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Chapter 7: Q 7.38 (page 293)

It's not obvious from Figure 7.19 how the Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the functionu(ϵ) for T = 3000 K and T = 6000 K (both on the same graph). Label the horizontal axis in electron-volts.

Short Answer

Expert verified

The function is:

u(ϵ)=8π(hc)3ϵ3eϵ/kT-1

Step by step solution

01

Given information

The Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the functionu(ϵ) for T = 3000 K and T = 6000 K

02

Explanation

The photon's Planck spectrum is given as:

u(ϵ)=8π(hc)3ϵ3eϵ/kT-1(1)

This function must be plotted at temperatures of T= 3000 K and T = 6000 K, with the constants in eV supplied by:

h=4.136×10-15eV·sk=8.62×10-5eV/Kc=3.00×108m/s

Using python to plot the function and the code is:

The graph is:

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Most popular questions from this chapter

Use the formula P=-(∂U/∂V)S,N to show that the pressure of a photon gas is 1/3 times the energy density (U/V). Compute the pressure exerted by the radiation inside a kiln at 1500 K, and compare to the ordinary gas pressure exerted by the air. Then compute the pressure of the radiation at the centre of the sun, where the temperature is 15 million K. Compare to the gas pressure of the ionised hydrogen, whose density is approximately 105 kg/m3.

Calculate the condensate temperature for liquid helium-4, pretending that liquid is a gas of noninteracting atoms. Compare to the observed temperature of the superfluid transition, 2.17K. ( the density of liquid helium-4 is 0.145g/cm3)

Consider a Bose gas confined in an isotropic harmonic trap, as in the previous problem. For this system, because the energy level structure is much simpler than that of a three-dimensional box, it is feasible to carry out the sum in equation 7.121 numerically, without approximating it as an integral.*

(a) Write equation 7.121 for this system as a sum over energy levels, taking degeneracy into account. Replace Tandμwith the dimensionless variables t=kT/hfandc=μ/hf.

(b) Program a computer to calculate this sum for any given values of tandc. Show that, for N=2000, equation 7.121 is satisfied at t=15provided that c=-10.534. (Hint: You'll need to include approximately the first 200 energy levels in the sum.)

(c) For the same parameters as in part (b), plot the number of particles in each energy level as a function of energy.

(d) Now reduce tto 14 , and adjust the value of cuntil the sum again equals 2000. Plot the number of particles as a function of energy.

(e) Repeat part (d) for t=13,12,11,and10. You should find that the required value of cincreases toward zero but never quite reaches it. Discuss the results in some detail.

Consider two single-particle states, A and B, in a system of fermions, where ϵA=μ-xand ϵB=μ+x; that is, level A lies below μ by the same amount that level B lies above μ. Prove that the probability of level B being occupied is the same as the probability of level A being unoccupied. In other words, the Fermi-Dirac distribution is "symmetrical" about the point where ϵ=μ.

For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is

(a) 1eVless than μ

(b) 0.01eVless than μ

(c) equal to μ

(d) 0.01eVgreater than μ

(e) 1eVgreater thanμ
See all solutions

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