The radiation emitted by a star `A` is `1000` times that of the sun. If the surface temperature of the sun and star `A` are `6000 K` and `2000 K` respectively. The ratio of the radii of the star `A` and the sun is:

Options

A. `300 : 1`
B. `600 : 1`
C. `900 : 1`
D. `1200 : 1`

Explanation

To solve the problem, we need to find the ratio of the radii of star A and the Sun based on the given information about their emitted radiation and surface temperatures.

### Step-by-Step Solution:

1. **Understand the Stefan-Boltzmann Law**:
The power radiated per unit area of a black body is given by the Stefan-Boltzmann law:
\[ E = \sigma T^4 \]
where \(E\) is the emissive power, \(\sigma\) is the Stefan-Boltzmann constant, and \(T\) is the temperature in Kelvin.

2. **Set up the relationship between the two stars**:
Let \(E_{\text{sun}}\) be the radiation emitted by the Sun and \(E_{\text{star A}}\) be the radiation emitted by star A. According to the problem:
\[ E_{\text{star A}} = 1000 \times E_{\text{sun}} \]

3. **Express the emissive powers**:
Using the Stefan-Boltzmann law, we can express the emissive powers in terms of their temperatures and radii:
\[ E_{\text{sun}} = \sigma T_{\text{sun}}^4 = \sigma (6000)^4 \]
\[ E_{\text{star A}} = \sigma T_{\text{star A}}^4 = \sigma (2000)^4 \]

4. **Relate the emissive powers to the radii**:
The total power radiated by a star is proportional to its surface area and can be expressed as:
\[ E_{\text{star A}} = \frac{P_{\text{star A}}}{A_{\text{star A}}} = \frac{P_{\text{star A}}}{4\pi R_{\text{star A}}^2} \]
\[ E_{\text{sun}} = \frac{P_{\text{sun}}}{4\pi R_{\text{sun}}^2} \]
Therefore, we can write:
\[ \frac{E_{\text{star A}}}{E_{\text{sun}}} = \frac{R_{\text{star A}}^2 T_{\text{star A}}^4}{R_{\text{sun}}^2 T_{\text{sun}}^4} \]

5. **Substituting known values**:
From the problem, we know:
\[ \frac{E_{\text{star A}}}{E_{\text{sun}}} = 1000 \]
Substituting the temperatures:
\[ 1000 = \frac{R_{\text{star A}}^2 (2000)^4}{R_{\text{sun}}^2 (6000)^4} \]

6. **Rearranging the equation**:
Rearranging gives:
\[ R_{\text{star A}}^2 = 1000 \cdot R_{\text{sun}}^2 \cdot \frac{(6000)^4}{(2000)^4} \]

7. **Calculating the temperature ratio**:
\[ \frac{(6000)^4}{(2000)^4} = \left(\frac{6000}{2000}\right)^4 = 3^4 = 81 \]
Therefore:
\[ R_{\text{star A}}^2 = 1000 \cdot R_{\text{sun}}^2 \cdot 81 \]
\[ R_{\text{star A}}^2 = 81000 \cdot R_{\text{sun}}^2 \]

8. **Finding the ratio of the radii**:
Taking the square root of both sides:
\[ \frac{R_{\text{star A}}}{R_{\text{sun}}} = \sqrt{81000} = 300 \]

### Final Answer:
The ratio of the radii of star A to the Sun is:
\[ \frac{R_{\text{star A}}}{R_{\text{sun}}} = 300:1 \]