Understanding Monochromatic Energy Density: A Temperature-Dependent Analysis

[zorro]

Homework Statement

Monochromatic energy density (Uλ)(i.e., per unit volume) of the radiation of
a blackbody is proportional to (where λ is wavelength)

(1) T⁴
(2) λ ^-5
(3) λ^5/2
(4) T^5/4

The Attempt at a Solution

A slight modification of Stefan-Boltzmann's law suggests that the answer should be (1), but the answer given is (2). Is my book wrong?

 

[G01]

Homework Statement

Monochromatic energy density (Uλ)(i.e., per unit volume) of the radiation of
a blackbody is proportional to (where λ is wavelength)

(1) T⁴
(2) λ ^-5
(3) λ^5/2
(4) T^5/4

The Attempt at a Solution

A slight modification of Stefan-Boltzmann's law suggests that the answer should be (1), but the answer given is (2). Is my book wrong?

Going off the information given here, I can see how you are getting confused. The problem refers to " monochromatic energy density" and is thus asking about the energy density per unit wavelength.

$U(\lambda)$ is the energy density per unit wavelength (energy/(volume*wavelength))and has the $\lambda^{-5}$ dependence. If you integrate that over all possible wavelengths, you get the energy density (energy/volume) which has the temperature dependence you expect.

 

[zorro]

The problem refers to " monochromatic energy density" and is thus asking about the energy density per unit wavelength.

Does the word 'monochromatic' make it obvious that it is energy density per unit wavelength.?

If you integrate that over all possible wavelengths, you get the energy density (energy/volume) which has the temperature dependence you expect.

Is energy density per unit wavelength not dependent on temperature?. Can you provide me a link which shows the derivation?