A closed metal box whose walls are at room temperature, K

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Origins of quantum theory · Origins of quantum theory

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Light in lumps: blackbody radiation, the photoelectric effect and Compton scattering

Problem

A closed metal box whose walls are at room temperature,

T=300

K, is filled with thermal radiation. The energy density per unit frequency interval is, according to Planck,

u(\nu)=\frac{8\pi h\nu^{3}}{c^{3}}\,\frac{1}{e^{h\nu/k_BT}-1},

while the classical law, giving each standing-wave pattern the average energy

k_BT

, is

u_{\rm cl}(\nu)=\frac{8\pi\nu^{2}k_BT}{c^{3}} .

Let

x=h\nu/k_BT

. Use

h=6.626\times10^{-34}

J s and

k_B=1.381\times10^{-23}

J K

^{-1}

. ((a)) Show that

u(\nu)/u_{\rm cl}(\nu)=x/(e^{x}-1)

, and use

e^{x}\approx1+x

for small

x

to show that the ratio tends to

1

when

h\nu\ll k_BT

. ((b)) Evaluate the ratio at

x=1

and at

x=5

, and say what each value means for the classical prediction. ((c)) Find

x

and the ratio for the microwave frequency

\nu=30

GHz. ((d)) Find

x

and the ratio for the infrared frequency

\nu=30

THz, and state which of the two frequencies a classical calculation would handle acceptably.