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==== Radiant Energy Density ====
**I. Definition**\\
**Radiant energy density** is defined as the __electromagnetic radiant energy stored per unit volume of space__, typically denoted as \(w\) or \(u\), describing the concentration of energy distribution of a radiation field in space.\\
The **SI unit** is: joules per cubic meter (\(J/m^3\)).\\
The **dimensional expression** is: \([M L^{-1} T^{-2}]\).\\
Its **mathematical expression** is:\\
$$w = \frac{dQ}{dV}$$
where \(dQ\) represents the radiant energy contained within the infinitesimal volume \(dV\).\\
**Application Example:**\\
{{ :yanding:成像基础知识:光学:辐射度学与光度学:微波炉.png?300|}}
**Radiant energy of the microwave oven:** $Q = 8000 J$\\
**Internal volume of the microwave oven:** $V= 23 L= 0.023 {m}^3$\\
**Radiant energy density:**  $w \approx 347\,826.09 {J/m}^3$\\


**II. Physical Nature and Characteristics**\\
**1. Composition of Electromagnetic Field Energy**\\
According to Maxwell's theory of electromagnetic fields, radiant energy density is the superposition of the electric field energy density and the magnetic field energy density in space. Its value is determined by the squared terms of the instantaneous electric field strength \(E\) (unit: V/m) and the magnetic flux density \(B\) (unit: T):\\
$$w = \frac{1}{2}\left(\varepsilon E^2 + \frac{1}{\mu} B^2\right)$$
This reveals that radiant energy is a physical energy residing in space in the form of alternating electromagnetic fields. In a plane electromagnetic wave propagating in a vacuum, the energy contributed by the electric and magnetic fields is equally divided.\\

**2. Distinction Between State and Process Quantities**\\
Radiant energy density is a state quantity, describing the "residence concentration" of energy in space at a specific moment; whereas irradiance (\(E\)) is a process quantity, describing the "transfer rate" of energy flowing through a surface. The two are linked through the speed of light \(c\), reflecting the unity of wave propagation characteristics and spatial energy distribution.\\
**Isotropic field (e.g., black-body cavity):** \(w = \dfrac{4E}{c}\)\\
**Directional radiation (e.g., parallel laser beam):** \(w = \dfrac{E}{c}\)\\

**3. Isotropy and Radiation Pressure**\\
In an isotropic radiation field in thermal equilibrium (such as a black-body cavity), the radiant energy density is uniformly distributed in all directions. Based on the law of conservation of momentum, this energy distribution generates an outward physical pressure, namely radiation pressure. For an isotropic field, there is a linear relationship between the radiant energy density and the radiation pressure \(P\): $P = \frac{1}{3} w$.\\

**III. Black-Body Radiation and Spectral Characteristics**\\
**1. Black-Body Radiation and Thermodynamics**\\
In thermal equilibrium, the radiant energy density of a black body depends only on the thermodynamic temperature T, following the Stefan-Boltzmann law:\\
$$w = a T^4$$
where \(a = \frac{4\sigma}{c}\) is the radiation constant, and \(\sigma\) is the Stefan-Boltzmann constant.\\

**2. Spectral Radiant Energy Density**\\
To describe the distribution of energy with respect to wavelength, the spectral radiant energy density \(w_\lambda\) is introduced:\\
$$w = \int_0^\infty w_\lambda d\lambda$$
According to Planck's law, its expression reveals the contribution of different spectral bands (such as visible light and infrared) to the total energy density.\\

**See Also**\\
[[yanding:成像基础知识:光学:辐射度学与光度学:辐射能]], [[yanding:成像基础知识:光学:辐射度学与光度学:辐射通量]]