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Ch. 4 Solutions

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Basic

Advanced

Basic

B-1. Verification of the Ultraviolet Catastrophe

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Problem:

Integrate the classical Rayleigh-Jeans law \(u(\nu) = \frac{8\pi \nu^2}{c^3} k_B T\) with respect to \(\nu\) from \(0\) to \(\infty\), and verify that the total energy density diverges.

Solution:

\[\int_0^\infty u(\nu)\,d\nu = \int_0^\infty \frac{8\pi \nu^2}{c^3} k_B T \,d\nu = \frac{8\pi k_B T}{c^3}\int_0^\infty \nu^2 \,d\nu\]

\(\int_0^\infty \nu^2 \,d\nu\) diverges (\(\nu^3/3 \to \infty\)).

\[\boxed{\text{Total energy density} = \infty}\]

Key point: The classical Rayleigh-Jeans law agrees with experiment at low frequencies, but diverges at high frequencies. Planck's quantum hypothesis \(E = h\nu\) resolved this divergence.

B-2. Threshold Frequency of the Photoelectric Effect

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Problem:

Light is incident on a metal with a work function \(W = 4.5\) eV. Calculate the minimum frequency \(\nu_{\min}\) required for electrons to be ejected using \(h\nu_{\min} = W\) (\(h = 6.626 \times 10^{-34}\) J·s, \(1\) eV \(= 1.602 \times 10^{-19}\) J).

Solution:

\[h\nu_{\min} = W = 4.5 \;\text{eV} = 4.5 \times 1.602 \times 10^{-19} = 7.209 \times 10^{-19} \;\text{J}\]
\[\nu_{\min} = \frac{W}{h} = \frac{7.209 \times 10^{-19}}{6.626 \times 10^{-34}} = \boxed{1.09 \times 10^{15} \;\text{Hz}}\]

This falls in the ultraviolet region. Visible light (\(\sim 4\text{--}8 \times 10^{14}\) Hz) cannot eject electrons.

Advanced

A-1. Mercury's Perihelion Precession and the Limits of the Newtonian Model

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Problem:

In Newtonian gravitation, a planetary orbit around the Sun forms a closed ellipse (the orbit does not rotate). However, observations show that Mercury's elliptical orbit rotates by approximately 5600 arcseconds per century, and even after subtracting all influences from other planets, a discrepancy of 43 arcseconds remains.

(a) Explain why "the influence of other planets" can be calculated within Newton's model, drawing a contrast with the discovery of Neptune discussed in Ch. 1.

(b) Two hypotheses can be considered for the 43-arcsecond discrepancy: "an unknown planet exists" and "Newton's model must be modified." State what predictions each hypothesis would make, and as a preview of Ch. 6, state which one actually turned out to be correct.

Note: The quantitative calculation using general relativity (derivation of \(\delta\phi\) and numerical verification) is treated in Exercise 6.4 of Ch. 6.

Solution (Report Format):

When all perturbations from other planets are calculated using Newton's model, the predicted perihelion precession of Mercury is approximately 532 arcseconds per century. The observed value is approximately 575 arcseconds. The difference is 43 arcseconds/century.

43 arcseconds corresponds to approximately 0.012 degrees. This is extremely small, but considering the precision of Newton's model (which can accurately calculate 532 arcseconds including perturbations from other planets), this residual is significant.

Einstein's general relativity explained this 43 arcseconds precisely without any additional parameters. This was the first experimental verification of general relativity, and a historic example of how a "small discrepancy" led to a revolution in our model.