Question

In the hydrogen spectrum the wavelength of $\text{H}_{\lambda }$ line is $656\,nm$ whereas, in the spectrum of a distant galaxy the $\text{H}_{\lambda }$ line has a wavelength of $706\,nm$ . The estimated speed of the galaxy with respect to the earth is

• A. $1.2\times 10^{7}m\,s^{- 1}$
• B. $2.2\times 10^{7}m\,s^{- 1}$
• C. $3\times 10^{7}m\,s^{- 1}$
• D. $1.9\times 10^{7}m\,s^{- 1}$

Answer 1

Answer: (B)

Since, the wavelength observed is more than the true wavelength, we can say that the galaxy is receding. Doppler effect in this case gives us
$\lambda^{\prime}=\lambda \sqrt{\frac{ c + v }{ c - v }} $
$706=656 \sqrt{\frac{ c + v }{ c - v }} $
$\text { or, } \frac{ c + v }{ c - v }=\left(\frac{706}{656}\right)^2=1.16$
$\text { or, } c+v=1.16 c-1.16 v $
$v=\frac{0.16 c}{2.16}=\frac{0.16 \times 3.0 \times 10^8}{2.16}=0.22 \times 10^8 ms ^{-1} $
$v=2.2 \times 10^7 m \,s ^{-1}$