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 8000 K and 2000 K respectively. The ratio of the radii of the star A and the sun is

Options

A. `1600:1`
B. `1800:1`
C. `16:1`
D. `25:1`

Answer

To solve the problem, we will use the Stefan-Boltzmann Law, which states that the power (or energy emitted) by a black body is proportional to the fourth power of its temperature. The formula for the energy emitted by a star is given by:

\[ E = \sigma A T^4 \]

Where:
- \( E \) is the energy emitted,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area,
- \( T \) is the temperature.

The surface area \( A \) of a star can be expressed in terms of its radius \( R \):

\[ A = 4\pi R^2 \]

Thus, the energy emitted can be rewritten as:

\[ E = \sigma (4\pi R^2) T^4 \]

Given:
- The energy emitted by star A is 10,000 times that of the Sun.
- The temperature of the Sun \( T_{sun} = 8000 \, K \)
- The temperature of star A \( T_A = 2000 \, K \)

### Step-by-Step Solution:

1. **Set up the ratio of energies:**
\[ \frac{E_A}{E_{sun}} = 10,000 \]

2. **Express the energies using the Stefan-Boltzmann Law:**
\[ \frac{\sigma (4\pi R_A^2) T_A^4}{\sigma (4\pi R_{sun}^2) T_{sun}^4} = 10,000 \]

The constants \( \sigma \) and \( 4\pi \) cancel out:
\[ \frac{R_A^2 T_A^4}{R_{sun}^2 T_{sun}^4} = 10,000 \]

3. **Rearrange the equation to find the ratio of the radii:**
\[ \frac{R_A^2}{R_{sun}^2} = 10,000 \cdot \frac{T_{sun}^4}{T_A^4} \]

4. **Substitute the known temperatures:**
\[ \frac{R_A^2}{R_{sun}^2} = 10,000 \cdot \frac{(8000)^4}{(2000)^4} \]

5. **Calculate \( \frac{(8000)^4}{(2000)^4} \):**
\[ \frac{(8000)^4}{(2000)^4} = \left(\frac{8000}{2000}\right)^4 = 4^4 = 256 \]

6. **Now substitute back into the equation:**
\[ \frac{R_A^2}{R_{sun}^2} = 10,000 \cdot 256 \]
\[ \frac{R_A^2}{R_{sun}^2} = 2,560,000 \]

7. **Take the square root to find the ratio of the radii:**
\[ \frac{R_A}{R_{sun}} = \sqrt{2,560,000} = 1600 \]

### Final Answer:
The ratio of the radii of star A to the Sun is:
\[ R_A : R_{sun} = 1600 : 1 \]