The rate of loss of heat by radiation from a body at `400^(@)C` is R. The radiation from it when the temperature rises to `800^(@)C` ?

Options

A. 2 R
B. 4 R
C. 16 R
D. None of the above

Explanation

To solve the problem, we need to determine the rate of heat loss by radiation from a body when its temperature rises from \(400^\circ C\) to \(800^\circ C\). The rate of heat loss by radiation is given by the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to the fourth power of its absolute temperature.

### Step-by-Step Solution:

1. **Identify Initial and Final Temperatures**:
- The initial temperature \(T_1\) is \(400^\circ C\).
- The final temperature \(T_2\) is \(800^\circ C\).

2. **Convert Celsius to Kelvin**:
- To use the Stefan-Boltzmann law, we need to convert the temperatures from Celsius to Kelvin.
- \(T_1 = 400 + 273 = 673 \, K\)
- \(T_2 = 800 + 273 = 1073 \, K\)

3. **Use the Stefan-Boltzmann Law**:
- The rate of heat loss by radiation is proportional to the fourth power of the absolute temperature:
\[ R \propto T^4 \]
- Therefore, we can write the relationship between the rates of heat loss at the two temperatures:
\[ \frac{R_1}{R_2} = \left(\frac{T_1}{T_2}\right)^4 \]

4. **Substitute Known Values**:
- Given \(R_1 = R\) (the rate of heat loss at \(400^\circ C\)) and we need to find \(R_2\) (the rate of heat loss at \(800^\circ C\)):
\[ \frac{R}{R_2} = \left(\frac{673}{1073}\right)^4 \]

5. **Calculate the Ratio**:
- First, calculate the ratio:
\[ \frac{673}{1073} \approx 0.626 \]
- Now raise this ratio to the power of 4:
\[ \left(0.626\right)^4 \approx 0.15 \]

6. **Find \(R_2\)**:
- Rearranging the equation gives:
\[ R_2 = \frac{R}{0.15} \approx 6.66 R \]

### Final Answer:
The radiation from the body when the temperature rises to \(800^\circ C\) is approximately \(6.66R\).