Question

Two heater wires, made of the same material and having the same length and the same radius, are first connected in series and then in parallel to a constant potential difference. If the rates of heat produced in the two cases are $H_s$ and $H_p$ respectively, then $H_s /H_p$ will be

• A. 1/2
• B. 2
• C. 1/4
• D. 4

Answer 1

Answer: (C)

When heater wires are connected in series, then equivalent resistance,
$R_{s}=R_{1}+R_{2}=2R\,\,\,\left(\because R_{1}=R_{2}\right)$
Rate of heat produced, $H_{s}=\frac{V^{2}}{ R_{s}}$
$H_{s}=\frac{V^{2}}{2 R}$...(i)
In second case, $R_{p}=\frac{R_{1}+R_{2}}{R_{1}+R_{2}}=\frac{R\times R}{2 R}$
or $R_{p}=\frac{R}{2}$
$\therefore $ Rate of heat produced, $H_{p}=\frac{V^{2}}{R_{p}}$
$H_{p}=\frac{2V^{2}}{R}$...(ii)
Dividing (i) by (ii), we get
$\frac{H_{s}}{H_{p}}=\frac{\left(V^{2}/2R\right)}{\left(2V^{2}/R\right)}$ $=\frac{V^{2}}{2 R}\times\frac{R}{2V^{2}}=\frac{1}{4}.$