Question

A nichrome wire of length l and area of cross-section $a/4$ has a resistance R. Another nichrome wire of length $3l$ and area of cross-section $a/2$ has a resistance of $R_1$. Find the ratio of $R_1:R$.

Answer 1

Step 1: Given data:

1. Initial resistance $R$
2. Length$=l$
3. Area $=a/4$
4. Length of another wire $=3l$
5. Area of another wire $=a/2$
6. Another Resistance $=R_1$

Step 2: Finding the ratio of $R_1:R$,

The formula for finding resistance is $R=\frac{\rho l}{A}$

Thus, the specific resistance in terms of resistance is,

$\rho =\frac{RA}{l}$

Both the wires are made up of nichrome. So they have the same specific resistance.

Substitute the given data in the formula.

$$\begin{gathered} \rho =\frac{R{\displaystyle \frac{a}{4}}}{l}=\frac{{\displaystyle R_1\frac{a}{2}}}{3l} \\ \frac{R}{4}=\frac{R_1}{6} \\ \frac{R_1}{R}=\frac{6}{4}=\frac{3}{2} \\ {R}_{1:}R=3:2 \end{gathered}$$

Thus, the ratio of $R_1:R$ is $3:2$.