The radiation emitted by a star `A` is `10000` 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 will use the Stefan-Boltzmann law, which states that the energy radiated per second by a black body is proportional to the fourth power of its absolute temperature.

### Step-by-step Solution:

1. **Understand the Given Information**:
- Radiation emitted by star A (E1) is 10,000 times that of the Sun (E2).
- Surface temperature of the Sun (T2) = 6000 K.
- Surface temperature of star A (T1) = 2000 K.

2. **Apply the Stefan-Boltzmann Law**:
The energy radiated per second (E) is given by:
\[ E = \sigma A T^4 \]
where \(A\) is the surface area and \(\sigma\) is the Stefan-Boltzmann constant.

3. **Express the Areas**:
The surface area of a sphere is given by \(A = 4\pi R^2\). Thus, for star A and the Sun:
- \(A_1 = 4\pi R_1^2\) (for star A)
- \(A_2 = 4\pi R_2^2\) (for the Sun)

4. **Set Up the Ratio of Energy Emitted**:
According to the problem, we have:
\[ E_1 = 10,000 \times E_2 \]
Substituting the expressions for energy:
\[ \sigma A_1 T_1^4 = 10,000 \times \sigma A_2 T_2^4 \]
The \(\sigma\) cancels out:
\[ A_1 T_1^4 = 10,000 \times A_2 T_2^4 \]

5. **Substituting Areas**:
Substitute \(A_1\) and \(A_2\):
\[ (4\pi R_1^2) T_1^4 = 10,000 \times (4\pi R_2^2) T_2^4 \]
The \(4\pi\) cancels out:
\[ R_1^2 T_1^4 = 10,000 \times R_2^2 T_2^4 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{R_1^2}{R_2^2} = 10,000 \times \frac{T_2^4}{T_1^4} \]

7. **Substituting the Temperatures**:
Substitute \(T_1 = 2000 K\) and \(T_2 = 6000 K\):
\[ \frac{R_1^2}{R_2^2} = 10,000 \times \left(\frac{6000}{2000}\right)^4 \]
Simplifying \(\frac{6000}{2000} = 3\):
\[ \frac{R_1^2}{R_2^2} = 10,000 \times 3^4 \]

8. **Calculating \(3^4\)**:
\[ 3^4 = 81 \]
Thus:
\[ \frac{R_1^2}{R_2^2} = 10,000 \times 81 = 810,000 \]

9. **Taking the Square Root**:
\[ \frac{R_1}{R_2} = \sqrt{810,000} = 900 \]

10. **Final Ratio**:
Therefore, the ratio of the radii of star A to the Sun is:
\[ R_1 : R_2 = 900 : 1 \]