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 an electric device when connected to a dc source of voltage \( V \) is proportional to its resistance.

• A. Both A and R are true and R explains A
• B. Both A and R are true but R does not explain A
• C. A is true but R is false
• D. A is false and R is false

Hint:

For constant voltage, higher resistance means lower power.

Answer 1

The Correct Option is C

Concept: The electrical power consumed by a resistor connected across a source of constant voltage \(V\) is given by:

\[ P=\frac{V^2}{R} \]

Thus, for a fixed voltage source, power is inversely proportional to resistance. When resistors are connected in series, their equivalent resistance increases, causing the current in the circuit to decrease. As a result, the total power consumed by the combination becomes smaller than the power consumed by an individual resistor connected directly across the same source.

Assertion Analysis:

Let the resistances of the two heaters be \(R_1\) and \(R_2\). Since the heaters are rated at powers \(P_1\) and \(P_2\) when connected across the same voltage \(V\),

\[ P_1=\frac{V^2}{R_1}, \qquad P_2=\frac{V^2}{R_2} \]

Given,

\[ P_2>P_1 \]

Therefore,

\[ R_2

When the heaters are connected in series, the equivalent resistance becomes

\[ R_s=R_1+R_2 \]

Since

\[ R_1+R_2>R_1, \]

the power consumed by the series combination is

\[ P_s=\frac{V^2}{R_1+R_2} \]

Because the denominator is larger, the power consumed by the series combination is less than the power consumed by either heater individually. Hence, the Assertion is true.

Reason Analysis:

The reason claims that power consumed by an electrical device connected across a DC voltage source is directly proportional to resistance.

For a constant voltage source,

\[ P=\frac{V^2}{R} \]

This clearly shows that power is inversely proportional to resistance. If resistance increases, power decreases; if resistance decreases, power increases. Therefore, the Reason is false.

Relationship Between Assertion and Reason:

The Assertion is true because connecting the heaters in series increases the equivalent resistance, which reduces the total power consumed. The Reason is false because power is not directly proportional to resistance for a constant voltage source.

Final Answer:

\[ \boxed{(\mathrm{C})} \]