The 6563 Å line emitted by hydrogen atom in a star is found to be red shifted by 5 Å. The speed with which the star is receding form the earth is

Options

A. `17.29xx10^(9)m//s`
B. `4.29xx10^(7)m//s`
C. `3.39xx10^(5)m//s`
D. `2.29xx10^(5)m//s`

Answer

To solve the problem of finding the speed at which the star is receding from the Earth based on the redshift of the hydrogen line, we can follow these steps:

### Step 1: Identify the Given Data
- The wavelength of the emitted light (λ) is 6563 Å.
- The redshift (Δλ) is 5 Å.

### Step 2: Convert the Wavelengths to Meters
- Convert the given wavelengths from Ångströms to meters:
- \( \lambda = 6563 \, \text{Å} = 6563 \times 10^{-10} \, \text{m} \)
- \( \Delta \lambda = 5 \, \text{Å} = 5 \times 10^{-10} \, \text{m} \)

### Step 3: Calculate the Observed Wavelength (λ')
- The observed wavelength (λ') can be calculated as:
\[ \lambda' = \lambda + \Delta \lambda = 6563 \times 10^{-10} \, \text{m} + 5 \times 10^{-10} \, \text{m} = 6568 \times 10^{-10} \, \text{m} \]

### Step 4: Use the Redshift Formula
- The formula relating redshift (z) to the speed of the star (v) is:
\[ z = \frac{\Delta \lambda}{\lambda} = \frac{v}{c} \]
Where:
- \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \))

### Step 5: Calculate the Redshift (z)
- Substitute the values into the redshift formula:
\[ z = \frac{\Delta \lambda}{\lambda} = \frac{5 \times 10^{-10}}{6563 \times 10^{-10}} = \frac{5}{6563} \]

### Step 6: Calculate the Speed (v)
- Rearranging the redshift formula gives:
\[ v = z \cdot c \]
- Substitute the value of \( z \) and \( c \):
\[ v = \left(\frac{5}{6563}\right) \cdot (3 \times 10^8) \]

### Step 7: Simplify and Calculate the Final Speed
- Calculate \( v \):
\[ v \approx \left(0.0007613\right) \cdot (3 \times 10^8) \approx 2.29 \times 10^5 \, \text{m/s} \]

### Final Answer
The speed at which the star is receding from the Earth is approximately \( 2.29 \times 10^5 \, \text{m/s} \).

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