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
$t=\frac{t_1{t}_2}{t_1+t_2}$