If a body coated black at 600 k surrounded by atmosphere at, 300 k has cooling rate `r`1 ​ the same body at 900 k surrounded by the same atmosphere , will have cooling rate equal to

Options

A. `16/3 r1`
B. `18/3 r0`
C. `16r0`
D. `4r0`

Answer

To find the cooling rate of a body at different temperatures, we can use the Stefan-Boltzmann law, which states that the rate of heat loss of a body is proportional to the fourth power of its absolute temperature minus the fourth power of the absolute temperature of its surroundings.

### Step-by-Step Solution:

1. **Identify the Cooling Rate Formula**:
The cooling rate \( R \) can be expressed as:
\[ R \propto T^4 - T_0^4 \]
where \( T \) is the temperature of the body and \( T_0 \) is the temperature of the surroundings.

2. **Define the Cooling Rates**:
For the first scenario (body at 600 K):
\[ R_1 = k (T_1^4 - T_0^4) \]
where \( T_1 = 600 \, K \) and \( T_0 = 300 \, K \).

For the second scenario (body at 900 K):
\[ R_2 = k (T_2^4 - T_0^4) \]
where \( T_2 = 900 \, K \).

3. **Substitute the Values**:
Substitute the values into the equations:
\[ R_1 = k (600^4 - 300^4) \]
\[ R_2 = k (900^4 - 300^4) \]

4. **Calculate the Ratios**:
To find the ratio of the cooling rates \( \frac{R_2}{R_1} \):
\[ \frac{R_2}{R_1} = \frac{900^4 - 300^4}{600^4 - 300^4} \]

5. **Simplify the Expression**:
We can factor out \( 300^4 \):
\[ 900^4 = (3 \times 300)^4 = 81 \times 300^4 \]
\[ 600^4 = (2 \times 300)^4 = 16 \times 300^4 \]
Thus, we have:
\[ R_2 = k (81 \times 300^4 - 300^4) = k (80 \times 300^4) \]
\[ R_1 = k (16 \times 300^4 - 300^4) = k (15 \times 300^4) \]

6. **Calculate the Final Ratio**:
Now substituting back:
\[ \frac{R_2}{R_1} = \frac{80 \times 300^4}{15 \times 300^4} = \frac{80}{15} = \frac{16}{3} \]

7. **Determine the Cooling Rate**:
Thus, we find:
\[ R_2 = \frac{16}{3} R_1 \]

### Final Answer:
The cooling rate \( R_2 \) when the body is at 900 K is:
\[ R_2 = \frac{16}{3} R_1 \]