Explanation Consider a source of light and the receiver; the velocity of receiver is v and the velocity of light is c . The source emits n waves during time interval T . The receiver is approaching the rest source; so the relative velocity of light with respect to receiver is v . The length of wave train is c T − v T and there are n waves emitted during interval T . Write the expression of wavelength: λ = c T − v T n Here, λ is the wavelength, T is time interval, v is the relative velocity, c is the velocity of light. Write the expression of frequency: f = c λ Substitute c T − v T n for λ in above expression. f = c n c T − v T (I) Here, f is the frequency of light. The source in rest emits n waves of frequency f 0 during proper time T 0 . Write the expression for n : n = f 0 T 0 (II) Here, n is the number of waves, f 0 is the natural frequency and T 0 is the proper time. Write the relation between proper time and time interval: T 0 = T γ Here, γ is the relativistic factor. Substitute T γ for T 0 in equation (II) n = f 0 T γ (III) Substitute f 0 T γ for n in equation (I). f = c ( f 0 T γ ) c T − v T = ( c f 0 T γ ) ⋅ 1 c T ( 1 − v c ) = 1 ( 1 − v c ) ⋅ f 0 γ Further substitute 1 ( 1 − v 2 c 2 ) for γ in above expression

Modern Physics for Scientists and Engineers

4th Edition

ISBN: 9781133103721

Author: Stephen T. Thornton, Andrew Rex

Publisher: Cengage Learning

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Chapter 2, Problem 49P

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To derive equation (2.32) when the source is fixed and the receiver approaches it with velocity v.

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Chapter 2, Problem 48P

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Modern Physics for Scientists and Engineers

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To derive equation (2.32) when the source is fixed and the receiver approaches it with velocity v .

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To derive equation (2.32) when the source is fixed and the receiver approaches it with velocity v.

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Chapter 2 Solutions