List of Physics Formulas Latex Code: Electricity & Magnetism (Graduate Level Physics)

physics_master 2022-12-24 #physics #electricity #magnetism

List of Physics Formulas Latex Code: Electricity & Magnetism (Graduate Level Physics)

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In this blog, we will introduce most popuplar physics formulas in Electricity & Magnetism. This blog covers topics, including the Maxwell equations, force and potential, Gauge Transformations, Energy of the Electromagnetic Field, etc.

• 1. Electricity & Magnetism
• 1.1 The Maxwell Equations-Integral Form
• 1.2 The Maxwell Equations-Differential Form
• 1.3 Force and potential
• 1.4 Gauge Transformations
• 1.5 Energy of the Electromagnetic Field

1. Electricity & Magnetism

• 1.1 The Maxwell Equations-Integral

  Equation

  
  
  

  Latex Code

              \oiint (\vec{D}\cdot \vec{n}) \mathrm{d}^{2}A=Q_{\text{free,included}}\\ \oiint (\vec{B}\cdot \vec{n}) \mathrm{d}^{2}A=0 \\ \oint \vec{E} \mathrm{d}\vec{s}=-\frac{\mathrm{d}\Phi}{\mathrm{d}t}\\ \oint \vec{H} \mathrm{d}\vec{s}=I_{\text{free,included}}+\frac{\mathrm{d}\Psi }{\mathrm{d}t}
          

  Explanation

  Latex code for integral form of the Maxwell Equations. I will briefly introduce the notations in this formulation.

  • : The electric displacement
  • : The electric field strength
  • : The magnetic flux density
  • : The magnetic field strength
  In the formulation, the first formula (1) describes the property of electric displacement . The second formula (2) describes the property of magnetic flux density . The third formula (3) describes how the variation in magnetic flux density influence the electric field strength . The fourth formula (4) describes how the variation in electric displacement influence the magnetic field strength . See this document for more details:

• 1.2 The Maxwell Equations-Differential Form

  Equation

  
  
  

  Latex Code

              \nabla \cdot \vec{D}=\rho_{free} \\
              \nabla \cdot \vec{B}=0 \\
              \nabla \times \vec{E}=-\frac{\partial{\vec{B}}}{\partial{t}} \\
              \nabla \times \vec{H}=\vec{J}_{free}+\frac{\partial{\vec{D}}}{\partial{t}}
          

  Explanation

  Latex code for integral form of the Maxwell Equations. I will briefly introduce the notations in this formulation.

  • : The electric displacement
  • : The electric field strength
  • : The magnetic flux density
  • : The magnetic field strength

• 1.3 Force and potential

  Equation

  
  
  

  Latex Code

              \vec{F}_{12}=\frac{Q_{1}Q{2}}{4\pi\epsilon_{0}\epsilon_{r}r^{2}}\vec{e_{r}} \\ \vec{E}=\frac{\vec{F}}{Q}
          

  Explanation

  The force and the electric field between 2 point charges are given by above equations. This formulation is a transformation of original Coulomb force:

• 1.4 Gauge Transformationsl

  Equation

  
  
  

  Latex Code

              \vec{A}^{'}=\vec{A}-\nabla f \\ V{'}=V+\frac{\partial{f}}{\partial{t}}
          

  Explanation

  The potentials of the electromagnetic fields transform as follows when a gauge transformation is applied. And \vec{E} and \vec{B} do not change.

• 1.5 Energy of the Electromagnetic Field

  Equation

  
  
  

  Latex Code

              \frac{\mathrm{d} W}{\mathrm{d} \text{Vol}}=\omega=\int H \mathrm{d}B + \int E \mathrm{d}D
          

  Explanation

  I will briefly introduce the notations in this formulation of the energy density of the electromagnetic field.

  • : The electric displacement
  • : The electric field strength
  • : The magnetic flux density
  • : The magnetic field strength
  
  

  Latex Code

              \omega_{mag}=\frac{1}{2}\int \vec{J} \cdot \vec{A} \mathrm{d}^{3} x \\ \omega_{elec}=\frac{1}{2}\int \rho V \mathrm{d}^{3} x
          

  Explanation

  The energy density can be expressed in the potentials and currents as follows.

  • : The electric displacement
  • : The electric field strength
  • : The magnetic flux density
  • : The magnetic field strength