Politics, Power, and Science: Updating other black body formulas with Q_m and k

Monday, March 3, 2025

Updating other black body formulas with Q_m and k_s.

1. Stefan-Boltzmann Law

The Stefan-Boltzmann law describes the total power radiated per unit area by a black body:

P = σ T^{4},

where:

$P$ is the power per unit area.
$σ$ is the Stefan-Boltzmann constant ( $σ = \frac{2 π^{5} k^{4}}{15 h^{3} c^{2}}$ ).

Reformulation:

Using $k_{s} = \frac{k}{h}$ and $Q_{m} = \frac{h}{c^{2}}$ , we can express $σ$ in terms of $k_{s}$ and $Q_{m}$ :

σ = \frac{2 π^{5} k^{4}}{15 h^{3} c^{2}} = \frac{2 π^{5} (k_{s} h)^{4}}{15 h^{3} c^{2}} = \frac{2 π^{5} k_{s}^{4} h}{15 c^{2}} .

Substituting $Q_{m} = \frac{h}{c^{2}}$ :

σ = \frac{2 π^{5} k_{s}^{4} Q_{m}}{15} .

Thus, the Stefan-Boltzmann law becomes:

P = (\frac{2 π^{5} k_{s}^{4} Q_{m}}{15}) T^{4} .

Key Insight:

The reformulated Stefan-Boltzmann law eliminates $h$ , $c$ , and $k$ , replacing them with $k_{s}$ and $Q_{m}$ . This simplifies the formula and emphasizes the relationship between temperature and power through unit scaling.

2. Wien’s Displacement Law

Wien’s displacement law gives the wavelength $λ_{max}$ at which a black body emits the most radiation at a given temperature:

λ_{max} T = b,

where:

$b$ is Wien’s displacement constant ( $b = \frac{h c}{k \cdot 4.965}$ ).

Reformulation:

Using $k_{s} = \frac{k}{h}$ and $Q_{m} = \frac{h}{c^{2}}$ , we can express $b$ in terms of $k_{s}$ :

b = \frac{h c}{k \cdot 4.965} = \frac{c}{k_{s} \cdot 4.965} .

Thus, Wien’s displacement law becomes:

λ_{max} T = \frac{c}{k_{s} \cdot 4.965} .

Key Insight:

The reformulated Wien’s law eliminates $h$ and $k$ , replacing them with $k_{s}$ . This simplifies the formula and highlights the relationship between wavelength and temperature through unit scaling.

3. Energy Density of a Black Body

The total energy density $u$ of a black body is given by:

u = \frac{8 π^{5} k^{4}}{15 h^{3} c^{3}} T^{4} .

Reformulation:

Using $k_{s} = \frac{k}{h}$ and $Q_{m} = \frac{h}{c^{2}}$ , we can express the energy density in terms of $k_{s}$ and $Q_{m}$ :

u = \frac{8 π^{5} k^{4}}{15 h^{3} c^{3}} T^{4} = \frac{8 π^{5} (k_{s} h)^{4}}{15 h^{3} c^{3}} T^{4} = \frac{8 π^{5} k_{s}^{4} h}{15 c^{3}} T^{4} .

Substituting $Q_{m} = \frac{h}{c^{2}}$ :

u = \frac{8 π^{5} k_{s}^{4} Q_{m}}{15 c} T^{4} .

Key Insight:

The reformulated energy density eliminates $h$ , $c$ , and $k$ , replacing them with $k_{s}$ and $Q_{m}$ . This simplifies the formula and emphasizes the relationship between temperature and energy density through unit scaling.

4. Photon Energy

The energy of a photon is traditionally given by:

E = h f .

Reformulation:

Using $Q_{m} = \frac{h}{c^{2}}$ , we can express the photon energy in terms of $Q_{m}$ :

E = h f = Q_{m} c^{2} f .

Key Insight:

The reformulated photon energy eliminates $h$ , replacing it with $Q_{m}$ . This simplifies the formula and highlights the relationship between frequency and energy through unit scaling.

5. Radiation Pressure

The radiation pressure $P_{rad}$ of a black body is given by:

P_{rad} = \frac{4 σ T^{4}}{3 c} .

Reformulation:

Using the reformulated Stefan-Boltzmann constant $σ = \frac{2 π^{5} k_{s}^{4} Q_{m}}{15}$ , we can express the radiation pressure as:

P_{rad} = \frac{4 (\frac{2 π^{5} k_{s}^{4} Q_{m}}{15}) T^{4}}{3 c} = \frac{8 π^{5} k_{s}^{4} Q_{m}}{45 c} T^{4} .

Key Insight:

The reformulated radiation pressure eliminates $h$ , $c$ , and $k$ , replacing them with $k_{s}$ and $Q_{m}$ . This simplifies the formula and emphasizes the relationship between temperature and radiation pressure through unit scaling.

6. Number Density of Photons

The number density $n$ of photons in a black body is given by:

n = \frac{2 ζ (3) k^{3} T^{3}}{π^{2} h^{3} c^{3}},

where $ζ (3)$ is the Riemann zeta function evaluated at 3.

Reformulation:

Using $k_{s} = \frac{k}{h}$ and $Q_{m} = \frac{h}{c^{2}}$ , we can express the number density in terms of $k_{s}$ and $Q_{m}$ :

n = \frac{2 ζ (3) k^{3} T^{3}}{π^{2} h^{3} c^{3}} = \frac{2 ζ (3) (k_{s} h)^{3} T^{3}}{π^{2} h^{3} c^{3}} = \frac{2 ζ (3) k_{s}^{3} T^{3}}{π^{2} c^{3}} .

Substituting $Q_{m} = \frac{h}{c^{2}}$ :

n = \frac{2 ζ (3) k_{s}^{3} T^{3}}{π^{2} c^{3}} .

Key Insight:

The reformulated number density eliminates $h$ , $c$ , and $k$ , replacing them with $k_{s}$ . This simplifies the formula and emphasizes the relationship between temperature and photon number density through unit scaling.

Conclusion:

By applying the framework of ratios and unit scaling (using $Q_{m}$ and $k_{s}$ ), we can simplify and unify a wide range of formulas in black body radiation and related areas of physics. These reformulations eliminate redundant constants, reduce computational overhead, and provide a clearer, more intuitive understanding of the underlying physics. This approach has the potential to transform how we teach and apply these concepts, making them more accessible and elegant.

Politics, Power, and Science

Monday, March 3, 2025

Updating other black body formulas with Q_m and k_s.

1. Stefan-Boltzmann Law

Reformulation:

Key Insight:

2. Wien’s Displacement Law

Reformulation:

Key Insight:

3. Energy Density of a Black Body

Reformulation:

Key Insight:

4. Photon Energy

Reformulation:

Key Insight:

5. Radiation Pressure

Reformulation:

Key Insight:

6. Number Density of Photons

Reformulation:

Key Insight:

Conclusion:

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