Chapter 19, Thermal radiation Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

(a) Give the SI units for the following quantities: energy density $u$ (that is, energy per unit volume), and spectral energy density $\rho$ (that is, energy per unit volume per unit frequency range).
(b) Write down the expression relating $u$ and $\rho$.
(c) If $\rho$ is the energy density per unit frequency range, and $\eta$ is the energy density per unit wavelength range, write down the expression relating $\rho$ and $\eta$ for electromagnetic radiation.
(d) Derive the relationship between energy density $u$ and energy flux $\phi$ for a collimated beam of light. (Energy flux is also called intensity, it is power per unit area.)

David Collins

Numerade Educator

03:46

Problem 2

Bearing in mind that glass is a good absorber of infrared radiation, sketch on the same, labelled graph:
(a) the spectral emissive power of a bathroom mirror at room temperature, as a function of wavelength, in the wavelength range $400 \mathrm{~nm}$ to $100 \mu \mathrm{m}$.
(b) the spectral energy density of radiation incident on a detector placed close to the mirror, when the mirror (still at room temperature) is used to reflect sunlight onto the detector.

Surendra Kumar

Numerade Educator

02:02

Problem 3

A confused student puts forward the following argument. 'Red paint appears red (under white light illumination), and this is to do with the fact that it absorbs other colours such as green and blue. Therefore, according to Kirchoff's law, it should be a good emitter of green and blue, and a relatively poor emitter of red. Therefore it should appear blue-green, and not red after all'. This is a contradiction. Where did the student go wrong?

Zulfiqar Ali

Numerade Educator

03:18

Problem 4

Find the temperature at which the number of photons in a chamber of volume 25 litres is equal to one mole.

Bhumika Jayee

Numerade Educator

03:48

Problem 5

Show that if the pressure of a simple compressible system is related to its energy density by $p=\Gamma u$, where $\Gamma$ is a constant, then the adiabatic index is $\gamma=\Gamma+1$.

Nathan Silvano

Numerade Educator

08:10

Problem 6

Show that the Helmholtz function of the cavity radiation in a cavity of volume $V$ is $F(T, V)=-a V T^4$, where $a$ is a constant. Hence obtain the equation of state and the heat capacities.

Mahnoor Amin

Numerade Educator

01:14

Problem 7

Show that the entropy of such radiation, as a function of its natural variables, is
$$
S=\frac{4}{3}\left(\frac{4 \sigma}{c}\right)^{1 / 4} V^{1 / 4} U^{3 / 4}
$$.

Ankur S

Numerade Educator

Problem 8

Show that a simple compressible system whose energy density and pressure each depend on temperature alone, i.e. $u=u(T), p=p(T)$, and such that $\mathrm{d} u / \mathrm{d} T \rightarrow 0$ and $\mathrm{d} p / \mathrm{d} T \rightarrow 0$ as $T \rightarrow 0$, must have entropy $S=V \mathrm{~d} p / \mathrm{d} T$ and chemical potential $\mu=0$. [Hint: generalize the argument of equations $(19.20)-(19.32)$.]

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01:14

Problem 9

Show that a simple compressible system whose chemical potential is always zero and whose energy density depends on temperature alone must have an entropy density that depends on temperature alone.

Ajay Singhal

Numerade Educator

02:08

Problem 10

Describe one kind of electromagnetic radiation which is not black body radiation, even approximately, and identify some features which distinguish it from black body radiation.

Meghan Hinton

Numerade Educator