Physlib.QuantumMechanics.Blackbody.PlancksLaw

The spectral radiance per unit frequency of blackbody radiation at frequency $\nu$ and temperature $T$, for a given speed of light $c$, is defined as: $$B(\nu, T) = \begin{cases} \frac{2h\nu^3}{c^2} \frac{1}{\exp\left(\frac{h\nu}{k_B T}\right) - 1} & \text{if } \nu > 0 \text{ and } T > 0 \\ 0 & \text{otherwise} \end{cases}$$ where $h$ is the Planck constant and $k_B$ is the Boltzmann constant. The definition is extended by zero outside the physical domain (where $\nu \le 0$ or $T \le 0$).

For a given speed of light $c$, frequency $\nu \in \mathbb{R}$, and temperature $T$, if the frequency $\nu$ is strictly positive and the temperature $T$ is strictly positive, then the spectral radiance of blackbody radiation $B(\nu, T)$ is strictly positive ($B(\nu, T) > 0$).

For any speed of light $c$ and frequency $\nu$, the spectral radiance of blackbody radiation $B(\nu, T)$ at absolute zero temperature ($T = 0$) is zero: $$B(\nu, 0) = 0$$

For any speed of light $c$ and frequency $\nu \in \mathbb{R}$, the spectral radiance of blackbody radiation $B(\nu, T)$ at absolute zero temperature ($T = 0$) is zero: $$B(\nu, 0) = 0$$

For any black body $B$ (at a given temperature $T$) and speed of light $c$, the spectral radiance per unit frequency $B(\nu, T)$ is zero when the frequency $\nu$ is zero: $$B(0, T) = 0$$

For a black body $B$ and speed of light $c$, the spectral radiance at frequency $\nu$, denoted $B(\nu, T)$, is equal to $0$ whenever the frequency is non-positive ($\nu \le 0$).

For a black body $B$ with temperature $T$ and a given speed of light $c$, the spectral radiance per unit wavelength $\lambda$ is defined as: $$B(\lambda, T) = \begin{cases} \dfrac{2 h c^2}{\lambda^5} \dfrac{1}{\exp\left(\frac{h c}{\lambda k_B T}\right) - 1} & \text{if } \lambda > 0 \\ 0 & \text{if } \lambda \le 0 \end{cases}$$ where $h$ is Planck's constant and $k_B$ is the Boltzmann constant.

For a black body $B$ with absolute temperature $T > 0$ and speed of light $c$, the spectral radiance per unit wavelength $B(\lambda, T)$ is strictly positive for any wavelength $\lambda > 0$.

For a black body $B$ with temperature $T > 0$ and speed of light $c$, the spectral radiance per unit wavelength $B(\lambda, T)$ is non-negative for any wavelength $\lambda > 0$, i.e., $$B(\lambda, T) \ge 0$$ where $B(\lambda, T)$ is the energy radiance per unit wavelength as defined by Planck's Law.

For any speed of light $c$ and any wavelength $\lambda \in \mathbb{R}$, the spectral radiance per unit wavelength $B(\lambda, T)$ of a black body at absolute zero temperature ($T = 0$) is zero.

For a black body $B$ and the speed of light $c$, the spectral radiance per unit wavelength $B(\lambda, T)$ is equal to $0$ when the wavelength $\lambda$ is $0$, i.e., $B(0, T) = 0$.

For a black body $B$ and the speed of light $c$, the spectral radiance per unit wavelength $B(\lambda, T)$ is equal to $0$ if the wavelength $\lambda$ is non-positive ($\lambda \le 0$).

For a black body $B$ at a positive temperature $T > 0$ and given the speed of light $c$, the spectral radiance per unit wavelength $B_{\lambda}(\lambda, T)$ for any wavelength $\lambda > 0$ is related to the spectral radiance per unit frequency $B_{\nu}(\nu, T)$ by: $$B_{\lambda}(\lambda, T) = \frac{c}{\lambda^2} B_{\nu}\left(\frac{c}{\lambda}, T\right)$$ where $\nu = \frac{c}{\lambda}$ is the frequency corresponding to wavelength $\lambda$.

Given the speed of light $c$, the first radiation constant $c_1$ is a real number defined as $c_1 = 2hc^2$, where $h$ is the Planck constant.

For any speed of light $c$, the first radiation constant $c_1$ is strictly positive ($c_1 > 0$).

For a given speed of light $c$, the first radiation constant $c_1$ is equal to $2 h c^2$, where $h$ is Planck's constant.

Given the speed of light $c$, the second radiation constant $c_2$ is defined as: $$c_2 = \frac{hc}{k_B}$$ where $h$ is Planck's constant and $k_B$ is the Boltzmann constant.

For any speed of light $c$, the second radiation constant $c_2$ is positive, i.e., $c_2 > 0$.

For a given speed of light $c$, the second radiation constant $c_2$ is equal to $\frac{hc}{k_B}$, where $h$ is Planck's constant and $k_B$ is the Boltzmann constant.

For a black body $B$ with temperature $T$ and speed of light $c$, if the wavelength $\lambda > 0$ and the temperature $T > 0$, then the spectral radiance per unit wavelength $B(\lambda, T)$ is given by $$B(\lambda, T) = \frac{c_1}{\lambda^5} \frac{1}{\exp\left(\frac{c_2}{\lambda T}\right) - 1}$$ where $c_1$ is the first radiation constant $2hc^2$ and $c_2$ is the second radiation constant $\frac{hc}{k_B}$.