \[R=\rho\frac{l}{a/4}\qquad\text{...(1)}\]

Case 2: \[R_{1}=?;\ \text{length}=3l;\ \text{area}=\frac{a}{2}\]

∴ \[R_{1}=\rho\frac{3l}{a/2}\qquad\text{...(2)}\]

Dividing (2) by (1)

\[\frac{R_{1}}{R}=\frac{\rho\cdot3l}{a/2}\times\frac{a/4}{\rho\cdot l}=\frac{3}{2}\]

∴ \[R_{1}:R=3:2\]