"Electromagnetics"의 두 판 사이의 차이

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51번째 줄: 51번째 줄:
 
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">electromagnetic field (four vector potential)</h5>
 
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">electromagnetic field (four vector potential)</h5>
  
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*  defined as follows<br><math>A_{\alpha} = \left( - \phi/c, \mathbf{A} \right)=(\phi,A_{x},A_{y},A_{z})</math><br><math>\phi</math> is the scalar potential<br><math>A</math>  is the vector potential.<br>
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*  defined as follows<br><math>A_{\alpha} = \left( - \phi, \mathbf{A} \right)=(-\phi,A_{x},A_{y},A_{z})</math><br><math>\phi</math> is the scalar potential<br><math>A</math>  is the vector potential.<br>
 
* gague field describing the photon
 
* gague field describing the photon
  
61번째 줄: 61번째 줄:
  
 
*  For any scalar field <math>\Lambda(x,y,z,t)</math>, the following transformation does not change any physical quantity<br><math>\mathbf{A} \to \mathbf{A} +\del \Lambda</math><br><math>\phi\to \phi-\frac{\partial\Lambda}{\partial t}</math><br>
 
*  For any scalar field <math>\Lambda(x,y,z,t)</math>, the following transformation does not change any physical quantity<br><math>\mathbf{A} \to \mathbf{A} +\del \Lambda</math><br><math>\phi\to \phi-\frac{\partial\Lambda}{\partial t}</math><br>
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*  
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* unchanged quantities<br><math>\mathbf{B}=\nabla \times \mathbf{A}</math><br><math>\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br>
  
 
*  the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field<br>
 
*  the electromagnetic potential is a connection on a U(1)-bundle on spacetime whose curvature is the electromagnetic field<br>
73번째 줄: 73번째 줄:
  
 
*  electromagnetic field strength<br><math>F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}</math><br><math>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)</math><br>
 
*  electromagnetic field strength<br><math>F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}</math><br><math>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)</math><br>
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* <math>F_{01}</math><br>
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*   <br>
  
 
 
 
 

2010년 5월 13일 (목) 21:09 판

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 \(\mathbf{B}\)
  • \({\rho} \)
  • \(\mathbf{J}\)
  • \(\mu_0\)
  • \(\varepsilon_0\)

 

 

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

 

 

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
    \(F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}\)
    \(F_{\alpha \beta} = \left( \begin{matrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & -B_z & B_y \\ \frac{-E_y}{c} & B_z & 0 & -B_x \\ \frac{-E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)\)
  • \(F_{01}\)
  •  

 

 

 

charge density and current density

 

 

 

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

 

 

메모

 

 

encyclopedia
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