Two bulbs of 500 watt and 200 watt are manufactured to operate on 220 volt line. The ratio of heat produced in `500 W` and `200 W`, in two cases, when firstly they are joined in parallel and secondly in series, will be

Options

A. `5/2, 2/5`
B. `5/2,5/2`
C. `2/5, 5/2`
D. `2/5, 2/5`

Answer

To solve the problem of finding the ratio of heat produced in two bulbs of 500 W and 200 W when connected in parallel and in series, we can follow these steps:

### Step 1: Calculate the Resistance of Each Bulb
The resistance \( R \) of each bulb can be calculated using the formula:
\[ R = \frac{V^2}{P} \]
where \( V = 220 \, \text{V} \) and \( P \) is the power rating of the bulb.

- For the 500 W bulb:
\[ R_1 = \frac{220^2}{500} \]

- For the 200 W bulb:
\[ R_2 = \frac{220^2}{200} \]

### Step 2: Calculate the Heat Produced in Series
When the bulbs are connected in series, the same current flows through both bulbs. The heat produced \( H \) in each bulb can be expressed as:
\[ H \propto R \]
Thus, the ratio of heat produced in the series connection is:
\[ \frac{H_1}{H_2} = \frac{R_1}{R_2} \]
Substituting the expressions for resistance:
\[ \frac{H_1}{H_2} = \frac{\frac{220^2}{500}}{\frac{220^2}{200}} = \frac{200}{500} = \frac{2}{5} \]

### Step 3: Calculate the Heat Produced in Parallel
When the bulbs are connected in parallel, the voltage across each bulb is the same. The heat produced in each bulb can be expressed as:
\[ H \propto \frac{1}{R} \]
Thus, the ratio of heat produced in the parallel connection is:
\[ \frac{H_1}{H_2} = \frac{R_2}{R_1} \]
Substituting the expressions for resistance:
\[ \frac{H_1}{H_2} = \frac{\frac{220^2}{200}}{\frac{220^2}{500}} = \frac{500}{200} = \frac{5}{2} \]

### Step 4: Final Ratios
From the calculations, we have:
- For series connection: \( \frac{H_1}{H_2} = \frac{2}{5} \)
- For parallel connection: \( \frac{H_1}{H_2} = \frac{5}{2} \)

### Conclusion
The final answer for the ratio of heat produced in the two cases is:
- Series: \( \frac{2}{5} \)
- Parallel: \( \frac{5}{2} \)

Thus, the answer is \( \frac{5}{2}, \frac{2}{5} \).