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

Options

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

Answer

To solve the problem, we will use the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to the fourth power of its absolute temperature. The formula is given by:

\[ P = \sigma A T^4 \]

where:
- \( P \) is the power radiated,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area of the object,
- \( T \) is the absolute temperature in Kelvin.

### Step-by-Step Solution:

1. **Convert the initial temperature from Celsius to Kelvin:**
The initial temperature of the object is given as \( 400^\circ C \). To convert Celsius to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
Therefore,
\[ T_1 = 400 + 273.15 = 673.15 \, K \]

2. **Set up the equation for the power radiated:**
The power radiated by the object at the initial temperature \( T_1 \) is:
\[ P_1 = \sigma A T_1^4 \]

3. **Determine the condition for the power to be twice:**
We want to find the new temperature \( T_2 \) at which the power radiated is twice that of \( P_1 \):
\[ P_2 = 2 P_1 \]
Using the Stefan-Boltzmann law, we can express this as:
\[ \sigma A T_2^4 = 2 \sigma A T_1^4 \]

4. **Simplify the equation:**
Since \( \sigma \) and \( A \) are constants, they can be canceled out from both sides:
\[ T_2^4 = 2 T_1^4 \]

5. **Solve for \( T_2 \):**
Taking the fourth root of both sides gives:
\[ T_2 = T_1 \cdot (2^{1/4}) \]
Substituting \( T_1 = 673.15 \, K \):
\[ T_2 = 673.15 \cdot (2^{1/4}) \]

6. **Calculate \( 2^{1/4} \):**
The value of \( 2^{1/4} \) is approximately \( 1.1892 \). Therefore:
\[ T_2 \approx 673.15 \cdot 1.1892 \approx 799.99 \, K \]

7. **Convert \( T_2 \) back to Celsius:**
To convert back to Celsius:
\[ T_2(°C) = T_2(K) - 273.15 \]
Thus,
\[ T_2(°C) \approx 799.99 - 273.15 \approx 526.84 \, °C \]

### Final Answer:
The temperature at which the object would radiate energy twice as fast is approximately \( 526.84^\circ C \).