Question

A uniform wire when connected directly across a 200 V line, produces heat $H$ per second. If the wire is divided into $n$ parts and all parts are connected in parallel across a 200 V line, the heat produced per second will be

• A. $H$
• B. $nH$
• C. $\frac{H}{n^2}$
• D. $n^{2}H$

Answer 1

Answer: (D)

$n^{2}H$

Initially the heat produced in wire is
$H=\frac{V^2}{R}$
when the wire is divided into n parts hence, resistance of each part will be $\frac{R}{n}$
Now, all parts are connected in parallel across a 200 V line, the equivalent heat is
$H_{eq}=H_1+H_2+........+H_n$
$H_{eq}=\frac{V^2}{R_1}+\frac{V^2}{R_2}+........+\frac{V^2}{R_n}$
$H_{eq}=V^2[\frac{1}{R_1}+\frac{1}{R_2}+........+\frac{1}{R_n}]$
$H_{eq}=V^2[\frac{1}{\frac{R_1}{n}}+\frac{1}{\frac{R_2}{n}}+........+\frac{1}{\frac{R_n}{n}}]$
$\Rightarrow H_{eq}=n{V}^2[\frac{1}{R_1}+\frac{1}{R_2}+........+\frac{1}{R_n}]$
$\Rightarrow H_{eq}=n\left(nH\right)=n^{2}H$