Question

Assertion (A) : Two electric heaters of power $P_1$ and $P_2$ ($> P_1$) are joined in series across a dc source of voltage V. The power consumed by the combination will be less than that consumed by $P_1$ when connected across the same source.
Reason (R) : The power consumed by a electric device when connected to a dc source of voltage V is proportional to its resistance.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false and Reason (R) is also false.

Hint:

Always aggressively verify whether a problem assumes a constant voltage (parallel/household circuits, use $P = V^2/R$) or a constant current (series circuits, use $P = I^2 R$) before jumping to proportionality conclusions about power and resistance.

Answer 1

The Correct Option is C

Concept:
• The rated power $P$ of any standard electrical heating device explicitly designed to operate at a specific fixed voltage $V$ is intrinsically inversely proportional to its internal physical resistance $R$, governed by the formula $P = \frac{V^2}{R}$.

• When multiple electrical resistors are physically joined end-to-end in a series configuration, their total effective equivalent resistance strictly increases, combining as $R_{eq} = R_1 + R_2$.

• A higher overall system resistance connected to the exact same fixed voltage source will inevitably draw much less total current, thereby drastically reducing the total power consumed by the entire system.

Step 1: Evaluate the Assertion (A)
Let the internal resistances of the two electric heaters be strictly $R_1$ and $R_2$.
Since their nominal rated powers at voltage $V$ are $P_1 = \frac{V^2}{R_1}$ and $P_2 = \frac{V^2}{R_2}$.
When these two heaters are violently joined in a series circuit, their new total effective resistance becomes $R_{eq} = R_1 + R_2$.
It is mathematically obvious that $R_{eq}$ is strictly greater than $R_1$ alone ($R_{eq} > R_1$).
The total combined power actually consumed by this new series combination when connected across the same source voltage $V$ is:
\[ P_{series} = \frac{V^2}{R_{eq}} \]
Because the denominator has significantly increased ($R_{eq} > R_1$), the resulting overall fraction must inevitably decrease:
\[ \frac{V^2}{R_{eq}} < \frac{V^2}{R_1} \implies P_{series} < P_1 \]
Therefore, the power consumed by the series combination is undeniably less than that consumed by heater 1 operating alone. The Assertion (A) is absolutely true.

Step 2: Evaluate the Reason (R)
The provided reason emphatically claims that the power consumed by an electrical device tightly connected to a fixed DC voltage source $V$ is strictly proportional to its internal resistance.
The foundational mathematical formula governing this exact scenario is $P = \frac{V^2}{R}$.
This formula clearly and undeniably demonstrates that, for any constant voltage source, the power consumed is strictly inversely proportional to the physical resistance ($P \propto \frac{1}{R}$), not directly proportional.
Therefore, the Reason (R) is fundamentally mathematically and physically false.

Step 3: Conclusion
Since the Assertion (A) is completely true but the corresponding Reason (R) contains a fatal physical error and is false, the correct choice is definitively option (C).