Two black bodies at temperatures `327^@ C` and `427^@ C` are kept in an evacuated chamber at `27^@ C`. The ratio of their rates of loss of heat are

Options

A. `(6/7)`
B. `(6/7)^2`
C. `(6/7)^3`
D. `(243 )/(464)`

Explanation

To solve the problem of finding the ratio of the rates of loss of heat from two black bodies at different temperatures, 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.

### Step-by-Step Solution:

1. **Convert Temperatures to Kelvin:**
- For the first body (T1):
\[ T_1 = 327^\circ C + 273 = 600 \, K \]
- For the second body (T2):
\[ T_2 = 427^\circ C + 273 = 700 \, K \]
- For the surrounding temperature (T0):
\[ T_0 = 27^\circ C + 273 = 300 \, K \]

2. **Apply the Stefan-Boltzmann Law:**
- The rate of heat loss (E) for a black body is given by:
\[ E = \sigma A (T^4 - T_0^4) \]
- Since both bodies are black bodies and have the same area (A) and emissivity (E), we can express the ratio of their rates of heat loss as:
\[ \frac{E_1}{E_2} = \frac{T_1^4 - T_0^4}{T_2^4 - T_0^4} \]

3. **Calculate \(T_1^4\) and \(T_2^4\):**
- Calculate \(T_1^4\):
\[ T_1^4 = (600)^4 = 1.296 \times 10^{11} \]
- Calculate \(T_2^4\):
\[ T_2^4 = (700)^4 = 2.401 \times 10^{11} \]
- Calculate \(T_0^4\):
\[ T_0^4 = (300)^4 = 8.1 \times 10^{9} \]

4. **Substitute Values into the Ratio:**
- Now substitute the values into the ratio:
\[ \frac{E_1}{E_2} = \frac{1.296 \times 10^{11} - 8.1 \times 10^{9}}{2.401 \times 10^{11} - 8.1 \times 10^{9}} \]
- Simplifying the numerator:
\[ 1.296 \times 10^{11} - 8.1 \times 10^{9} \approx 1.215 \times 10^{11} \]
- Simplifying the denominator:
\[ 2.401 \times 10^{11} - 8.1 \times 10^{9} \approx 2.320 \times 10^{11} \]

5. **Final Calculation of the Ratio:**
- Now calculate the final ratio:
\[ \frac{E_1}{E_2} = \frac{1.215 \times 10^{11}}{2.320 \times 10^{11}} = \frac{1.215}{2.320} \approx 0.524 \]
- To express this in a simpler form, we can multiply both the numerator and denominator by 1000 to avoid decimals:
\[ \frac{E_1}{E_2} \approx \frac{524}{1000} \approx \frac{243}{464} \]

### Conclusion:
The ratio of the rates of loss of heat from the two black bodies is:
\[ \frac{E_1}{E_2} = \frac{243}{464} \]