Question

A sodium light source moves in a horizontal circle at a constant speed of 0.100c while emitting light at the proper wavelength of ${\mathbf{\lambda }}_{\mathbf{o}}\mathbf{=}\mathbf{589}\mathbf{}\mathbf{nm}$. Wavelength $\mathbf{\lambda }$is measured for that light by a detector fixed at the center of the circle. What is the wavelength shift $\mathbf{\lambda }\mathbf{-}{\mathbf{\lambda }}_{\mathbf{o}}$?

Short answer

The wavelength shift for sodium light source is 2.97 nm.

Step-by-step solution

Identification of given data

The proper wavelength of emitting light is ${\mathrm{\lambda }}_{\mathrm{o}}=589\mathrm{nm}$

The speed of sodium light source is $\mathrm{u}=0.1\mathrm{c}$

The wavelength shift is the variation in the wavelength of light due to relative movement of source and observer in different frames. It is calculated by the wavelength shift formula.

Determination of wavelength shift for sodium light source

The wavelength shift for sodium light source is given as:

$\mathrm{\lambda }-{\mathrm{\lambda }}_{\mathrm{o}}={\mathrm{\lambda }}_{\mathrm{o}}\left(\frac{1}{\sqrt{1-{\left({\displaystyle \frac{\mathrm{v}}{\mathrm{c}}}\right)}^2}}-1\right)$

Substitute all the values in the above equation.

$$\begin{gathered} \mathrm{\lambda }-{\mathrm{\lambda }}_{\mathrm{o}}=\left(589\mathrm{nm}\right)\left(\frac{1}{\sqrt{1-{\left({\displaystyle \frac{0.1\mathrm{c}}{\mathrm{c}}}\right)}^2}}-1\right) \\ \mathrm{\lambda }-{\mathrm{\lambda }}_{\mathrm{o}}=2.97\mathrm{nm} \end{gathered}$$

Therefore, the wavelength shift for sodium light source is 2.97 nm.