## 1 Gaussian Units


### 1.1 SI Units


**Definition 1.1.1 **  The **SI unit system** is the most popular system of units that uses the fundamental units of [seconds](https://kaedon.net/l/^ahc1#h5t8), [meters](https://kaedon.net/l/^ahc1#8dza), [kilograms](https://kaedon.net/l/^ahc1#wahe), [ampere](https://kaedon.net/l/^ahc1#613r) and [Kelvin](https://kaedon.net/l/^ahc1#ncp0) to derive a system of units to describe the universe.




**Definition 1.1.2 **  A **second** (**s**) is the SI unit of time that is exactly 9192631770 hyperfine transitions of a Caesium-133 atom.

**Definition 1.1.3 **  A **meter** (**m**) is the SI unit of distance that is exactly the distance light travels in $1/299792458$ seconds.

**Definition 1.1.4 **  A **kilogram** (**kg**) is the SI unit of mass defined exactly by fixing Plank's constant $h=6.62607015 \times 10^{−34}\text{kg } \text{m}^2\text{s}^{−1}$.

**Definition 1.1.5 **  An **ampere** (**A**) is the SI unit of current that is exactly the flow of $10^{19}/1.602176634$ protons per second.

**Definition 1.1.6 **  A **Kelvin** (**K**) is the SI unit of absolute temperature defined exactly by fixing Boltzmann's constant $k=1.380649\times 10^{-23}\text{kg }\text{m}^2 \text{s}^{-2}\text{K}^{-1}$.

**Definition 1.1.7 **  A **Coulomb** (**C**) is the SI unit of charge defined by $\text{C} = \text{A s}$ or exactly $10^{19}/1.602176634$ protons.

**Definition 1.1.8 **  A **Newton** (**N**) is the SI unit of force defined by $\text{N} = \text{kg m}/\text{s}^{2}$.

**Definition 1.1.9 **  A **Joule** (**J**) is the SI unit of energy defined by $\text{J} = \text{N m} = \text{kg }\text{m}^{2}/\text{s}^{2}$.

**Definition 1.1.10 **  A **Watt** (**w**) is the SI unit of power defined by $\text{w} = \text{J}/\text{s}$.

**Definition 1.1.11 **  A **Pascal** (**Pa**) is the SI unit of pressure defined by $\text{Pa}=\text{N}/\text{m}^2 = \text{J}/\text{m}^3 = \text{kg }\text{m}^{-1}\text{s}^{-2}$.

**Definition 1.1.12 **  A **Volt** (**V**) is the SI unit of electric potential defined by $\text{V} = \text{J}/\text{C} = \text{w}/\text{A} = \text{kg }\text{m}^2\text{s}^{-3}\text{A}^{-1}$.

**Definition 1.1.13 **  A **Volt per Meter** (**V/m**) is the SI unit of electric field defined by $\text{V}/\text{m} = \text{N}/\text{C} = \text{kg m}\text{s}^{-2}\text{A}^{-1}$.

**Definition 1.1.14 **  A **Telsa** (**T**) is the SI unit of magnetic field defined by $\text{T} = \text{V s}/\text{m}^2 = \text{kg }\text{s}^{-2}\text{A}^{-1}$.

**Definition 1.1.15 **&nbsp; The **fine structure constant** denoted $\alpha$ is a dimensionless experimentally determined constant defined below. In any system of units, the fine structure constant is dimensionless and therefore has the same value<sup class="footnote-ref"><a href="#fn-32ca" id="fnref-32ca" data-footnote-ref="">1</a></sup>:
\[\alpha = \frac{\mu_0 e^2 c}{2 h} = \frac{e^2}{2\varepsilon_0 h c} \approx 0.0072973525643 \approx 1/137.035999177\]

**Law 1.1.16 **&nbsp; **Maxwell's Equations** are a set of coupled differential equations that form the foundations of classical electromagnetism.
\[\nabla\cdot \vec{E} = \frac{\rho}{\varepsilon_0}\]
\[\nabla\cdot \vec{B} = 0\]
\[\nabla\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}\]
\[\nabla\times \vec{B} = \mu_0\left( \vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t} \right)\]

**Definition 1.1.17 **&nbsp; The **vacuum permittivity** $\varepsilon_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\varepsilon_0 = \frac{e^2}{2\alpha hc}\]

**Definition 1.1.18 **&nbsp; The **vacuum permeability** $\mu_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\mu_0 = \frac{2\alpha h}{e^2c}\]

**Result 1.1.19 **&nbsp; The **product of vacuum permittivity and vacuum permeability** is the reciprocal of the speed of light squared.
\[\varepsilon_0\mu_0 = \frac{1}{c^2}\]

**Definition 1.2 **&nbsp; The **Gaussian unit system** is an alternate systems of units that uses the fundamental units of [centimeters](https://kaedon.net/l/^ahc1#8dza), [grams](https://kaedon.net/l/^ahc1#wahe), [seconds](https://kaedon.net/l/^ahc1#h5t8) and [Kelvin](https://kaedon.net/l/^ahc1#ncp0) to derive a system of units to describe the universe.

**Definition 1.3 **&nbsp; A **second** (**s**) is the SI unit of time that is exactly 9192631770 hyperfine transitions of a Caesium-133 atom.

**Definition 1.4 **&nbsp; A **centimeter** (**cm**)  is the Gaussian unit of distance that is exactly the distance light travels in $1/29979245800$ seconds.

**Definition 1.5 **&nbsp; A **gram** (*g*) is the Gaussian unit of mass defined exactly by fixing Plank's constant $h=6.62607015 \times 10^{−27}\text{g } \text{cm}^2\text{s}^{−1}$.

**Definition 1.6 **&nbsp; A **statcoulomb** (**statC**) is the Gaussian unit of charge defined by $\text{statC} = \text{g}^{1/2}\text{cm}^{1/2}/\text{s}$ or exactly the amount of charge that results in one $\text{dyn}$ of repulsion between two particles of that charge separated by one $\text{cm}$.

**Corollary 1.7 **&nbsp; The **vacuum permitivity in Gaussian units** denoted $\varepsilon_0$ is unitless and $\varepsilon_0 = \frac{1}{4\pi}$.

**Corollary 1.8 **&nbsp; The **vacuum permeability in Gaussian units** denoted $\mu_0$ is unitless and $\mu_0=4\pi/c^2$.

**Definition 1.9 **&nbsp; The **fine structure constant** denoted $\alpha$ is a dimensionless experimentally determined constant defined below. In any system of units, the fine structure constant is dimensionless and therefore has the same value<sup class="footnote-ref"><a href="#fn-32ca" id="fnref-32ca" data-footnote-ref="">1</a></sup>:
\[\alpha = \frac{\mu_0 e^2 c}{2 h} = \frac{e^2}{2\varepsilon_0 h c} \approx 0.0072973525643 \approx 1/137.035999177\]

**Result 1.10 **&nbsp; **Maxwell's equations in Gaussian Units**  are a set of coupled differential equations that form the foundations of classical electromagnetism.
\[\nabla\cdot\vec{E} = 4\pi \rho\]
\[\nabla\cdot\vec{B} = 0\]
\[\nabla\times\vec{E} = -\frac{1}{c} \frac{\partial \vec{B}}{\partial t}\]
\[\nabla\times\vec{B} = \frac{1}{c}\frac{\partial \vec{E}}{\partial t} + \frac{4\pi}{c}\vec{J}\]