Earth is moving towards a stationary star with a velocity `100 kms^(-1)` . If the wavelength of light emitted by the star is 5000Å, then the apparent change in wavelength observed by the observer on earth will be

Options

A. 0.67Å
B. 1.67Å
C. 16.7Å
D. 167Å

Answer

To solve the problem of the apparent change in wavelength observed by an observer on Earth as it moves towards a stationary star, we will use the Doppler effect formula for light. Here are the steps to derive the solution:

### Step 1: Understand the parameters
- The velocity of the Earth (observer) towards the star (source) is given as \( v = 100 \, \text{km/s} \).
- The wavelength of light emitted by the star is given as \( \lambda_0 = 5000 \, \text{Å} \) (which is \( 5000 \times 10^{-10} \, \text{m} \)).

### Step 2: Convert the velocity to meters per second
- Convert \( v \) from kilometers per second to meters per second:
\[ v = 100 \, \text{km/s} = 100 \times 10^3 \, \text{m/s} = 100000 \, \text{m/s} \]

### Step 3: Use the Doppler effect formula
- The formula for the change in wavelength due to the Doppler effect for light is given by:
\[ \Delta \lambda = \lambda_0 \left( \frac{v}{c} \right) \]
where \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 4: Substitute the values into the formula
- Substitute \( \lambda_0 = 5000 \times 10^{-10} \, \text{m} \), \( v = 100000 \, \text{m/s} \), and \( c = 3 \times 10^8 \, \text{m/s} \):
\[ \Delta \lambda = 5000 \times 10^{-10} \, \text{m} \left( \frac{100000 \, \text{m/s}}{3 \times 10^8 \, \text{m/s}} \right) \]

### Step 5: Calculate the fraction
- Calculate the fraction:
\[ \frac{100000}{3 \times 10^8} = \frac{1}{3000} \approx 0.0003333 \]

### Step 6: Calculate the change in wavelength
- Now calculate \( \Delta \lambda \):
\[ \Delta \lambda = 5000 \times 10^{-10} \, \text{m} \times 0.0003333 \approx 1.67 \times 10^{-6} \, \text{m} = 1.67 \, \text{Å} \]

### Step 7: Final result
- The apparent change in wavelength observed by the observer on Earth is:
\[ \Delta \lambda \approx 1.67 \, \text{Å} \]

### Summary
The apparent change in wavelength observed by the observer on Earth is approximately \( 1.67 \, \text{Å} \).

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