Two bulbs which consume powers `P_(1) " and P_(2)` are connected in series. The power consumed by the combination is

Options

A. `P_(1)+P_(2)`
B. `sqrt(P_(1)P_(2)`
C. `(P_(1)P_(2))/(P_(1)+P_(2))`
D. `(2P_(1)P_(2))/(P_(1)+P_(2))`

Explanation

To find the power consumed by two bulbs connected in series, we can follow these steps:

### Step 1: Understand the Power Formula
The power consumed by a bulb can be expressed using the formula:
\[ P = \frac{V^2}{R} \]
where \( P \) is the power, \( V \) is the voltage across the bulb, and \( R \) is the resistance of the bulb.

### Step 2: Identify the Resistances
Let the resistances of the two bulbs be \( R_1 \) and \( R_2 \). The power consumed by the first bulb is:
\[ P_1 = \frac{V_1^2}{R_1} \]
and for the second bulb:
\[ P_2 = \frac{V_2^2}{R_2} \]

### Step 3: Series Connection of Bulbs
In a series connection, the total voltage \( V \) across the combination is distributed across the two bulbs. The total resistance in series is:
\[ R_{\text{total}} = R_1 + R_2 \]

### Step 4: Relate Voltage and Power
Since the bulbs are in series, the same current \( I \) flows through both. The power consumed by the combination can be expressed as:
\[ P = \frac{V^2}{R_{\text{total}}} = \frac{V^2}{R_1 + R_2} \]

### Step 5: Express the Resistances in Terms of Power
From the power formulas, we can express the resistances in terms of the powers:
\[ R_1 = \frac{V_1^2}{P_1} \]
\[ R_2 = \frac{V_2^2}{P_2} \]

### Step 6: Use the Voltage Division Rule
Using the voltage division rule in series circuits, we have:
\[ V_1 = \frac{R_1}{R_1 + R_2} V \]
\[ V_2 = \frac{R_2}{R_1 + R_2} V \]

### Step 7: Substitute Back to Find Total Power
Substituting \( R_1 \) and \( R_2 \) back into the power formula for the total power:
\[ P = \frac{V^2}{R_1 + R_2} = \frac{V^2}{\frac{V_1^2}{P_1} + \frac{V_2^2}{P_2}} \]

### Step 8: Final Expression for Power
Using the relationship between the powers and resistances, we can derive:
\[ \frac{1}{P} = \frac{1}{P_1} + \frac{1}{P_2} \]
Cross-multiplying gives:
\[ P = \frac{P_1 P_2}{P_1 + P_2} \]

### Conclusion
Thus, the power consumed by the combination of the two bulbs connected in series is:
\[ P = \frac{P_1 P_2}{P_1 + P_2} \]