Two heaters $A$ and $B$ have power rating of 1 kW and 2 kW , respectively. Those two are first conne

Question

Two heaters $A$ and $B$ have power rating of 1 kW and 2 kW , respectively. Those two are first connected in series and then in parallel to a fixed power source. The ratio of power outputs for these two cases is

Professor PC Thomas Classes · Answer

(b) : Given that, $P_A=1 \mathrm{~kW}, P_B=2 \mathrm{~kW}$ For the series combination of two heaters $A$ and $B$ of same voltage equivalent resistance is, $$ \begin{equation*} R_s=R_A+R_B 	ag{i} \end{equation*} $$ Power across both heaters, $P_A=\frac{V^2}{R_A}, P_B=\frac{V^2}{R_B}$ $$ \begin{align*} & 	herefore \frac{V^2}{P}=\frac{V^2}{P_A}+\frac{V^2}{P_B}  	ag{i}\ & 	herefore \frac{1}{P_s}=\frac{1}{P_A}+\frac{1}{P_B} \quad 	herefore P_s=\frac{P_A \cdot P_B}{P_A+P_B} \quad 	herefore P_s=\frac{1 	imes 2}{3}=\frac{2}{3} \mathrm{~kW} \end{align*} $$ For the parallel combination of the two heaters, \ $ \begin{aligned} & \frac{1}{R_P}=\frac{1}{R_A}+\frac{1}{R_B} ; \frac{P_P}{V^2}=\frac{P_A}{V^2}+\frac{P_B}{V^2} \ & 	herefore \quad P_P=P_A+P_B=3 \mathrm{~kW} \end{aligned} $ So ratio of power output in series to power output in parallel  \ $ 	ext { is }=\frac{2}{3} 	imes \frac{1}{3}=\frac{2}{9} $