Electromagnetics

gauge invariance

• 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)
  

 

 

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?

 

 

 

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 \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)
• THOMAS YU LAGRANGIAN FORMULATION OF THE ELECTROMAGNETIC FIELD
• 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 \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)

 

 

 

 

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