Two bulbs consume the same power when operated at 200 V and 300 V respectively. When these bulbs are connected in series across a DC source of 400 V, then the ratio of power consumed across them is

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `(4)/(9)`
D. `(9)/(4)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have two bulbs that consume the same power when operated at different voltages:
- Bulb 1 operates at 200 V
- Bulb 2 operates at 300 V
Both bulbs consume power \( P \).

### Step 2: Calculate the resistances of the bulbs
Using the formula for power:
\[ P = \frac{V^2}{R} \]
We can rearrange this to find the resistance:
\[ R = \frac{V^2}{P} \]

For Bulb 1:
\[ R_1 = \frac{(200)^2}{P} = \frac{40000}{P} \]

For Bulb 2:
\[ R_2 = \frac{(300)^2}{P} = \frac{90000}{P} \]

### Step 3: Set up the series circuit
When the two bulbs are connected in series across a 400 V DC source, the same current \( I \) flows through both bulbs. The total voltage across the series circuit is:
\[ V_{total} = V_1 + V_2 = 400 V \]

### Step 4: Express power consumed by each bulb
The power consumed by each bulb in a series circuit can be expressed as:
\[ P = I^2 R \]

Thus, the power consumed by Bulb 1:
\[ P_1 = I^2 R_1 = I^2 \left(\frac{40000}{P}\right) \]

And for Bulb 2:
\[ P_2 = I^2 R_2 = I^2 \left(\frac{90000}{P}\right) \]

### Step 5: Find the ratio of power consumed
To find the ratio of power consumed by both bulbs:
\[ \frac{P_1}{P_2} = \frac{I^2 R_1}{I^2 R_2} = \frac{R_1}{R_2} \]

Substituting the values of \( R_1 \) and \( R_2 \):
\[ \frac{P_1}{P_2} = \frac{\frac{40000}{P}}{\frac{90000}{P}} = \frac{40000}{90000} = \frac{4}{9} \]

### Final Answer
The ratio of power consumed across the two bulbs when connected in series across a 400 V source is:
\[ \frac{P_1}{P_2} = \frac{4}{9} \]