Question

The 6563 $\buildrel _\circ \over {\mathrm{A}}$ $H_\alpha$ line emitted by hydrogen in a star is found to be red-shifted by 15 $\buildrel _\circ \over {\mathrm{A}}$. The speed with which the star is receding from the earth is
A. $3.2 \times 10^5 ms^{-1}$
B. $6.87 \times 10^5 ms^{-1}$
C. $2 \times 10^5 ms^{-1}$
D. $12.74 \times 10^5 ms^{-1}$

Answer 1

Hint: When a source of light appears to be moving away from the observer, the observer notes an increase in the wavelength and a decrease in the frequency of the light from the source. This is due to Doppler Effect and the term for the shift in frequency is red shift.

Formula used:
If the difference in the wavelength is $\Delta \lambda$ and source has a wavelength $\lambda$ then,
$\dfrac{\Delta \lambda}{\lambda} = \dfrac{v}{c}$

Complete step-by-step answer:
We are given that the spectral line from hydrogen spectra with wavelength 6563 $\buildrel _\circ \over {\mathrm{A}}$, appears to have changed its wavelength by 15 $\buildrel _\circ \over {\mathrm{A}}$. Since it is also given that we observe a red shift, the wavelength of the light appears to be increased to us. Therefore we observe a wavelength of 6578 $\buildrel _\circ \over {\mathrm{A}}$.
Now, we know that the Doppler shift formula for red shift is:
$\lambda ' = \lambda \left(1+ \dfrac{v}{c} \right)$.
Where $\lambda '$ is the wavelength that we observe after the red shift, v is the velocity with which the source is moving away from us and c is the speed of the light.
A better and (more useful for us) version of the formula is:
$\dfrac{\Delta \lambda}{\lambda} = \dfrac{v}{c}$
In this, we are already given that $\Delta \lambda = 15 \buildrel _\circ \over {\mathrm{A}}$. So, we just plug in all the values. We may write:
$( \dfrac{15}{6563} ) \times 3 \times 10^8 m/s = v$
Therefore we get,
$v= 6.85 \times 10^5$ m/s

So, the correct answer is “Option B”.

Note: A minus sign would have appeared in front of v/c if the wavelength was shrinking. Though, the magnitude would have remained the same in that case too. To better remember the difference between redshift and blueshift, one should keep in mind that red color has low frequency and larger wavelength. As the source will move away from us, the light from it will have to travel more distance, so increased wavelength is observed. This effect is apparent and occurs due to the relative motion between source and the observer. The frequency of light is apparently changing and not actually changing (just depends on observation).