The radiation emitted by a star A is 10,000 times that of the sun. If the surface temperatures of the sun and the 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`

Answer

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 radiation and temperatures. We can use the Stefan-Boltzmann law, which states that the power radiated per unit area of a black body is proportional to the fourth power of its temperature.

### Step-by-Step Solution:

1. **Identify the given values:**
- Radiation emitted by star A (U_A) = 10,000 * Radiation emitted by the sun (U_S)
- Temperature of the sun (T_S) = 6000 K
- Temperature of star A (T_A) = 2000 K

2. **Apply the Stefan-Boltzmann law:**
The total power radiated by a black body is given by:
\[ U = \sigma A T^4 \]
where \(A\) is the surface area and \(\sigma\) is the Stefan-Boltzmann constant.

3. **Express the surface area in terms of radius:**
The surface area \(A\) of a sphere is given by:
\[ A = 4\pi r^2 \]
Therefore, we can write the power emitted by the sun and star A as:
\[ U_S = \sigma (4\pi r_S^2) T_S^4 \]
\[ U_A = \sigma (4\pi r_A^2) T_A^4 \]

4. **Set up the equation for the ratio of the radiations:**
Since \(U_A = 10,000 U_S\), we can write:
\[ \sigma (4\pi r_A^2) T_A^4 = 10,000 \cdot \sigma (4\pi r_S^2) T_S^4 \]
The \(\sigma\) and \(4\pi\) terms cancel out:
\[ r_A^2 T_A^4 = 10,000 r_S^2 T_S^4 \]

5. **Substitute the temperatures:**
Substitute \(T_A = 2000\) K and \(T_S = 6000\) K into the equation:
\[ r_A^2 (2000)^4 = 10,000 r_S^2 (6000)^4 \]

6. **Simplify the equation:**
Rearranging gives:
\[ \frac{r_A^2}{r_S^2} = \frac{10,000 \cdot (6000)^4}{(2000)^4} \]

7. **Calculate the ratio:**
\[ \frac{r_A^2}{r_S^2} = 10,000 \cdot \left(\frac{6000}{2000}\right)^4 = 10,000 \cdot 3^4 = 10,000 \cdot 81 = 810,000 \]
Therefore,
\[ \frac{r_A}{r_S} = \sqrt{810,000} = 900 \]

8. **Final result:**
The ratio of the radii of star A to the sun is:
\[ \frac{r_A}{r_S} = 900 \]