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. 1:04
• B. 2:01
• C. $1: \sqrt 2 $
• D. 1:01

Answer 1

The Correct Option is B

Given $\hspace5mm \frac {A_1}{A_2}= \frac {1}{2} $
and $\hspace5mm i_1=i_2=i $
$\hspace5mm l_1=l_2=l $
and $\hspace5mm \rho _1= \rho _2= \rho $
We know that the heat produced
$\hspace15mm H=i^2Rt $
and heat produced per second
$\hspace15mm H=i^2R\times 1 \hspace15mm \bigg (But \, R= \frac {\rho l}{A}\bigg ) $
$\Rightarrow \hspace10mm H=i^2 \rho \frac {l}{A} \times 1 $
$\Rightarrow \hspace10mm H=\frac {\rho li^2}{A} $
So, the ratio of heat produced per second in both the wires
$\hspace10mm \frac {H_1}{H_2}= \frac {\rho _1}{\rho _2} \frac {l_1i_1^2}{l_2i_2^2} \times \frac {A_2}{A_1}$
On putting the values
$\hspace10mm \frac {H_1}{H_2}= \frac {\rho}{\rho}\times \frac {l}{l} \times \frac {i^2}{i^2}\times \frac {2}{1} $
$\Rightarrow \hspace5mm \frac {H_1}{H_2}= \frac {2}{1} $
$\Rightarrow \hspace5mm H_1:H_2=2:1 $