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Ch. 4 Solutions

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Table of Contents

Basic

Advanced

Basic

B-1. Verification of the Ultraviolet Catastrophe

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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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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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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.