Assertion : Two identical bulbs when connected across a battery, produce a total power P. When they are connected across the same battery in series total power consumed will be `(P)/(4)`.
Reason : In parallel, `P=P_(1)+P_(2)` and in series `P=(P_(1)P_(2))/(P_(1)+P_(2))`

Options

A. If both Assertion and Reason are true and Reason is the correct explanation of Assertion.
B. If both Assertion and Reason are true and Reason is not correct explanation of Assertion.
C. If Assertion is true but Reason is false.
D. If Assertion is false but Reason is true.

Answer

To solve the given problem, we need to analyze the assertion and reason provided regarding the power consumed by two identical bulbs when connected in parallel and in series across a battery.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- When two identical bulbs are connected in parallel across a battery, they produce a total power \( P \).
- When the same bulbs are connected in series across the same battery, the total power consumed will be \( \frac{P}{4} \).

2. **Power in Parallel**:
- Let the power consumed by each bulb when connected in parallel be \( P_1 \) and \( P_2 \).
- Since the bulbs are identical, \( P_1 = P_2 = \frac{P}{2} \).
- The total power in parallel is given by:
\[ P = P_1 + P_2 = \frac{P}{2} + \frac{P}{2} = P \]
- This confirms that the assertion holds true for parallel connection.

3. **Power in Series**:
- When the two bulbs are connected in series, the effective resistance \( R_{\text{eff}} \) is:
\[ R_{\text{eff}} = R + R = 2R \]
- The current \( I \) flowing through the circuit can be calculated using Ohm's law:
\[ I = \frac{E}{R_{\text{eff}}} = \frac{E}{2R} \]
- The power consumed in the series configuration is given by:
\[ P_{\text{series}} = I^2 R_{\text{eff}} = \left(\frac{E}{2R}\right)^2 (2R) = \frac{E^2}{4R} \]
- Since \( P = \frac{E^2}{R} \), we can express \( P_{\text{series}} \) as:
\[ P_{\text{series}} = \frac{P}{4} \]
- This confirms that the total power consumed when the bulbs are connected in series is indeed \( \frac{P}{4} \).

4. **Understanding the Reason**:
- The reason states the formulas for power in parallel and series:
- In parallel: \( P = P_1 + P_2 \)
- In series: \( P = \frac{P_1 P_2}{P_1 + P_2} \)
- For identical bulbs, substituting \( P_1 = P_2 = \frac{P}{2} \) into the series formula gives:
\[ P = \frac{\left(\frac{P}{2}\right) \left(\frac{P}{2}\right)}{\frac{P}{2} + \frac{P}{2}} = \frac{\frac{P^2}{4}}{P} = \frac{P}{4} \]
- This confirms that the reason is a correct explanation of the assertion.

### Conclusion:
Both the assertion and the reason are true, and the reason correctly explains the assertion.