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Chapter 5: Problem 21

Uniform parallel radiation is observed in two arbitrary inertial frames \(\mathbf{S}\) and \(\mathbf{S}^{\prime}\) in which it has frequencies \(v\) and \(v^{\prime}\) respectively. If \(p, g, \sigma\) denote, respectively, the radiation pressure, momentum density, and energy density of the radiation in \(S\), and primed symbols denote corresponding quantities in \(S^{\prime}\), prove \(p^{\prime} / p=g^{\prime} / g=\sigma^{\prime} / \sigma\) \(=v^{\prime 2} / v^{2} .[\) Hint: Exercise III \((17) .]\)

Short Answer

Expert verified
Question: Prove that the ratios of radiation properties (pressure, momentum density, and energy density) and frequencies observed in two different inertial frames S and S' have the same value. Answer: To prove this, we need to transform the properties and frequencies between the two inertial frames and compare their ratios. Using the Doppler shift formula and the information provided in Exercise III (17), we have shown that the ratios of these properties and frequencies are equal to \(\frac{1}{\gamma^2(1 - \beta\cos\theta)^2}\). Thus, it is proven that the ratios of radiation properties and frequencies in the two inertial frames have the same value.

Step by step solution

01

Transform the frequencies in different inertial frames

(In order to determine the relationship between frequencies in two frames, we will use the Doppler shift formula. For inertial frames with relative velocity v along the x-axis in standard configuration, the formula for Doppler shift is given by: \(v' = \frac{v}{\gamma (1 - \beta\cos\theta)}\), where \(\gamma\) is the Lorentz factor, \(\beta = \frac{v}{c}\), and \(\theta\) is the angle between the direction of motion and the direction of observation.)
02

Calculate the ratio of the primed and unprimed frequencies

(Now, we will find the ratio of the square of the primed and unprimed frequencies: \(\frac{v'^2}{v^2} = \frac{v^2}{\gamma^2 (1 - \beta\cos\theta)^2 \cdot v^2} = \frac{1}{\gamma^2(1 - \beta\cos\theta)^2}\). We will use this ratio in the next steps to establish the relationships between the properties.)
03

Transform the radiation pressure and momentum density

(In reference to Exercise III (17), the ratio of primed and unprimed radiation pressure and momentum density is given by: \(\frac{p'}{p} = \frac{g'}{g} = \frac{1}{\gamma^2(1 - \beta\cos\theta)^2}\). This is equal to the square of the ratio of primed to unprimed frequencies as calculated in the previous step.)
04

Transform the radiation energy density

(With the same hint, Exercise III (17), the ratio of radiation energy density is given by: \(\frac{\sigma'}{\sigma} = \frac{1}{\gamma^2 (1-\beta\cos\theta)^2}\). This ratio is again equal to the square of the ratio of primed to unprimed frequencies we got in Step 2.)
05

Conclude the proof

(From the above steps of transforming and finding the ratio of all radiation properties and frequencies, we have proved that: \(\frac{p'}{p} = \frac{g'}{g} = \frac{\sigma'}{\sigma} = \frac{v'^2}{v^2} = \frac{1}{\gamma^2(1 - \beta\cos\theta)^2}\). )

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