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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:
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.
