star emitting light of wavelength `5896 Å` is moving away from the earth with a speed of 3600 km / sec . The wavelength of light observed on earth will
(`c= 3 xx 10 ^(8) m//sec` is the speed of light)

Options

A. Decrease by 5825.25 `overset(o)(A)`
B. Increase by 5966.75 `overset(o)(A)`
C. Decreaseby 70.75`overset(o)(A)`
D. Increaseby 70.75`overset(o)(A)`

Answer

To solve the problem, we need to determine the observed wavelength of light from a star that is moving away from the Earth. We will use the formula for the shift in wavelength due to the Doppler effect for light.

### Step-by-Step Solution:

1. **Identify the given values:**
- Wavelength of light emitted by the star, \( \lambda = 5896 \, \text{Å} \) (which is \( 5896 \times 10^{-10} \, \text{m} \))
- Speed of the star moving away from the Earth, \( v = 3600 \, \text{km/s} = 3600 \times 10^3 \, \text{m/s} \)
- Speed of light, \( c = 3 \times 10^8 \, \text{m/s} \)

2. **Use the formula for the change in wavelength:**
The change in wavelength \( \Delta \lambda \) due to the Doppler effect is given by:
\[ \Delta \lambda = \frac{v \cdot \lambda}{c} \]

3. **Substitute the values into the formula:**
\[ \Delta \lambda = \frac{(3600 \times 10^3 \, \text{m/s}) \cdot (5896 \times 10^{-10} \, \text{m})}{3 \times 10^8 \, \text{m/s}} \]

4. **Calculate \( \Delta \lambda \):**
- First, calculate the numerator:
\[ 3600 \times 10^3 \cdot 5896 \times 10^{-10} = 2.123776 \times 10^{-3} \, \text{m} \quad (\text{after calculating the multiplication}) \]
- Now, divide by the speed of light:
\[ \Delta \lambda = \frac{2.123776 \times 10^{-3}}{3 \times 10^8} = 7.07592 \times 10^{-12} \, \text{m} \]

5. **Convert \( \Delta \lambda \) back to Ångströms:**
- Since \( 1 \, \text{Å} = 10^{-10} \, \text{m} \):
\[ \Delta \lambda = 7075.92 \, \text{Å} \approx 7076 \, \text{Å} \]

6. **Calculate the observed wavelength:**
The observed wavelength \( \lambda' \) is given by:
\[ \lambda' = \lambda + \Delta \lambda \]
\[ \lambda' = 5896 \, \text{Å} + 7076 \, \text{Å} = 12972 \, \text{Å} \]

### Final Answer:
The wavelength of light observed on Earth will be approximately \( 12972 \, \text{Å} \).