In hydrogen spectrum the wavelength `H_(a)` of line is 656 nm whereas in the spectrum of a distant galaxy. `H_(a)` line wavelength is 706 nm. Estimated speed of the galaxy with respect to earth is:

Options

A. `2 xx 10^(8) m//s`
B. `2 xx 10^(7) m//s`
C. `2 xx 10^(6) m//s`
D. `2 xx 10^(5) m//s`

Answer

B

Explanation

Since, the wavelength `(lambda)` is increasing we can say that the galaxy is receding. Doppler efect can be given by-
`lambda'=lambdasqrt((c+v)/(c-v))implies 706=656 sqrt((c+v)/(c-v))` ...(i)
`implies (c+v)/(c-v)=(706/656)^(2)=1.16`
`:. c+v=1.16c-1.16 v`
`:. v=(0.16 c)/(2.16)=(0.16xx3.0xx10^(8))/2.16 m//s`
`v~~2.2xx10^(7) m//s`
If we take the approximation then equation (i) can be written as `Delta lambda=lambda(v/c)` ...(iii)
From here `v=((Delta lambda)/lambda).c=((706-656)/656)(3xx10^(8)) m//s`
`implies v=0.23xx10^(8) m//s`
Which is almost equal to the previous answer. So, we use equation (ii) also.