Editing Plasmoid Tech (section)

= Equations and Formulas =

== Plasmoid Formation ==
Plasmoids, coherent toroidal structures of plasma, are essential for initiating and sustaining the energy release process. The equations presented in this table elucidate the fundamental principles governing plasmoid formation, shedding light on the intricate dynamics at play within the Thunderstorm Generator.

== Plasmoid Formation Equations ==
{| class="wikitable"
|+ Plasmoid Formation Equations
|-
! Equation !! Description
|-
| <math>P = \frac{{T \cdot V}}{{n \cdot R}}</math> || Ideal gas law where <math>P</math> is pressure, <math>T</math> is temperature, <math>V</math> is volume, <math>n</math> is the number of moles, and <math>R</math> is the ideal gas constant.
|-
| <math>F = q(E + v \times B)</math> || Lorentz force equation where <math>F</math> is the force, <math>q</math> is the charge, <math>E</math> is the electric field, <math>v</math> is the velocity, and <math>B</math> is the magnetic field.
|-
| <math>m = \frac{{m_0}}{{\sqrt{1 - \frac{{v^2}}{{c^2}}}}}</math> || Relativistic mass equation where <math>m</math> is the relativistic mass, <math>m_0</math> is the rest mass, <math>v</math> is the velocity, and <math>c</math> is the speed of light.
|-
| <math>E = mc^2</math> || Energy-mass equivalence equation from Einstein's theory of relativity where <math>E</math> is energy, <math>m</math> is mass, and <math>c</math> is the speed of light.
|-
| <math>v_f = v_i + at</math> || Kinematic equation for final velocity where <math>v_f</math> is the final velocity, <math>v_i</math> is the initial velocity, <math>a</math> is acceleration, and <math>t</math> is time.
|-
| <math>I = \frac{V}{R}</math> || Ohm's law where <math>I</math> is current, <math>V</math> is voltage, and <math>R</math> is resistance.
|-
| <math>F_{\text{buoyant}} = \rho \cdot g \cdot V</math> || Buoyant force equation where <math>F_{\text{buoyant}}</math> is the buoyant force, <math>\rho</math> is the density of the fluid, <math>g</math> is the acceleration due to gravity, and <math>V</math> is the volume of the displaced fluid.
|-
| <math>P_{\text{mech}} = P_{\text{hydro}} + P_{\text{static}} + P_{\text{dynamic}}</math> || Mechanical power equation where <math>P_{\text{mech}}</math> is the mechanical power, <math>P_{\text{hydro}}</math> is the hydrostatic pressure, <math>P_{\text{static}}</math> is the static pressure, and <math>P_{\text{dynamic}}</math> is the dynamic pressure.
|}

== Ideal Gas Law ==
The ideal gas law, given by the equation:
<math>P = \frac{{nRT}}{{V}}</math>
describes the behavior of gases under various conditions of pressure, volume, and temperature. 

===== Alternative formulations include: =====
* Van der Waals equation: <math>(P + \frac{{n^2a}}{{V^2}})(V - nb) = nRT</math>
* Combined gas law: <math>\frac{{P_1V_1}}{{T_1}} = \frac{{P_2V_2}}{{T_2}}</math>
Related formulas in the same application context include:
* Boyle's law: <math>P_1V_1 = P_2V_2</math>
* Gay-Lussac's law: <math>\frac{{P_1}}{{T_1}} = \frac{{P_2}}{{T_2}}</math>

===== This equation is fundamental in understanding the properties of gases and their interactions in real-world applications such as: =====
* Gas turbine engines
* Air conditioning systems
* Weather forecasting models

== Lorentz Force Equation ==
The Lorentz force equation, expressed as:
<math>F = q(E + v \times B)</math>
is essential in describing the electromagnetic force experienced by charged particles moving through electric and magnetic fields. 

===== Alternative formulations include: =====
* Magnetic force on a current-carrying wire: <math>F = IL \times B</math>
* Force on a charged particle in an electric field: <math>F = qE</math>
Related formulas in the same application context include:
* Ampère's law: <math>\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}</math>
* Lorentz transformation equations: <math>x' = \gamma(x - vt)</math>, <math>t' = \gamma(t - vx/c^2)</math>

===== This equation finds applications in: =====
* Particle accelerators
* Plasma physics experiments
* Magnetic confinement fusion research

== Relativistic Mass Equation ==
The relativistic mass equation, given by:
<math>m = \frac{{m_0}}{{\sqrt{1 - \frac{{v^2}}{{c^2}}}}}</math>
relates the relativistic mass of an object to its rest mass and velocity. 

