Question

Two heaters designed for the same voltage $V$ have different power ratings. When connected individually across a source of voltage $V$, they produce $H$ amount of heat each in times $t_{1}$ and $t_{2}$ respectively. When used together across the same source, they produce $H$ amount of heat in time $t$.

• A. If they are in series, then $t=t_{1}+t_{2}$
• B. If they are in series, then $t=2\left(t_{1}+t_{2}\right)$
• C. If they are in parallel, then $t=\frac{t_{1} \times t_{2}}{\left(t_{1}+t_{2}\right)}$
• D. If they are in parallel, then $t=\frac{t_{1} t_{2}}{2\left(t_{1}+t_{2}\right)}$

Answer 1

Answer: (C)

As, $H=\frac{V^{2}}{R_{1}} t_{1}=\frac{V^{2}}{R_{2}} t_{2}$
or $R_{1}=\frac{H}{V^{2} t_{1}}$
and $R_{2}=\frac{H}{V^{2} t_{2}}$
When heaters are in series,
$H=\frac{V^{2} t}{\left(R_{1}+R_{2}\right)}$ .....(i)
When heaters are in parallel,
$H=\left(\frac{V^{2}}{R_{1}}+\frac{V^{2}}{R_{2}}\right) t$ .......(ii)
Putting the values of $R_{1}$ and $R_{2}$ in Eq. (i), we have
$t=t_{1}+t_{2}$
When the values of $R_{1}$ and $R_{2}$ is put in Eq. (ii), we get
$t=\frac{t_{1} t_{2}}{t_{1}+t_{2}}$