Question

The ratio of cross-sectional areas of two conducting wires made up of same material and having same length is $1:2$. What will be the ratio of heat produced per second in the wires, when same current is flowing in them?

• A. $1:4$
• B. $2:1$
• C. $1:\sqrt{2}$
• D. $1:1$

Answer 1

Answer: (B)

Given $\frac{A_1}{A_2}=\frac{1}{2}$
and $i_1=i_2=i$
$l_1=l_2=l$
and ${\rho }_1={\rho }_2=\rho$
We know that the heat produced
$H=i^{2}Rt$
and heat produced per second
$H=i^{2}R\times 1$ (But $R=\frac{\rho l}{A}$)
$\Rightarrow H=i^2\rho \frac{l}{A}\times 1$
$\Rightarrow H=\frac{\rho l{i}^2}{A}$
So, the ratio of heat produced per second in both the wires
$\frac{H_1}{H_2}=\frac{{\rho }_1}{{\rho }_2}\frac{l_1{i}_1^2}{l_2{i}_2^2}\times \frac{A_2}{A_1}$
On putting the values
$\frac{H_1}{H_2}=\frac{\rho }{\rho }\times \frac{l}{l}\times \frac{i^2}{i^2}\times \frac{2}{1}$
$\Rightarrow \frac{H_1}{H_2}=\frac{2}{1}$
$\Rightarrow H_1:H_2=2:1$