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Diffstat (limited to 'source/know/concept/plancks-law')
| -rw-r--r-- | source/know/concept/plancks-law/index.md | 38 |
1 files changed, 19 insertions, 19 deletions
diff --git a/source/know/concept/plancks-law/index.md b/source/know/concept/plancks-law/index.md index 2db783d..6c2cd2e 100644 --- a/source/know/concept/plancks-law/index.md +++ b/source/know/concept/plancks-law/index.md @@ -18,14 +18,14 @@ and photons are bosons (see [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/)), this system must obey the [Bose-Einstein distribution](/know/concept/bose-einstein-distribution/), -with a chemical potential $\mu = 0$ (due to the freely varying population): +with a chemical potential $$\mu = 0$$ (due to the freely varying population): $$\begin{aligned} f_B(E) = \frac{1}{\exp(\beta E) - 1} \end{aligned}$$ -Each photon has an energy $E = \hbar \omega = \hbar c k$, +Each photon has an energy $$E = \hbar \omega = \hbar c k$$, so the [density of states](/know/concept/density-of-states/) is as follows in 3D: @@ -37,9 +37,9 @@ $$\begin{aligned} = \frac{8 \pi V E^2}{h^3 c^3} \end{aligned}$$ -Where the factor of $2$ accounts for the photon's polarization degeneracy. -We thus expect that the number of photons $N(E)$ -with an energy between $E$ and $E + \dd{E}$ is given by: +Where the factor of $$2$$ accounts for the photon's polarization degeneracy. +We thus expect that the number of photons $$N(E)$$ +with an energy between $$E$$ and $$E + \dd{E}$$ is given by: $$\begin{aligned} N(E) \dd{E} @@ -47,16 +47,16 @@ $$\begin{aligned} = \frac{8 \pi V}{h^3 c^3} \frac{E^2}{\exp(\beta E) - 1} \dd{E} \end{aligned}$$ -By substituting $E = h \nu$, we find that the number of photons $N(\nu)$ -with a frequency between $\nu$ and $\nu + \dd{\nu}$ must be as follows: +By substituting $$E = h \nu$$, we find that the number of photons $$N(\nu)$$ +with a frequency between $$\nu$$ and $$\nu + \dd{\nu}$$ must be as follows: $$\begin{aligned} N(\nu) \dd{\nu} = \frac{8 \pi V}{c^3} \frac{\nu^2}{\exp(\beta h \nu) - 1} \dd{\nu} \end{aligned}$$ -Multiplying by the energy $h \nu$ yields the distribution of the radiated energy, -which we divide by the volume $V$ to get Planck's law, +Multiplying by the energy $$h \nu$$ yields the distribution of the radiated energy, +which we divide by the volume $$V$$ to get Planck's law, also called the **Plank distribution**, describing a black body's radiated spectral energy density per unit volume: @@ -70,8 +70,8 @@ $$\begin{aligned} ## Wien's displacement law -The Planck distribution peaks at a particular frequency $\nu_{\mathrm{max}}$, -which can be found by solving the following equation for $\nu$: +The Planck distribution peaks at a particular frequency $$\nu_{\mathrm{max}}$$, +which can be found by solving the following equation for $$\nu$$: $$\begin{aligned} 0 @@ -81,7 +81,7 @@ $$\begin{aligned} = 3 \nu^2 (\exp(\beta h \nu) - 1) - \nu^3 \beta h \exp(\beta h \nu) \end{aligned}$$ -By defining $x \equiv \beta h \nu_{\mathrm{max}}$, +By defining $$x \equiv \beta h \nu_{\mathrm{max}}$$, this turns into the following transcendental equation: $$\begin{aligned} @@ -98,14 +98,14 @@ $$\begin{aligned} } \end{aligned}$$ -Which states that the peak frequency $\nu_{\mathrm{max}}$ -is proportional to the temperature $T$. +Which states that the peak frequency $$\nu_{\mathrm{max}}$$ +is proportional to the temperature $$T$$. ## Stefan-Boltzmann law -Because $u(\nu)$ represents the radiated spectral energy density, -we can find the total radiated energy $U$ per unit volume by integrating over $\nu$: +Because $$u(\nu)$$ represents the radiated spectral energy density, +we can find the total radiated energy $$U$$ per unit volume by integrating over $$\nu$$: $$\begin{aligned} U @@ -116,9 +116,9 @@ $$\begin{aligned} = \frac{8 \pi}{\beta^4 h^3 c^3} \int_0^\infty \frac{x^3}{\exp(x) - 1} \dd{x} \end{aligned}$$ -This definite integral turns out to be $\pi^4/15$, +This definite integral turns out to be $$\pi^4/15$$, leading us to the **Stefan-Boltzmann law**, -which states that the radiated energy is proportional to $T^4$: +which states that the radiated energy is proportional to $$T^4$$: $$\begin{aligned} \boxed{ @@ -126,7 +126,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where $\sigma$ is the **Stefan-Boltzmann constant**, which is defined as follows: +Where $$\sigma$$ is the **Stefan-Boltzmann constant**, which is defined as follows: $$\begin{aligned} \sigma |