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:

The correct answer is 2:9.

Step-by-step Explanation:

Let the rated voltage for both heaters A and B be $V_0$.
The power rating of heater A is given as $P_A=1\text{kW}$ and for heater B is $P_B=2\text{kW}$.

The resistance of an electrical appliance is related to its rated power and rated voltage by the formula:
$R=\frac{{V_0}^2}{P}$

Therefore, the resistance of heater A ($R_A$) and heater B ($R_B$) are:
$R_A=\frac{{V_0}^2}{1}$
$R_B=\frac{{V_0}^2}{2}$

From the above relations, we can see that:
$R_A=2R_B$

Let us define $R_B=R$. This means $R_A=2R$.

Now, let both heaters be connected to a fixed power source of voltage $V$.

Case 1: Series Connection
When the heaters are connected in series, the equivalent resistance ($R_s$) is the sum of their individual resistances:
$R_s=R_A+R_B=2R+R=3R$

The total power output in series ($P_s$) is given by:
$P_s=\frac{V^2}{R_s}=\frac{V^2}{3R}$

Case 2: Parallel Connection
When the heaters are connected in parallel, the equivalent resistance ($R_p$) is:
$R_p=\frac{R_A\times R_B}{R_A+R_B}=\frac{2R\times R}{2R+R}=\frac{2R}{3}$

The total power output in parallel ($P_p$) is given by:
$P_p=\frac{V^2}{R_p}=\frac{V^2}{\left(\frac{2R}{3}\right)}=\frac{3V^2}{2R}$

Ratio of Power Outputs:
The ratio of the power output in series to that in parallel is:
$\frac{P_s}{P_p}=\frac{\left(\frac{V^2}{3R}\right)}{\left(\frac{3V^2}{2R}\right)}$

Simplifying this ratio:
$\frac{P_s}{P_p}=\frac{1}{3}\times \frac{2}{3}=\frac{2}{9}$

Thus, the ratio of power outputs in series to parallel is 2:9.