/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q 7.40 Starting from equation 7.83, der... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q 7.40 (page 293)

Starting from equation 7.83, derive a formula for the density of states of a photon gas (or any other gas of ultra relativistic particles having two polarisation states). Sketch this function.

Short Answer

Expert verified

Hence, the formula for density of states of a photon gas isg(系)=8蟺V系2(hc)3

Step by step solution

01

Given information

The equation 7.83 is

UV=8蟺(hc)3鈭�0鈭�系3e系/kT-1d系

02

Explanation

The equation 7.83 is:

UV=8蟺(hc)3鈭�0鈭�系3e系/kT-1d系

We can write the equation as:

localid="1647752992962">U=鈭�0鈭�系8蟺V系2(hc)31e系/kT-1d系(1)

Distribution function for Planck's constant is given as:

n炉Pl=1e系/kT-1

Substituting this into (1)

U=鈭�0鈭�系8蟺V系2(hc)3n炉Pld系

Hence the energy density for Planck's constant is

g(系)=8蟺V系2(hc)3

Using Python to solve this function, the code is:

The graph is:

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In this problem you will model helium-3 as a non-interacting Fermi gas. Although He3liquefies at low temperatures, the liquid has an unusually low density and behaves in many ways like a gas because the forces between the atoms are so weak. Helium-3 atoms are spin-1/2 fermions, because of the unpaired neutron in the nucleus.

(a) Pretending that liquid 3He is a non-interacting Fermi gas, calculate the Fermi energy and the Fermi temperature. The molar volume (at low pressures) is 37cm3鈥�

(b)Calculate the heat capacity for T<<Tf, and compare to the experimental result CV=(2.8K-1)NkT(in the low-temperature limit). (Don't expect perfect agreement.)

(c)The entropy of solid H3ebelow 1 K is almost entirely due to its multiplicity of nuclear spin alignments. Sketch a graph S vs. T for liquid and solid H3eat low temperature, and estimate the temperature at which the liquid and solid have the same entropy. Discuss the shape of the solid-liquid phase boundary shown in Figure 5.13.

Explain in some detail why the three graphs in Figure 7.28 all intercept the vertical axis in about the same place, whereas their slopes differ considerably.

Repeat the previous problem, taking into account the two independent spin states of the electron. Now the system has two "occupied" states, one with the electron in each spin configuration. However, the chemical potential of the electron gas is also slightly different. Show that the ratio of probabilities is the same as before: The spin degeneracy cancels out of the saha equation.

The heat capacity of liquid H4ebelow 0.6Kis proportional to T3, with the measured valueCV/Nk=(T/4.67K)3. This behavior suggests that the dominant excitations at low temperature are long-wavelength photons. The only important difference between photons in a liquid and photons in a solid is that a liquid cannot transmit transversely polarized waves-sound waves must be longitudinal. The speed of sound in liquid He4is 238m/s, and the density is 0.145g/cm3. From these numbers, calculate the photon contribution to the heat capacity ofHe4in the low-temperature limit, and compare to the measured value.

Number of photons in a photon gas.

(a) Show that the number of photons in equilibrium in a box of volume V at temperature T is

N=8蟺VkThc3鈭�0鈭�x2ex-1dx

The integral cannot be done analytically; either look it up in a table or evaluate it numerically.

(b) How does this result compare to the formula derived in the text for the entropy of a photon gas? (What is the entropy per photon, in terms of k?)

(c) Calculate the number of photons per cubic meter at the following temperatures: 300 K; 1500 K (a typical kiln); 2.73 K (the cosmic background radiation).
See all solutions

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Read Explanation

Electromagnetism

Read Explanation

Fields in Physics

Read Explanation

Turning Points in Physics

Read Explanation

Wave Optics

Read Explanation

Oscillations

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.