Q. In hydrogen spectrum, the wavelength of $H_{\alpha}$ line is 656 nm, whereas in the spectrum of a distant galaxy, $H_{\alpha}$ wavelength is 706 nm. Estimated speed of the galaxy with respect to earth is
A) $2 \times 10^{8}\ {ms}^{- 1}$
B) $2 \times 10^{7}\ {ms}^{- 1}$
C) $2 \times 10^{6}\ {ms}^{- 1}$
D) $2 \times 10^{5}\ {ms}^{- 1}$
Explanation
(b) The wavelength of light emitted by a moving object is shifted. This effecd is called the Doppler’s shift given by
$\lambda = \lambda_{0}\left( 1 + \frac{v}{c} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(i)$
Where $\lambda$ is perceived wavelength, $v$ the velocity and $c$ the speed of light.
From Eq. (i) $v = c\left( \frac{\lambda - \lambda_{0}}{\lambda_{0}} \right)$
Given, $\lambda_{0} = 656\ nm,\ \lambda = 706\ nm,$
$c = 3 \times 10^{8}\ {ms}^{- 1}$
$v = 3 \times 10^{8}\left( \frac{706 - 656}{656} \right)$
$v = 3 \times 10^{8} \times \frac{50}{656} \approx 2 \times 10^{7}\ {ms}^{- 1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.