Light Doppler Effect
According to Doppler’s effect, whenever there is a relative motion between a source of light and observer, the apparent frequency of light received by observer is different from the true frequency of light emitted actually from the source of light. The apparent frequency of light increases when the distance between source of light and observer is decreasing and the apparent frequency of light decreases if the distance between the source of light and observer is increasing.
Expression for apparent frequency of light
Suppose a source of light emits waves of frequency v and wavelength λ . If c is velocity of light in vacuum, then
λ = c/v (1)
Let the source and observer approach each other with a velocity v along the direction of propagation of light. In one second, the two come closer by a distance v.
Therefore, apparent frequency = number of light waves received per second by the observer i.e.
v’ = number of light waves emitted/sec by source plus number of light waves contained in a distance v
v’ = v + v/ λ = v/(c/v) = v + v × v/c
v’ = v (1 + v/c) (2)
Clearly, v’ > v
Similarly, when the source of light and observer are moving away from each other with a velocity v, then the apparent frequency is obtained by replacing v by –v in eqn. 2
v’ = v (1 – v/c) (3)
Clearly, v’ < v
We may combine eqns. 2 and 3 and rewrite
V’ = v (1 ± v/c) (4)
+ sign when source and observer are approaching each other, and – sign when source and observer are receding away from each other.
From (4), we find that,
V’ – v = ± vv/c
Δ v = (± v/c) v
Or, v = (± Δ v/v) c (5)
From (1), v = c/ λ
Differentiating w.r.t. λ , we get
dv/d λ = - (c/ λ2)
Or d λ = ( λ 2/c) dv
i.e. Δλ = -( λ2/c) × Δ v = λ2/c [±(v/c) v]
= ± (v/c) λ (using c = v λ )
Hence in terms of wavelength, we can write (5) as
Δ λ = ± (v/c) λ
Or v = ± Δλ /λ (6)
When source and observer approach each other,
?v is + i.e. apparent frequency increases or apparent wavelength decreases. This is called Blue shift.
When source and observer recede away from each other, Δ v is negative i.e. apparent frequency decreases or apparent wavelength increases. This is called red shift.
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