A star emits a light of wavelength I and it is receding from the earth with a velocity `v_(s)`. The shift in wavelength of spectral line observed on earth is

Options

A. `c lambda//v_(s)`
B. `-c lambda//v_(s)`
C. `-v_(s) lambda//c`
D. `v_(s) lambda//c`

Answer

To solve the problem of determining the shift in wavelength of light emitted from a star that is receding from the Earth, we can use the Doppler effect for light. Here’s a step-by-step solution:

### Step 1: Understand the Doppler Effect
The Doppler effect describes how the observed wavelength of light changes when the source of the light is moving relative to an observer. If the source is moving away from the observer, the observed wavelength increases (redshift).

### Step 2: Define the Variables
- Let \( \lambda \) be the original wavelength emitted by the star.
- Let \( v_s \) be the velocity of the star receding from the Earth.
- Let \( c \) be the speed of light in a vacuum.

### Step 3: Use the Doppler Effect Formula
The formula for the shift in wavelength due to the Doppler effect when the source is moving away from the observer is given by:

\[ \frac{\Delta \lambda}{\lambda} = \frac{v_s}{c} \]

Where:
- \( \Delta \lambda \) is the change in wavelength (shift in wavelength).
- \( \lambda \) is the original wavelength.
- \( v_s \) is the velocity of the source (star) moving away.
- \( c \) is the speed of light.

### Step 4: Rearranging the Formula
From the formula, we can express the change in wavelength \( \Delta \lambda \):

\[ \Delta \lambda = \lambda \cdot \frac{v_s}{c} \]

### Step 5: Conclusion
The shift in wavelength observed on Earth due to the star receding is:

\[ \Delta \lambda = \lambda \cdot \frac{v_s}{c} \]

This indicates that the observed wavelength increases as the star moves away from the Earth.

### Final Answer
The shift in wavelength of the spectral line observed on Earth is:

\[ \Delta \lambda = \lambda \cdot \frac{v_s}{c} \]

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