Two bodies A and B are kept in an evacuated chamber at `27^@C`. The temperatures of A and B are `327^@ and 427^@C` and respectively. The ratio of rates of loss of heat from A and B will be–

Options

A. 0.52
B. 0.25
C. 1.52
D. 2.52

Answer

To solve the problem of finding the ratio of rates of heat loss from bodies A and B, we can use Stefan's law of thermal radiation. The power loss (rate of heat loss) from a body due to radiation is given by:

\[ P = \sigma A e (T^4 - T_s^4) \]

where:
- \( P \) is the power loss,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area,
- \( e \) is the emissivity,
- \( T \) is the absolute temperature of the body,
- \( T_s \) is the absolute temperature of the surroundings.

### Step 1: Convert temperatures to Kelvin
The temperatures of bodies A and B and the surroundings need to be converted from Celsius to Kelvin.

- Temperature of A: \( 327^\circ C = 327 + 273 = 600 \, K \)
- Temperature of B: \( 427^\circ C = 427 + 273 = 700 \, K \)
- Temperature of surroundings: \( 27^\circ C = 27 + 273 = 300 \, K \)

### Step 2: Write the expressions for power loss
Assuming the area \( A \) and emissivity \( e \) are the same for both bodies, we can write:

\[ P_A = \sigma A e (T_A^4 - T_s^4) \]
\[ P_B = \sigma A e (T_B^4 - T_s^4) \]

### Step 3: Set up the ratio of power losses
The ratio of the power loss from A to that from B can be expressed as:

\[ \frac{P_A}{P_B} = \frac{T_A^4 - T_s^4}{T_B^4 - T_s^4} \]

### Step 4: Substitute the temperatures
Substituting the temperatures we calculated:

\[ \frac{P_A}{P_B} = \frac{600^4 - 300^4}{700^4 - 300^4} \]

### Step 5: Calculate the powers
Now we need to calculate \( 600^4 \), \( 300^4 \), and \( 700^4 \):

- \( 600^4 = 129600000000 \)
- \( 300^4 = 8100000000 \)
- \( 700^4 = 240100000000 \)

### Step 6: Substitute and simplify
Now substituting these values into the ratio:

\[ \frac{P_A}{P_B} = \frac{129600000000 - 8100000000}{240100000000 - 8100000000} \]
\[ = \frac{121500000000}{232000000000} \]

### Step 7: Calculate the final ratio
Now, simplifying this ratio gives:

\[ \frac{P_A}{P_B} = \frac{1215}{2320} \approx 0.52 \]

### Conclusion
Thus, the ratio of rates of loss of heat from A and B is approximately \( 0.52 \).