The relationship between the modes of vibration in a cube and the radiation emitted by a black body, as described by the Rayleigh-Jeans law, is based on the principles of classical physics and the assumption that energy is continuously distributed among all possible modes of vibration.
The Rayleigh-Jeans law is an approximation that describes the spectral radiance (intensity) of electromagnetic radiation emitted by a black body at thermal equilibrium across a wide range of wavelengths. According to this law, the spectral radiance is directly proportional to the temperature of the black body and increases as the wavelength of the radiation becomes larger.
In the context of a cube, the modes of vibration refer to the different vibrational patterns that can exist within the cube. For simplicity, let's consider a cube with three mutually perpendicular axes (x, y, and z). Each axis can support vibrational modes, such as longitudinal and transverse modes, which correspond to different patterns of motion of the cube's molecules or atoms.
According to classical physics, the energy associated with each vibrational mode is quantized and can take on specific discrete values. In the Rayleigh-Jeans law, the energy associated with each mode is assumed to be continuous and can have any value. This assumption is known as the equipartition of energy.
The Rayleigh-Jeans law states that the average energy per mode of vibration is directly proportional to the temperature of the black body and to the Boltzmann constant (k). Mathematically, this can be expressed as:
E = k * T
Where: E is the average energy per mode of vibration, k is the Boltzmann constant, and T is the temperature of the black body.
The radiated power per unit area per unit wavelength (spectral radiance) as described by the Rayleigh-Jeans law is then given by:
B(λ, T) = (2 * k * T) / λ^4
Where: B(λ, T) is the spectral radiance at a given wavelength (λ) and temperature (T).
In summary, the Rayleigh-Jeans law relates the modes of vibration in a cube (or any black body) to the energy distribution and spectral radiance of the radiation it emits. It assumes that energy is continuously distributed among all possible modes, and the average energy per mode is directly proportional to the temperature.