Editing MHD Core (section)

== Theoretical Foundations ==

=== Quantum Field Theorist's Equations ===

'''Zero-Point Energy of a Quantum Harmonic Oscillator:'''

<math>E = \frac{1}{2} \hbar \omega</math>

* ''E'': Zero-point energy
* ''<math>\hbar</math>'': Reduced Planck's constant
* ''<math>\omega</math>'': Angular frequency

'''Casimir Effect Force Between Two Plates:'''

<math>F_{\text{Casimir}} = \frac{\pi^2 \hbar c}{240} \frac{A}{L^4}</math>

* ''F''<sub>Casimir</sub>: Casimir force
* ''c'': Speed of light
* ''A'': Area of the plates
* ''L'': Separation between the plates

'''Dynamic Casimir Effect Photon Generation Rate:'''

<math>\Gamma = \frac{\pi \omega_{\text{cavity}}^2}{3c^2} \left( \frac{\Delta L}{L} \right)^2</math>

* ''<math>\Gamma</math>'': Photon generation rate
* ''<math>\omega</math>''<sub>cavity</sub>: Resonant frequency of the cavity
* ''<math>\Delta L</math>'': Modulation amplitude of cavity length
* ''<math>L</math>'': Original cavity length

'''Expectation Value of the Energy-Momentum Tensor:'''

<math>\langle T_{\mu\nu} \rangle = -\frac{\hbar c}{720 \pi^2} \frac{1}{L^4} g_{\mu\nu}</math>

* ''T''<sub>μν</sub>: Energy-momentum tensor
* ''g''<sub>μν</sub>: Metric tensor of spacetime

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=== Electromagnetic Field Specialist's Equations ===

'''Modified Wave Equation with Scalar Potential:'''

<math>\nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\epsilon_0}</math>

* ''<math>\phi</math>'': Scalar potential
* ''<math>\rho</math>'': Charge density
* ''<math>\epsilon_0</math>'': Vacuum permittivity

'''Magnetic Flux Quantum:'''

<math>\Phi_0 = \frac{h}{2e}</math>

* ''<math>\Phi</math>''<sub>0</sub>: Magnetic flux quantum
* ''h'': Planck's constant
* ''e'': Elementary charge

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