A star is moving away from the earth with a velocity of `100 km//s`. If the velocity of light is `3 xx 10^(8) m//s` then the shift of its spectral line of wavelength `5700 A` due to Doppler effect is

Options

A. `0.63Å`
B. `1.90 Å`
C. `3.80 Å`
D. `5.70 Å`

Explanation

To solve the problem of the spectral line shift due to the Doppler effect for a star moving away from the Earth, we can follow these steps:

### Step 1: Understand the Doppler Effect for Light
The Doppler effect describes how the wavelength of light changes due to the relative motion of the source and the observer. When the source is moving away from the observer, the observed wavelength increases, resulting in a redshift.

### Step 2: Identify the Given Values
- Velocity of the star, \( V = 100 \, \text{km/s} \)
- Wavelength of the spectral line, \( \lambda = 5700 \, \text{Å} \) (1 Å = \( 10^{-10} \, \text{m} \))
- Speed of light, \( c = 3 \times 10^8 \, \text{m/s} \)

### Step 3: Convert Units
Convert the velocity of the star from km/s to m/s:
\[ V = 100 \, \text{km/s} = 100 \times 10^3 \, \text{m/s} = 1 \times 10^5 \, \text{m/s} \]

### Step 4: Use the Doppler Shift Formula
The formula for the change in wavelength (\( \Delta \lambda \)) due to the Doppler effect is given by:
\[ \frac{\Delta \lambda}{\lambda} = \frac{V}{c} \]
Rearranging this gives:
\[ \Delta \lambda = \lambda \cdot \frac{V}{c} \]

### Step 5: Substitute the Values
Now substitute the values into the equation:
\[ \Delta \lambda = 5700 \, \text{Å} \cdot \frac{1 \times 10^5 \, \text{m/s}}{3 \times 10^8 \, \text{m/s}} \]

### Step 6: Calculate \( \Delta \lambda \)
First, calculate the fraction:
\[ \frac{1 \times 10^5}{3 \times 10^8} = \frac{1}{3 \times 10^3} \approx 3.33 \times 10^{-4} \]
Now calculate \( \Delta \lambda \):
\[ \Delta \lambda = 5700 \, \text{Å} \cdot 3.33 \times 10^{-4} \approx 1.9 \, \text{Å} \]

### Step 7: Final Result
The shift of the spectral line due to the Doppler effect is approximately:
\[ \Delta \lambda \approx 1.9 \, \text{Å} \]