Correct Option: Assertion (A) is True, but Reason (R) is False.
Step 1 (Formula / Governing Law): [1/3 Mark]
The power consumed by a resistor connected to a DC source of voltage $V$ is given by:
$$P = \frac{V^2}{R} \implies R = \frac{V^2}{P}$$For two heaters in series, the equivalent resistance $R_{eq}$ is $R_1 + R_2$. The total power consumed in series is $P_s = \frac{V^2}{R_1 + R_2}$.
Step 2 (Step-by-Step Mathematical Working): [1/3 Mark]
- Given $P_2 > P_1$, therefore $R_1 > R_2$ (since $R \propto 1/P$).
- Power of heater 1 alone: $P_1 = \frac{V^2}{R_1}$.
- Power of the series combination: $P_s = \frac{V^2}{R_1 + R_2}$.
- Since $(R_1 + R_2) > R_1$, it follows that $\frac{V^2}{R_1 + R_2} < \frac{V^2}{R_1}$.
- Thus, $P_s < P_1$. Assertion (A) is True.
- Reason (R) states $P \propto R$, but the formula $P = \frac{V^2}{R}$ shows $P \propto \frac{1}{R}$. Reason (R) is False.
Step 3 (Final Concluded Answer): [1/3 Mark]
The Assertion is correct because the series combination increases the total resistance, thereby decreasing the total power. The Reason is incorrect because power is inversely proportional to resistance for a fixed voltage. Therefore, (A) is True and (R) is False.
💡 EL10 Concept Insight:
Think of resistance like a narrow pipe that slows down water flow; because power is inversely related to resistance, a higher-power heater actually has a "wider" pipe (lower resistance). When you put two heaters in series, you are adding their resistances together, which makes the "pipe" even narrower and restricts the total power more than just using the single heater alone.