An object is at temperature of `400^(@)C`. At what approximate temperature would it radiate energy twice as first ? The temperature of surroundings may be assumed to be negligible:

Options

A. `200 ^(@)C`
B. 200K
C. `800^(@)C`
D. 800 K

Explanation

To solve the problem of finding the temperature at which an object radiates energy twice as much as it does at 400°C, we can follow these steps:

### Step-by-Step Solution:

1. **Convert the initial temperature to Kelvin**:
The initial temperature of the object is given as 400°C. To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273 \]
Therefore,
\[ T_1 = 400 + 273 = 673 \, K \]

2. **Understand the relationship between power and temperature**:
The power radiated by an object is given by the Stefan-Boltzmann law:
\[ P \propto T^4 \]
This means that the power radiated is proportional to the fourth power of the absolute temperature.

3. **Set up the equation for the new power**:
If at temperature \( T_1 \) the power is \( P_1 \), and we want to find the temperature \( T_2 \) at which the power is \( 2P_1 \), we can write:
\[ P_1 \propto T_1^4 \quad \text{and} \quad 2P_1 \propto T_2^4 \]

4. **Formulate the relationship**:
From the above proportionalities, we can express the relationship as:
\[ \frac{2P_1}{P_1} = \frac{T_2^4}{T_1^4} \]
This simplifies to:
\[ 2 = \left(\frac{T_2}{T_1}\right)^4 \]

5. **Solve for \( T_2 \)**:
Taking the fourth root of both sides, we have:
\[ \frac{T_2}{T_1} = 2^{1/4} \]
Therefore,
\[ T_2 = T_1 \cdot 2^{1/4} \]
Substituting \( T_1 = 673 \, K \):
\[ T_2 = 673 \cdot 2^{1/4} \]

6. **Calculate \( 2^{1/4} \)**:
The value of \( 2^{1/4} \) is approximately \( 1.1892 \). Thus,
\[ T_2 \approx 673 \cdot 1.1892 \approx 800 \, K \]

7. **Convert back to Celsius (if needed)**:
If you need the temperature in Celsius, convert it back:
\[ T_2(°C) = T_2(K) - 273 \approx 800 - 273 \approx 527°C \]

### Final Answer:
The approximate temperature at which the object would radiate energy twice as much as at 400°C is **800 K**.