Unit 2 Electromagnetic Waves
2.2 Propagation of Electromagnetic Waves
2.2.3 Doppler Effect of Electromagnetic Waves
The normal Doppler shift for waves such as sound which move with velocities v much less than c is given by the expression:
f_{observed} = f_{source}{^{V}/_{[V±Vs]}} (source moving and stationary observer)
A positive value is used for motion of the source away from the observer.
Light waves travel in empty space so the Doppler effect for light is analyzed in terms of the motion of the source relative to the listener who is always considered to be at rest. If the source is moving away from the listener, its velocity v is positive, but if it is moving toward the listener, then the v is negative. The speed of light c is always considered positive.
At nonrelativistic speeds (c >> Vs we calculate Vrelative using the formula
f_{observed} = f_{source}{^{c}/_{[c±Vs]}}
At relativistic speeds we calculate Vrelative using the formula
V_{relative} = [Vs + Vp] /{[1 + VsVp] /c^{2}}
so adjusting the equation f_{observed} = f_{source}{^{V}/_{[V±Vs]}}
we get f_{observed} = f_{source}{1/√[1β^{2}]} where β = ^{Vs}/_{c}
If the source is moving away from the observer then f_{observed} < f_{source}.
If the source is moving towards the observer then f_{observed} > f_{source}  In the visible spectrum we get a shift to the red end of the spectrum called a red shift and
If the source is moving towards the observer then f_{observed} > f_{source} In the visible spectrum we get a shift to the blue end of the spectrum called a blue shift.
Wavelength is easier to measure than frequency so we often work with wavelength.
Example 1: A galaxy is moving away from the Earth at 2.6x10^{7}ms^{1}. Calculate the wavelength and frequency change of a 650 nm line in its spectrum. Take c = 3x10^{8}ms^{1}.
f_{source} = ^{c}/_{λ}
f_{source} = 4.6x10^{14}Hz
f_{observed} = f_{source}{^{c}/_{[c±Vs]}}
f_{observed} = 4.6x10^{14}Hz{(3x10^{8}ms^{1})/[(3x10^{8}ms^{1})+(2.6x10^{7}ms^{1})]}
f_{observed} =4.2x10^{14}Hz
λ_{observed} = ^{c}/f_{observed}
λ_{observed} = 709 nm
The frequency has shifted from red to infrared.
Example 2: A distant galaxy is moving away from us at approximately 5x10^{7}ms^{1} and we approximate the speed of light as c = 3x10^{8}ms^{1}. (a) What is the resulting wavelength of the Hydrogen spectral line of λ = 434 nm? (b) what would be the shift if the galaxy was moving towards earth?
(a) f_{source} = ^{c}/_{λ}
f_{source} = 6.91x10^{14}Hz
f_{observed} = f_{source}{^{c}/_{[c±Vs]}}
f_{observed} = 6.91x10^{14}Hz{(3x10^{8}ms^{1})/[(3x10^{8}ms^{1})+(5.0x10^{7}ms^{1})]}
f_{observed} =5.92x10^{14}Hz
λ_{observed} = ^{c}/f_{observed}
λ_{observed} = 507 nm
The frequency has shifted from violet to green.
(b) f_{observed} = f_{source}{^{c}/_{[c±Vs]}}
f_{observed} = 6.91x10^{14}Hz{(3x10^{8}ms^{1})/[(3x10^{8}ms^{1})(5.0x10^{7}ms^{1})]}
f_{observed} =8.29x10^{14}Hz
λ_{observed} = ^{c}/f_{observed}
λ_{observed} = 362 nm
The frequency has shifted from violet to ultraviolet.
