Electromagnetics

Lorentz force

• almost all forces in mechanics are conservative forces, those that are functions only 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

• charge density \({\rho} \) (for point charge, density will be a Dirac delta function)

 

electromagnetic field (four vector potential)

• 포벡터 포텐셜과 맥스웰 방정식
• in covariant formulation, this is a 1-form
  \(A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}\)
   
  

 

conserved four-current

• this is necessary for Maxwell equations with sources
• describes the distribution and motion of charged particles
• charge density \({\rho} \) (for point charge, density will be a Dirac delta function)
• current density \(\mathbf{J}=(J_x,J_y,J_z)\)
• charge density and current density
  \(J^a = \left(c \rho, \mathbf{J} \right)\)
  
• four vector is called a conserved current if \(\partial_{a}J^{a}=0\)
  
• in covariant formulation, this is a 3-form
  \(J=\rho dx\wedge dy \wedge dz - J_{z}dx\wedge dy \wedge dt -J_{x}dy\wedge dz\wedge dt-J_{y}dz\wedge dx\wedge dt\)
  

 

 

covariant formulation using differential form

• electromagnetic field strength
  \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!\)
  
• In Gauge theory, we regard F as 2-form, A as 1-form
  
• \(A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}\)
  
• \(F=F_{01}dx^{0}\wedge dx^{1}+F_{02}dx^{0}\wedge dx^{2}+\cdots\)
  
• \(J=(-\rho,J_1,J_2,J_3)\)
  
• Maxwell's equation can be recast into
  
  • \(dF=0\) (\(\nabla \cdot \mathbf{B} = 0\), \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\))
    
  • \(d*F=J\) (\(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\),  \(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \))
     
    
• broken Maxwell's Equations in Terms of Dierential Forms
  
• Maxwell Theory and Differential Forms
  
  • http://www22.pair.com/csdc/
    
  • Maxwell Faraday and Maxwell Ampere Equations
    
• http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.7828&rep=rep1&type=pdf
• Geometrical Concepts in Teaching Electromagnetics

 

 

 

Lagrangian formulation

• Lagrangian for a charged particle in an electromagnetic field
  \(L=T-V\)
  \(L(q,\dot{q})=m||\dot{q}||-e\phi+eA_{i}\dot{q}^{i}\)
  
• 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
  

 

 

Hamiltonian formulation

• total energy of a charge particle in an electromagnetic field
  \(E=\frac{1}{2m}(p_j-eA_{j})(p_j-eA_j)+q\phi\)
  
• replace the momentum with the canonical momentum
  
  • similar to covariant derivative
    

 

 

force on a particle

• force on a particle is same as \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)

 

 

 

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)
  

 

 

 

 

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
  
• QED
  

 

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

 

 

books

ELECTROMAGNETIC THEORY AND COMPUTATION