Electromagnetics

Lorentz force

• almost all forces in mechanics are conservative forces, those that are functions nly of positions, and certainly not functions of velocities
• Lorentz force is a rare example of velocity dependent force

 

 

polarization of light

• has two possibilites
  
  • what does this mean?

 

 

notations

• vector potential \(\mathbf{A}(x,y,z,t)=(A_{x},A_{y},A_{z})\)
• electrostatic potential \(\phi(x,y,z,t)\) (scalar)
• electric field \(\mathbf{E}\)
• magnetic field \(\mathbf{B}\)
• charge density \({\rho} \) (for point charge, density will be a Dirac delta function)
• \(\mathbf{J}\)
• \(\mu_0\)
• \(\varepsilon_0\)

 

 

Maxwell's equations

• using vector calculus notation
  \(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\)
  \(\nabla \cdot \mathbf{B} = 0\)
  \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\)
  \(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \)
  

 

potentials

• vector potential \(A\)
  from \(\nabla \cdot \mathbf{B} = 0\), we can find a vector potential such that \(\mathbf{B}=\nabla \times \mathbf{A}\)
  
• scalar potential \(\phi\)
  \(\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi \)
  

 

electromagnetic field (four vector potential)

• defined as follows
  \(A_{\alpha} = \left( - \phi, \mathbf{A} \right)=(-\phi,A_{x},A_{y},A_{z})\)
  \(\phi\) is the scalar potential
  \(A\)  is the vector potential.
  
• gague field describing the photon

 

 

gauge transformation

• For any scalar field \(\Lambda(x,y,z,t)\), the following transformation does not change any physical quantity
  \(\mathbf{A} \to \mathbf{A} +\del \Lambda\)
  \(\phi\to \phi-\frac{\partial\Lambda}{\partial t}\)
  
• unchanged quantities
  \(\mathbf{B}=\nabla \times \mathbf{A}\)
  \(\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi \)
  

• the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field
  
• the electromagnetism is a gauge field theory with structure group U(1)
  

 

 

Covariant formulation

• electromagnetic field strength
  \(F_{\alpha \beta} = \partial_{\beta} A_{\alpha}-\partial_{\alpha} A_{\beta}\)
  \(F_{\alpha \beta} = \left( \begin{matrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & -B_z & B_y \\ \frac{-E_y}{c} & B_z & 0 & -B_x \\ \frac{-E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)\)
  
• \(F_{01}=-\partial_{0} A_{1} + \partial_{1} A_{0}=-\partial_{t} A_{x} -\partial_{x} \phi=E_{x}\)
  
• \(F_{12}=-\partial_{1} A_{2} + \partial_{2} A_{1}=-\partial_{x} A_{y} + \partial_{y} A_{x}=-B_{z}\)
  
• In Gauge theory, we regard F as 2-form, A as 1-form
  
• \(A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}\)
  
• \(F=F_{01}dx^{0}\wedge dx^{1}+F_{02}dx^{0}\wedge dx^{2}+\cdots\)
  

 

 

Lagrangian formulation

• Lagrangian
  \(L(q,\dot{q})=m||\dot{q}||+eA_{i}\dot{q}^{i}\)
  
• action
  \(S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x\)
  
• Euler-Lagrange equations
  \(p_{i}=\frac{\partial{L}}{\partial{\dot{q}^{i}}}=m\frac{\dot{q}_{i}}{||\dot{q}_{i}||}+eA_{i}=mv_{i}+eA_{i}\)
  \(F_{i}=\frac{\partial{L}}{\partial{{q}^{i}}}=\frac{\partial}{\partial{{q}^{i}}}(eA_{j}\dot{q}^{j})=e\frac{\partial{A_{j}}}{\partial{q}^{i}}\dot{q}^{j}}}\)
  
• equation of motion
  \(\dot{p}=F\) Therefore we get
  \(m\frac{dv_{i}}{dt}=eF_{ij}\dot{q}^{j}\). This is what we call the Lorentz force law.
  
• force on a particle is same as \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\).
   
  

• http://dexterstory.tistory.com/888
  
• path integral
  

 

 

 

 

 

 

 

charge density and current density

• this is necessary for Maxwell equations with sources
• ρ the charge density
• j the conventional current density.

 

 

conserved four-current

• charge density and current density
  

\[J^a = \left(c \rho, \mathbf{j} \right)\] where

c is the speed of light

ρ the charge density

j the conventional current density.

a labels the space-time dimensions

• four vector is called a conserved current if \(\partial_{a}J^{a}=0\)

 

 

메모

• http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf
  
• Feynman's proof of Maxwell equations and Yang's unification of electromagnetic and gravitational Aharonov–Bohm effects
  

 

 

related items

• Gauge theory
  

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Classical_electromagnetism
• http://en.wikipedia.org/wiki/Maxwell's_equations
• http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism
• http://en.wikipedia.org/wiki/electrical_current
• http://en.wikipedia.org/wiki/Four-current