Two electric bulbs rated `P_(1)` watt and V volt, are connected in series, across V-volt supply. The total power consumed is

Options

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

Answer

To solve the problem of finding the total power consumed by two electric bulbs rated \( P_1 \) watts and connected in series across a \( V \)-volt supply, we will follow these steps:

### Step 1: Determine the Resistance of Each Bulb
The resistance \( R \) of an electric bulb can be calculated using the formula:
\[ R = \frac{V^2}{P} \]
For the first bulb rated \( P_1 \) watts:
\[ R_1 = \frac{V^2}{P_1} \]
For the second bulb (assuming it has a power rating \( P_2 \)):
\[ R_2 = \frac{V^2}{P_2} \]

### Step 2: Calculate the Total Resistance in Series
When resistances are connected in series, the total resistance \( R_{\text{total}} \) is the sum of the individual resistances:
\[ R_{\text{total}} = R_1 + R_2 = \frac{V^2}{P_1} + \frac{V^2}{P_2} \]

### Step 3: Simplify the Total Resistance
To simplify the expression for \( R_{\text{total}} \):
\[ R_{\text{total}} = V^2 \left( \frac{1}{P_1} + \frac{1}{P_2} \right) \]
Taking the common denominator:
\[ R_{\text{total}} = \frac{V^2 (P_2 + P_1)}{P_1 P_2} \]

### Step 4: Calculate the Total Power Consumed
The total power \( P \) consumed in a circuit can be calculated using the formula:
\[ P = \frac{V^2}{R_{\text{total}}} \]
Substituting the expression for \( R_{\text{total}} \):
\[ P = \frac{V^2}{\frac{V^2 (P_1 + P_2)}{P_1 P_2}} = \frac{P_1 P_2}{P_1 + P_2} \]

### Final Answer
Thus, the total power consumed by the two bulbs connected in series across a \( V \)-volt supply is:
\[ P = \frac{P_1 P_2}{P_1 + P_2} \]