JEE Main & Advanced Physics Wave Optics / तरंग प्रकाशिकी Doppler's Effect of Light

The phenomenon of apparent change in frequency (or wavelength) of the light due to relative motion between the source of light and the observer is called Doppler's effect.

If \[\nu =\] actual frequency, \[\nu '=\]Apparent frequency, v = speed of source w.r.t stationary observer, c = speed of light

(1) Source of light moves towards the stationary observer : When a light source is moving towards an observer with a relative velocity v then the  apparent frequency (n') is greater than the actual frequency (n) of light. Thus apparent wavelength \[(\lambda ')\] is lesser the actual wavelength \[(\lambda )\].

\[\nu '=\nu \sqrt{\frac{(1+v/c)}{(1-v/c)}}\] and \[\lambda '=\lambda \sqrt{\frac{(1-v/c)}{(1+v/c)}}\]

For v << c :

(i) Apparent frequency \[{\nu }'=\nu \,\left( 1+\frac{v}{c} \right)\] and

(ii) Apparent wavelength \[{\lambda }'=\lambda \,\left( 1-\frac{v}{c} \right)\]

(iii) Doppler's shift : Apparent wavelength < actual wavelength,

So spectrum of the radiation from the source of light shifts towards the violet end of spectrum. This is called violet shift

Doppler's shift \[\Delta \lambda =\lambda .\frac{v}{c}\]

(iv) The fraction decrease in wavelength \[=\frac{\Delta \lambda }{\lambda }=\frac{v}{c}\]

(2) Source of light moves away from the stationary observer : In this case \[v'<v\] and \[\lambda '>\lambda \]

\[\nu '=\nu \sqrt{\frac{(1-v/c)}{(1+v/c)}}\] and \[\lambda '=\lambda \sqrt{\frac{(1+v/c)}{(1-v/c)}}\]

For v << c :

(i) Apparent frequency  \[{\nu }'=\nu \,\left( 1-\frac{v}{c} \right)\] and

(ii) Apparent wavelength \[{\lambda }'=\lambda \,\left( 1+\frac{v}{c} \right)\]

(iii) Doppler's shift : Apparent wavelength > actual wavelength,

So spectrum of the radiation from the source of light shifts towards the red end of spectrum. This is called red shift

Doppler's shift \[\Delta \lambda =\lambda .\frac{v}{c}\]

(iv) The fractional increase in wavelength \[=\frac{\Delta \lambda }{\lambda }=\frac{v}{c}\].

(3) Doppler broadening : For a gas in a discharge tube, atoms are moving randomly in all directions. When spectrum of light emitted from these atoms is analyzed, then due to Doppler effect (because some atoms are moving towards detector, some atoms are moving away from detector), the frequency of a spectral line is not observed as having one value, but is spread over a range

\[\pm \Delta \nu =\pm \frac{v}{c}\nu \],   \[\pm \,\Delta \lambda =\,\pm \,\frac{v}{c}\lambda \]

This broadens the spectral line by an amount \[(2\Delta \lambda )\]. It is called Doppler broadening. The Doppler broadening is proportional to v, which in turn is proportional to \[\sqrt{T},\] where T is the temperature in Kelvin.

(4) Radar : Radar is a system for locating distant object by means of reflected radio waves, usually of microwave frequencies. Radar is used for navigation and guidance of aircraft, ships etc.,.

Radar employs the Doppler effect to distinguish between stationary and moving targets. The change in frequency between transmitted and received waves is measured. If v is the velocity of the approaching target, then the change in frequency is

\[\Delta \nu =\frac{2v}{c}\nu \]. (The factor of 2 arises due to refection of waves).

For a receding target \[\Delta \nu =-\frac{2v}{c}\nu \]. (The minus sign indicates decrease in frequency).

(5) Applications of Doppler effect

(i) Determination of speed of moving bodies (aeroplane, submarine etc) in RADAR and SONAR.

(ii) Determination of the velocities of stars and galaxies by spectral shift.

(iii) Determination of rotational motion of sun.

(iv) Explanation of width of spectral lines.

(v) Tracking of satellites. (vi) In medical sciences in echo cardiogram, sonography etc.