Question

The ratio of lengths of two wires made up of same material is 2 : 3 and the ratio of areas of cross section 3 : 2 . The ratio of their resistances is

• A. 9 : 4
• B. 4 : 9
• C. 1 : 3
• D. 3 : 1

Answer 1

Answer: (B)

Step 1: Given that:

The ratio of the length of two wires of the same material= $2:3$

The ratio of the area of cross-section of the wires= $3:2$

Step 2: Calculation of the ratio of their resistances:

Let the resistances of the wires are $R_1$ and $R_2$ and the resistivity of the material of the wire is ρ, then

we have

$\text{ρ}=\frac{RA}{l}$

Where $l$ is the length of the wire and $A$ is the area of cross-section of the wire.

Now,

For first wire

$\text{ρ}=\frac{R_1{A}_1}{l_1}$ ...........(1)

And for the second wire;

$\text{ρ}=\frac{R_2{A}_2}{l_2}$ ..........(2)

From equations 1) and 2), we get

$\frac{R_1{A}_1}{l_1}=\frac{R_2{A}_2}{l_2}$

$\frac{R_1}{R_2}=\frac{l_1}{l_2}\times \frac{A_2}{A_1}$

$Since,\frac{A_1}{A_2}=\frac{3}{2}then$

$\frac{A_2}{A_1}=\frac{2}{3}$

$Thus$

$\frac{R_1}{R_2}=\frac{2}{3}\times \frac{2}{3}$

$\frac{R_1}{R_2}=\frac{4}{9}$

$R_1:R_2=4:9$

Thus,

The ratio of their resistances is $4:9$ .