Earth is moving towards a fixed star with a velocity of `30 kms^(-1)`. An observer on earth observes a shift of `0.58 Å` in wavelength of light coming from star. What is the actual wavelength of light emitted by star ?

Options

A. 5800 Ã…
B. 2400 Ã…
C. 12000 Ã…
D. 6000 Ã…

Explanation

To find the actual wavelength of light emitted by the star, we can use the formula related to the Doppler effect for light. Here's a step-by-step solution:

### Step 1: Identify the given values
- Velocity of Earth towards the star, \( V = 30 \, \text{km/s} \)
- Shift in wavelength, \( \Delta \lambda = 0.58 \, \text{Å} \)

### Step 2: Convert the velocity to meters per second
Since \( 1 \, \text{km} = 1000 \, \text{m} \), we convert the velocity:
\[ V = 30 \, \text{km/s} = 30 \times 10^3 \, \text{m/s} = 30000 \, \text{m/s} \]

### Step 3: Use the Doppler shift formula
The formula relating the shift in wavelength to the actual wavelength is given by:
\[ \Delta \lambda = \frac{V}{c} \lambda \]
where:
- \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \)
- Rearranging the formula to find \( \lambda \):
\[ \lambda = \frac{c \Delta \lambda}{V} \]

### Step 4: Substitute the known values into the equation
Substituting \( c = 3 \times 10^8 \, \text{m/s} \), \( \Delta \lambda = 0.58 \, \text{Å} = 0.58 \times 10^{-10} \, \text{m} \), and \( V = 30000 \, \text{m/s} \):
\[ \lambda = \frac{(3 \times 10^8) \times (0.58 \times 10^{-10})}{30000} \]

### Step 5: Calculate the value of \( \lambda \)
Calculating the numerator:
\[ 3 \times 10^8 \times 0.58 \times 10^{-10} = 1.74 \times 10^{-2} \]
Now, divide by \( 30000 \):
\[ \lambda = \frac{1.74 \times 10^{-2}}{30000} = 5.8 \times 10^{-7} \, \text{m} \]
Converting \( \lambda \) back to Ångstroms (since \( 1 \, \text{Å} = 10^{-10} \, \text{m} \)):
\[ \lambda = 5.8 \times 10^{-7} \, \text{m} = 5800 \, \text{Å} \]

### Final Answer
The actual wavelength of light emitted by the star is:
\[ \lambda = 5800 \, \text{Å} \]