A star is moving away from the earth with a velocity of `10^(5)` m/sec. If wavelength of its spectral line be `5700 Å`, the Doppler's shift will be

Options

A. `1.9Å` towards violet end
B. `1.9Å` towards red end
C. `3.8Å` towards violet end
D. `3.8Å` towards red end

Explanation

To solve the problem of finding the Doppler shift for a star moving away from Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Velocity of the star, \( v = 10^5 \, \text{m/s} \)
- Wavelength of the spectral line, \( \lambda = 5700 \, \text{Å} = 5700 \times 10^{-10} \, \text{m} \) (since \( 1 \, \text{Å} = 10^{-10} \, \text{m} \))
- Speed of light, \( c = 3 \times 10^8 \, \text{m/s} \)

2. **Use the Doppler Shift Formula:**
The formula for the Doppler shift in wavelength when the source is moving away is given by:
\[ \Delta \lambda = \frac{v}{c} \cdot \lambda \]

3. **Substitute the Values into the Formula:**
\[ \Delta \lambda = \frac{10^5 \, \text{m/s}}{3 \times 10^8 \, \text{m/s}} \cdot (5700 \times 10^{-10} \, \text{m}) \]

4. **Calculate the Fraction:**
\[ \frac{10^5}{3 \times 10^8} = \frac{1}{3 \times 10^3} = \frac{1}{3000} \approx 0.0003333 \]

5. **Calculate the Doppler Shift:**
\[ \Delta \lambda = 0.0003333 \cdot (5700 \times 10^{-10}) \, \text{m} \]
\[ \Delta \lambda = 0.0003333 \cdot 5700 \times 10^{-10} \approx 1.9 \times 10^{-6} \, \text{m} \]

6. **Convert the Result Back to Angstroms:**
\[ \Delta \lambda \approx 1.9 \, \text{Å} \]

7. **Conclusion:**
The Doppler shift of the star's spectral line is approximately \( 1.9 \, \text{Å} \).

### Final Answer:
The Doppler shift will be \( 1.9 \, \text{Å} \).

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