A black body with surface area 0.001 `m^(2)` is heated upto a temperature 400 K and is suspended in a room temperature 300K. The intitial rate of loss of heat from the body to room is

Options

A. 10 W
B. 1W
C. 0.1W
D. 0.5 W

Explanation

To solve the problem of calculating the initial rate of heat loss from a black body to its surroundings, we can use the Stefan-Boltzmann law. The law states that the power radiated by a black body per unit area is proportional to the fourth power of its absolute temperature.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Surface area of the black body, \( A = 0.001 \, m^2 \)
- Temperature of the black body, \( T = 400 \, K \)
- Temperature of the surroundings (room temperature), \( T_0 = 300 \, K \)
- Stefan-Boltzmann constant, \( \sigma = 5.67 \times 10^{-8} \, W/m^2K^4 \)

2. **Calculate the Rate of Heat Emission from the Black Body**:
The rate of heat emission from the black body can be calculated using the formula:
\[ U = \sigma A T^4 \]
Substituting the known values:
\[ U = (5.67 \times 10^{-8}) \times (0.001) \times (400)^4 \]

3. **Calculate the Rate of Heat Absorption by the Surroundings**:
The rate of heat absorption by the surroundings can be calculated using the same formula, but with the room temperature:
\[ U_0 = \sigma A T_0^4 \]
Substituting the known values:
\[ U_0 = (5.67 \times 10^{-8}) \times (0.001) \times (300)^4 \]

4. **Calculate the Initial Rate of Heat Loss**:
The initial rate of heat loss from the black body to the surroundings is given by:
\[ \text{Rate of heat loss} = U - U_0 \]
Substitute the values calculated in the previous steps.

5. **Perform the Calculations**:
- Calculate \( U \):
\[ U = (5.67 \times 10^{-8}) \times (0.001) \times (400)^4 = (5.67 \times 10^{-8}) \times (0.001) \times 25600000 = 0.145 \, W \]
- Calculate \( U_0 \):
\[ U_0 = (5.67 \times 10^{-8}) \times (0.001) \times (300)^4 = (5.67 \times 10^{-8}) \times (0.001) \times 8100000 = 0.046 \, W \]
- Calculate the rate of heat loss:
\[ \text{Rate of heat loss} = 0.145 - 0.046 = 0.099 \, W \]

6. **Final Answer**:
The initial rate of loss of heat from the body to the room is approximately \( 0.099 \, W \).