A star emitting light of wavelength 6000 Å is moving towards the earth with a speed of ` 3 xx 10^(6) m//s `. What is the apparent wavelength of the emitted light ?

Options

A. a. `6060Å`
B. b. `5900Å`
C. c. `5940Å`
D. d. `5850Å`

Explanation

To find the apparent wavelength of light emitted by a star moving towards the Earth, we can use the Doppler effect for light. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- Wavelength of light emitted by the star, \( \lambda = 6000 \, \text{Å} \) (angstroms).
- Speed of the star moving towards the Earth, \( v = 3 \times 10^6 \, \text{m/s} \).
- Speed of light, \( c = 3 \times 10^8 \, \text{m/s} \).

### Step 2: Convert Wavelength to Meters
Since we need to work with standard units, we convert the wavelength from angstroms to meters:
\[ 1 \, \text{Å} = 10^{-10} \, \text{m} \]
Thus,
\[ \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} = 6 \times 10^{-7} \, \text{m} \]

### Step 3: Use the Doppler Effect Formula
For a source moving towards the observer, the apparent wavelength \( \lambda' \) can be calculated using the formula:
\[ \lambda' = \lambda \left(1 - \frac{v}{c}\right) \]
Substituting the known values:
\[ \lambda' = 6 \times 10^{-7} \left(1 - \frac{3 \times 10^6}{3 \times 10^8}\right) \]

### Step 4: Calculate the Fraction
Calculate the fraction \( \frac{v}{c} \):
\[ \frac{v}{c} = \frac{3 \times 10^6}{3 \times 10^8} = 0.01 \]

### Step 5: Substitute Back into the Formula
Now substitute this value back into the equation:
\[ \lambda' = 6 \times 10^{-7} \left(1 - 0.01\right) = 6 \times 10^{-7} \times 0.99 \]

### Step 6: Calculate the Apparent Wavelength
Now calculate \( \lambda' \):
\[ \lambda' = 6 \times 10^{-7} \times 0.99 = 5.94 \times 10^{-7} \, \text{m} \]

### Step 7: Convert Back to Angstroms
Convert the apparent wavelength back to angstroms:
\[ \lambda' = 5.94 \times 10^{-7} \, \text{m} = 5.94 \times 10^{-7} \times 10^{10} \, \text{Å} = 5940 \, \text{Å} \]

### Final Answer
The apparent wavelength of the emitted light is \( \lambda' = 5940 \, \text{Å} \).

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