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Ch. 4 Problems

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Basic

Advanced

Basic

B-1. Verification of the Ultraviolet Catastrophe

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.

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B-2. Threshold Frequency of the Photoelectric Effect

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).

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Advanced

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

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.

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