==== Alternative formulations include: ====
* Energy-momentum relation: <math>E^2 = (pc)^2 + (mc^2)^2</math>
* Lorentz factor: <math>\gamma = \frac{{1}}{{\sqrt{1 - \frac{{v^2}}{{c^2}}}}}</math>
Related formulas in the same application context include:
* Time dilation equation: <math>t' = \frac{{t}}{{\sqrt{1 - \frac{{v^2}}{{c^2}}}}}</math>
* Length contraction equation: <math>L' = L\sqrt{1 - \frac{{v^2}}{{c^2}}}</math>

===== This equation has implications in: =====
* High-energy particle physics
* Astrophysics and cosmology
* Particle collider experiments

== Energy-Mass Equivalence Equation ==
The energy-mass equivalence equation, represented as:
<math>E = mc^2</math>
demonstrates the equivalence between mass and energy, as predicted by Einstein's theory of relativity. 

==== Alternative formulations include: ====
* Mass-energy-momentum relation: <math>E^2 = (pc)^2 + (mc^2)^2</math>
* Einstein's mass-energy equation: <math>\Delta E = \Delta m c^2</math>
Related formulas in the same application context include:
* Photon energy equation: <math>E = hf</math>
* De Broglie wavelength equation: <math>\lambda = \frac{{h}}{{p}}</math>

===== This equation is utilized in: =====
* Nuclear energy generation
* Particle physics research
* Astrophysical phenomena like black holes and supernovae

== Kinematic Equation for Final Velocity ==
The kinematic equation for final velocity, expressed as:
<math>v_f = v_i + at</math>
relates the final velocity of an object to its initial velocity, acceleration, and time. 

==== Alternative formulations include: ====
* Kinematic equation for displacement: <math>d = v_i t + \frac{{1}}{{2}} a t^2</math>
* Kinematic equation for average velocity: <math>v_{\text{avg}} = \frac{{v_i + v_f}}{{2}}</math>
Related formulas in the same application context include:
* Newton's second law: <math>F = ma</math>
* Kinetic energy equation: <math>KE = \frac{{1}}{{2}} mv^2</math>

===== This equation is applicable in various scenarios including: =====
* Projectile motion calculations
* Vehicle dynamics and braking systems
* Spacecraft maneuvering and orbital mechanics

== Ohm's Law ==
Ohm's law, defined by the equation:
<math>V = IR</math>
relates the voltage across a conductor to the current flowing through it and its resistance. 

==== Alternative formulations include: ====
* Conductance equation: <math>G = \frac{{1}}{{R}}</math>
* Current density equation: <math>J = \sigma E</math>
Related formulas in the same application context include:
* Power equation: <math>P = IV</math>
* Kirchhoff's voltage law: <math>\sum V_{\text{loop}} = 0</math>

===== This equation is foundational in: =====
* Electrical circuit analysis and design
* Electronic device operation
* Power distribution systems

== Buoyant Force Equation ==
The buoyant force equation, given by:
<math>F_{\text{buoyant}} = \rho \cdot g \cdot V</math>
describes the upward force exerted on an object submerged in a fluid. 

==== Alternative formulations include: ====
* Archimedes' principle: <math>F_{\text{buoyant}} = \text{weight of fluid displaced}</math>
* Hydrostatic pressure equation: <math>P = \rho g h</math>
Related formulas in the same application context include:
* Pascal's law: <math>P_{\text{fluid}} = P_{\text{ext}}</math>
* Continuity equation: <math>A_1v_1 = A_2v_2</math>

===== This equation finds application in: =====
* Ship and submarine design
* Hot air balloon flight
* Hydrodynamic simulations and modeling

== Mechanical Power Equation ==
The mechanical power equation, represented as:
<math>P_{\text{mech}} = P_{\text{hydro}} + P_{\text{static}} + P_{\text{dynamic}}</math>
describes the total mechanical power in a fluid system, comprising hydrostatic, static, and dynamic components. 

==== Alternative formulations include: ====
* Pump power equation: <math>P_{\text{pump}} = \rho gQH</math>
* Turbine power equation: <math>P_{\text{turbine}} = \dot{m} \Delta h</math>
Related formulas in the same application context include:
* Bernoulli's equation: <math>\frac{{\rho v^2}}{{2}} + \rho gh + P = \text{constant}</math>
* Reynolds number equation: <math}\

===== This equation is useful in: =====
* Fluid mechanics and hydraulics
* Pump and turbine design
* HVAC systems and fluid flow control