Question

A uniform wire connected across a supply produces heat H per second. If the wire is cut into n equal parts and all the parts are connected in parallel across the same supply, the heat produced per second will be

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

Answer 1

The correct option is B $n^{2}H$

A uniform wire connected across a supply produces heat H per second is,
$H=\frac{V^2}{R}$
If the wire is cut into n equal parts the resistance of each part is
$R=\frac{R}{n}$
and when all the parts are connected in parallel across the same supply,
$\frac{1}{R_{eq}}=\frac{R}{n}+\frac{R}{n}$ + ...........n times
$\frac{1}{R_{eq}}=\frac{R}{n^2}$
heat produced per second will be
$H^{\prime }=H_1+H_2+H_3+............+H_n$
$H^{\prime }=\frac{V^2}{R_1}+\frac{V^2}{R_2}+\frac{V^2}{R_3}+........+\frac{V^2}{R_n}$
$H^{\prime }=V^2[\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+........+\frac{1}{R_n}]$
$H^{\prime }=\frac{V^2}{R_{eq}}=\frac{V^2}{\frac{R}{n^2}}$
$H^{\prime }=n^2\frac{V^2}{R}=n^{2}H$