Question

The wavelength of ${{H}_{\alpha }}$ line in the hydrogen spectrum was found to be $6563{{A}^{0}}$ in the laboratory. If the wavelength of the same line in the spectrum of a milky way is observed to be $6568{{A}^{0}}$ then recession velocity of milky way will be?
A. $105 m/s$
B. $1.05\times {{10}^{6}}m/s$
C. $10.5\times {{10}^{6}}m/s$
D. $0.105\times {{10}^{6}}m/s$

Answer 1

Hint: This is a problem which can be solved by using Doppler’s effect of light. Doppler effect is predominant in case of sound. In case of sound when the source and the detector move apart, the detector. As a result, the detector encounters fewer cycles of a wave in each second, and therefore lower frequency.

Complete step by step answer:
According to Doppler's effect of light, the wavelength shift is given by the formula,
$\Delta \lambda =\dfrac{v\lambda }{c}$
Now we know that wavelength of ${{H}_{\alpha }}$ line in hydrogen spectrum was found to be $6563{{A}^{0}}$ in the laboratory while the same in case of a milky way is observed to be $6568{{A}^{0}}$. So the difference in wavelength comes out to be $=6568-6563=5{{A}^{0}}$
Putting the value in the formula we get,
$v=\dfrac{\Delta \lambda }{\lambda }\times c$
$\Rightarrow v=\dfrac{5}{6563}\times 3\times {{10}^{8}}$
$\therefore v=1.05\times {{10}^{6}}m/s$
Here, we have used the value of speed of light denoted by $c$.

So, the correct option is B.

Note:The wavelength is dependent upon the frequency and the speed of the propagating wave. Frequency is the characteristic of the source which is producing the wave. The SI unit of the wavelength is metre and velocity of the wave is m/s while for the frequency it is to be taken in Hertz (hz) always. Due to relativistic Doppler effect the phenomenon of red shifts and Blue shifts occur.