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^2Rt $
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 li^2}{A} $
So, the ratio of heat produced per second in both the wires
$\frac {H_1}{H_2}= \frac {\rho _1}{\rho _2} \frac {l_1i_1^2}{l_2i_2^2} \times \frac {A_2}{A_1}$
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 $