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Question: In the hydrogen spectrum, the wavelength of \[H\alpha \] line is \[656nm\], whereas in the spectrum ...

In the hydrogen spectrum, the wavelength of HαH\alpha line is 656nm656nm, whereas in the spectrum of a distant galaxy, HαH\alpha line wavelength is 706nm706nm. Estimated speed of the galaxy with respect to the earth is:
A. 2×108ms−12\times{{10}^{8}}m{{s}^{-1}}
B. 2×107ms−12\times{{10}^{7}}m{{s}^{-1}}
C. 2×106ms−12\times{{10}^{6}}m{{s}^{-1}}
D. 2×105ms−12\times{{10}^{5}}m{{s}^{-1}}

Explanation

Solution

To find the apparent wavelength (observed wavelength) we use Doppler’s wavelength shift formula. When the source of light is in relative motion with respect to the observer then there is a shift in wavelength and the frequency of the light.

Complete step by step solution:
When the source is moving towards the observer, then the wavelengths are shifted to shorter wavelengths, that is, towards the blue. When the source is moving away from the observer, the wavelengths are shifted to longer wavelengths, that is, towards the red. This phenomenon of shifting of wavelength of the light is explained by Doppler’s shift.

When the hydrogen atom is at rest relative to the observer then the observed wavelength of Hα{{H}_{\alpha }} line is 656nm656nm.
Whereas the wavelength Hα{{H}_{\alpha }} line from the distant galaxy is 706nm706nm which is longer than the wavelength when the source was at rest relative to the observer. This implies that the source of the light is moving away from the observer.
According to Doppler’s shift formula the wavelength shift is given as,
Δλ=vcλ\Delta \lambda =\dfrac{v}{c}\lambda
Where,
Δλ=\Delta \lambda = Shift in wavelength
v=v=Velocity of the source of light
c=c=Speed of the light
λ=\lambda =The wavelength of the light when source is at rest
Hence, v=(Δλ)cλv=\dfrac{\left( \Delta \lambda \right)c}{\lambda }
Speed of the distance galaxy is calculated as,
v=(706−656)×10−9×3×108656×10−9m/s =50×3×108656m/s =2.3×107m/s\begin{aligned} & v=\dfrac{\left( 706-656 \right)\times {{10}^{-9}}\times 3\times {{10}^{8}}}{656\times {{10}^{-9}}}m/s \\\ & =\dfrac{50\times 3\times {{10}^{8}}}{656}m/s \\\ & =2.3\times {{10}^{7}}m/s \end{aligned}

Hence, the speed of the galaxy is approximately equal to 2×107m/s2\times {{10}^{7}}m/s.

Note: - When source of light is moving away from the observer then wavelength of the light shifts towards red wavelength.
- When the source of light is moving towards the observer then the wavelength of the light shifts towards blue wavelength.