In hydrogen spectrum the wavelength of `H_(a)` line is `656nm, ` where in the spectrum of a distance galaxy `H_(a)` line wavelength is `706 nm` . Estimated speed of the galaxy with respect to earth is ,

Options

A. `2 xx10^(8) m s^(-1)`
B. `2 xx10^(7) m s^(-1)`
C. `2 xx10^(6) m s^(-1)`
D. `2 xx10^(5) m s^(-1)`

Explanation

To solve the problem of estimating the speed of a distant galaxy with respect to Earth based on the observed wavelengths of the hydrogen `Hα` line, we can follow these steps:

### Step 1: Identify the given data
- Wavelength of `Hα` line in the hydrogen spectrum (stationary source):
\[ \lambda_0 = 656 \, \text{nm} = 656 \times 10^{-9} \, \text{m} \]
- Wavelength of `Hα` line in the spectrum of the distant galaxy:
\[ \lambda = 706 \, \text{nm} = 706 \times 10^{-9} \, \text{m} \]

### Step 2: Calculate the change in wavelength (Δλ)
The change in wavelength (Δλ) is calculated as:
\[ \Delta \lambda = \lambda - \lambda_0 = 706 \times 10^{-9} \, \text{m} - 656 \times 10^{-9} \, \text{m} \]
\[ \Delta \lambda = (706 - 656) \times 10^{-9} \, \text{m} = 50 \times 10^{-9} \, \text{m} \]

### Step 3: Use the Doppler effect formula for light
The formula for the velocity (v) of a galaxy moving away from us (redshift) is given by:
\[ v = \frac{\Delta \lambda}{\lambda_0} \cdot c \]
where \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 4: Substitute the values into the formula
Now we can substitute the values we have:
\[ v = \frac{50 \times 10^{-9} \, \text{m}}{656 \times 10^{-9} \, \text{m}} \cdot (3 \times 10^8 \, \text{m/s}) \]

### Step 5: Calculate the fraction
Calculating the fraction:
\[ \frac{50 \times 10^{-9}}{656 \times 10^{-9}} = \frac{50}{656} \approx 0.0763 \]

### Step 6: Calculate the velocity
Now, substituting this back into the equation for velocity:
\[ v \approx 0.0763 \cdot (3 \times 10^8 \, \text{m/s}) \approx 2.289 \times 10^7 \, \text{m/s} \]

### Final Answer
The estimated speed of the galaxy with respect to Earth is approximately:
\[ v \approx 2.29 \times 10^7 \, \text{m/s} \]

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