Consider a cubical box of side L with no matter in its...

Consider a cubical box of side $L$ with no matter in its interior. The walls are fixed at absolute temperature $T$, and they are in thermal equilibrium with the electromagnetic radiation field in the interior. (a) Find the mean electromagnetic energy per unit volume in the frequency range from $\omega$ to $\omega+d \omega$ as a function of $\omega$ and $T$. (If you wish to start with a known distribution function - e.g., Maxwell-Boltzmann, Planck, etc. - you need not derive that function.) (b) Find the temperature dependence of the total electromagnetic energy per unit volume. (Hint: you do not have to actually carry out the integration of the result of part (a) to answer this question.)

Consider a cubical box of side $L$ with no matter in its interior. The walls are fixed at absolute temperature $T$, and they are in thermal equilibrium with the electromagnetic radiation field in the interior.
(a) Find the mean electromagnetic energy per unit volume in the frequency range from $\omega$ to $\omega+d \omega$ as a function of $\omega$ and $T$. (If you wish to start with a known distribution function - e.g., Maxwell-Boltzmann, Planck, etc. - you need not derive that function.)
(b) Find the temperature dependence of the total electromagnetic energy per unit volume. (Hint: you do not have to actually carry out the integration of the result of part (a) to answer this question.)

Key Concepts

Blackbody Radiation

Blackbody radiation refers to the idealized emission of electromagnetic radiation by an object that absorbs all incident radiation, regardless of frequency or angle of incidence. In thermal equilibrium, the radiation within a cavity is solely determined by the temperature of the walls, leading to a characteristic spectrum that depends only on temperature. This concept underpins the study of electromagnetic energy in confined systems such as a cubical box at thermal equilibrium.

Planck Distribution

The Planck distribution describes the spectral energy density of blackbody radiation, providing the mean electromagnetic energy per unit volume at a given frequency and temperature. It corrects the shortcomings of classical theories by introducing the concept of quantized energy levels, and it forms the basis for calculating the distribution of electromagnetic energy over frequency ranges in thermal systems.

Density of States

The density of states quantifies the number of available electromagnetic modes per unit frequency interval in a confined space, such as a cubical cavity. This concept is crucial when deriving expressions for the energy density, as it accounts for the possible configurations or modes that the electromagnetic field can occupy, depending on the geometry of the system.

Stefan-Boltzmann Law

The Stefan-Boltzmann law establishes that the total energy density (or radiated energy per unit area) of blackbody radiation is proportional to the fourth power of the absolute temperature. Although derived through integration of the Planck distribution over all frequencies, this law succinctly encapsulates the overall temperature dependence of the energy density in thermal electromagnetic fields.

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