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Chapter 7: Q 7.37 (page 293)

Prove that the peak of the Planck spectrum is at x = 2.82.

Short Answer

Expert verified

Hence proved that plank's spectrum is at x = 2.82.

Step by step solution

01

Given information

The peak of the Planck spectrum is at e = 2.82.

02

Explanation

Set the derivative of x3/ex-1with respect to x equals to zero to get the maximum, then solve for:

鈭�鈭�xx3ex-1=03x2ex-1-x3exex-12=03x2ex-1-x3ex=03x2ex-3x2-x3ex=03ex-3-xex=0

Using matlab to solve the above equation: The code is:

Therefore,x=2.8214

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Most popular questions from this chapter

A black hole is a blackbody if ever there was one, so it should emit blackbody radiation, called Hawking radiation. A black hole of mass M has a total energy of Mc2, a surface area of 16蟺G2M2/c4, and a temperature ofhc3/16蟺2kGM(as shown in Problem 3.7).

(a) Estimate the typical wavelength of the Hawking radiation emitted by a one-solar-mass (2 x 1030 kg) black hole. Compare your answer to the size of the black hole.

(b) Calculate the total power radiated by a one-solar-mass black hole.

(c) Imagine a black hole in empty space, where it emits radiation but absorbs nothing. As it loses energy, its mass must decrease; one could say it "evaporates." Derive a differential equation for the mass as a function of time, and solve this equation to obtain an expression for the lifetime of a black hole in terms of its initial mass.

(d) Calculate the lifetime of a one-solar-mass black hole, and compare to the estimated age of the known universe (1010 years).

(e) Suppose that a black hole that was created early in the history of the universe finishes evaporating today. What was its initial mass? In what part of the electromagnetic spectrum would most of its radiation have been emitted?

The argument given above for why Cv鈭�Tdoes not depend on the details of the energy levels available to the fermions, so it should also apply to the model considered in Problem 7.16: a gas of fermions trapped in such a way that the energy levels are evenly spaced and non-degenerate.

(a) Show that, in this model, the number of possible system states for a given value of q is equal to the number of distinct ways of writing q as a sum of positive integers. (For example, there are three system states for q = 3, corresponding to the sums 3, 2 + 1, and 1 + 1 + 1. Note that 2 + 1 and 1 + 2 are not counted separately.) This combinatorial function is called the number of unrestricted partitions of q, denoted p(q). For example, p(3) = 3.

(b) By enumerating the partitions explicitly, compute p(7) and p(8).

(c) Make a table of p(q) for values of q up to 100, by either looking up the values in a mathematical reference book, or using a software package that can compute them, or writing your own program to compute them. From this table, compute the entropy, temperature, and heat capacity of this system, using the same methods as in Section 3.3. Plot the heat capacity as a function of temperature, and note that it is approximately linear.

(d) Ramanujan and Hardy (two famous mathematicians) have shown that when q is large, the number of unrestricted partitions of q is given approximately by

p(q)鈮�e蟺2q343q

Check the accuracy of this formula for q = 10 and for q = 100. Working in this approximation, calculate the entropy, temperature, and heat capacity of this system. Express the heat. capacity as a series in decreasing powers of kT/畏, assuming that this ratio is large and keeping the two largest terms. Compare to the numerical results you obtained in part (c). Why is the heat capacity of this system independent of N, unlike that of the three dimensional box of fermions discussed in the text?

A white dwarf star (see Figure 7.12) is essentially a degenerate electron gas, with a bunch of nuclei mixed in to balance the charge and to provide the gravitational attraction that holds the star together. In this problem you will derive a relation between the mass and the radius of a white dwarf star, modeling the star as a uniform-density sphere. White dwarf stars tend to be extremely hot by our standards; nevertheless, it is an excellent approximation in this problem to set T=0.

(a) Use dimensional analysis to argue that the gravitational potential energy of a uniform-density sphere (mass M, radius R) must equal

Ugrav=-(constant)GM2R

where (constant) is some numerical constant. Be sure to explain the minus sign. The constant turns out to equal 3/5; you can derive it by calculating the (negative) work needed to assemble the sphere, shell by shell, from the inside out.

(b) Assuming that the star contains one proton and one neutron for each electron, and that the electrons are nonrelativistic, show that the total (kinetic) energy of the degenerate electrons equals

Ukinetic=(0.0086)h2M53memp53R2

Figure 7.12. The double star system Sirius A and B. Sirius A (greatly overexposed in the photo) is the brightest star in our night sky. Its companion, Sirius B, is hotter but very faint, indicating that it must be extremely small-a white dwarf. From the orbital motion of the pair we know that Sirius B has about the same mass as our sun. (UCO /Lick Observatory photo.)

( c) The equilibrium radius of the white dwarf is that which minimizes the total energy Ugravity+Ukinetic路 Sketch the total energy as a function of R, and find a formula for the equilibrium radius in terms of the mass. As the mass increases, does the radius increase or decrease? Does this make sense?

( d) Evaluate the equilibrium radius for M=2脳1030kg, the mass of the sun. Also evaluate the density. How does the density compare to that of water?

( e) Calculate the Fermi energy and the Fermi temperature, for the case considered in part (d). Discuss whether the approximation T = 0 is valid.

(f) Suppose instead that the electrons in the white dwarf star are highly relativistic. Using the result of the previous problem, show that the total kinetic energy of the electrons is now proportional to 1 / R instead of 1R2鈥� Argue that there is no stable equilibrium radius for such a star.

(g) The transition from the nonrelativistic regime to the ultra relativistic regime occurs approximately where the average kinetic energy of an electron is equal to its rest energy, mc2Is the nonrelativistic approximation valid for a one-solar-mass white dwarf? Above what mass would you expect a white dwarf to become relativistic and hence unstable?

Consider a free Fermi gas in two dimensions, confined to a square area A=L2鈥�

(a) Find the Fermi energy (in terms of Nand A), and show that the average energy of the particles is 鈭�F2.

(b) Derive a formula for the density of states. You should find that it is a constant, independent of 鈭�.

(c) Explain how the chemical potential of this system should behave as a function of temperature, both when role="math" localid="1650186338941" kT鈮�鈭�Fand when Tis much higher.

(d) Because g鈭�is a constant for this system, it is possible to carry out the integral 7.53 for the number of particles analytically. Do so, and solve for 渭as a function of N. Show that the resulting formula has the expected qualitative behavior.

(e) Show that in the high-temperature limit, kT鈮�鈭�F, the chemical potential of this system is the same as that of an ordinary ideal gas.

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.
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