Two uniform solid spheres `A` and `B` of same material, painted completely black and placed in free space separately. Their radii are `R` and `2R` respectively and the domainating wavelengths in their spectrum are observed to be in the ratio `1:2`, which of the following is not correct.

Options

A. Ratio of their rates of heat loss is `4:1`
B. Ratio of their rates of cooling is `32:1`
C. Ratio of their temperatures is `2:1`
D. Ratio of their emissive powers is `4:1`

Explanation

To solve the problem, we need to analyze the properties of the two spheres A and B based on the given information about their sizes and the ratio of their dominating wavelengths.

### Step-by-step Solution:

1. **Understanding the Problem**:
- We have two spheres, A with radius \( R \) and B with radius \( 2R \).
- The ratio of their dominating wavelengths (\( \lambda_1 \) for sphere A and \( \lambda_2 \) for sphere B) is given as \( \frac{\lambda_1}{\lambda_2} = \frac{1}{2} \).

2. **Using Wien's Displacement Law**:
- According to Wien's Displacement Law, the temperature \( T \) of a black body is inversely proportional to its dominating wavelength:
\[ T \propto \frac{1}{\lambda} \]
- Therefore, we can express the temperatures of the two spheres as:
\[ \frac{T_1}{T_2} = \frac{\lambda_2}{\lambda_1} = \frac{2}{1} = 2 \]
- This means \( T_1 = 2T_2 \).

3. **Calculating the Rate of Heat Loss**:
- The rate of heat loss \( Q \) for a black body is given by:
\[ Q = 4 \pi r^2 \sigma T^4 \]
- For spheres A and B, we can write:
\[ Q_1 = 4 \pi R^2 \sigma T_1^4 \]
\[ Q_2 = 4 \pi (2R)^2 \sigma T_2^4 = 16 \pi R^2 \sigma T_2^4 \]

4. **Finding the Ratio of Heat Loss**:
- The ratio of heat loss rates \( \frac{Q_1}{Q_2} \) can be expressed as:
\[ \frac{Q_1}{Q_2} = \frac{4 \pi R^2 \sigma T_1^4}{16 \pi R^2 \sigma T_2^4} = \frac{T_1^4}{4T_2^4} \]
- Substituting \( T_1 = 2T_2 \):
\[ \frac{Q_1}{Q_2} = \frac{(2T_2)^4}{4T_2^4} = \frac{16T_2^4}{4T_2^4} = 4 \]
- Thus, \( \frac{Q_1}{Q_2} = 4 \), which means the ratio of heat loss is \( 4:1 \).

5. **Calculating the Rate of Cooling**:
- The rate of cooling is proportional to the heat loss, so:
\[ \frac{dQ/dt_1}{dQ/dt_2} = \frac{Q_1}{Q_2} \cdot \frac{R_2}{R_1^3} \]
- Using \( R_1 = R \) and \( R_2 = 2R \):
\[ \frac{dQ/dt_1}{dQ/dt_2} = \frac{4}{1} \cdot \frac{2R}{R^3} = 8 \]

6. **Final Analysis**:
- The ratio of temperatures \( \frac{T_1}{T_2} = 2 \).
- The ratio of heat loss \( \frac{Q_1}{Q_2} = 4 \).
- The ratio of cooling rates \( \frac{dQ/dt_1}{dQ/dt_2} = 8 \).
- The ratio of emissive powers is \( 16 \) (since \( \frac{E_1}{E_2} = \frac{T_1^4}{T_2^4} = 16 \)).

### Conclusion:
The statement that is not correct among the options provided is the one that contradicts the calculations above.