"Electromagnetics"의 두 판 사이의 차이
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42번째 줄: | 42번째 줄: | ||
* vector potential <math>A</math><br> from <math>\nabla \cdot \mathbf{B} = 0</math>, we can find a vector potential such that <math>\mathbf{B}=\nabla \times \mathbf{A}</math><br> | * vector potential <math>A</math><br> from <math>\nabla \cdot \mathbf{B} = 0</math>, we can find a vector potential such that <math>\mathbf{B}=\nabla \times \mathbf{A}</math><br> | ||
* scalar potential <math>\phi</math><br><math>\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br> | * scalar potential <math>\phi</math><br><math>\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t} - \nabla \phi </math><br> | ||
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81번째 줄: | 79번째 줄: | ||
<h5 style="margin: 0px; line-height: 2em;">force on a particle</h5> | <h5 style="margin: 0px; line-height: 2em;">force on a particle</h5> | ||
− | + | <math>\mathbf{F}=q\mathbf{E}+q\mathbf{v}\times \mathbf{B}</math> | |
2010년 9월 21일 (화) 15:26 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 \(A\)
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
- \(A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}\)
- \(F=F_{01}dx^{0}\wedge dx^{1}+F_{02}dx^{0}\wedge dx^{2}+\cdots\)
force on a particle
\(\mathbf{F}=q\mathbf{E}+q\mathbf{v}\times \mathbf{B}\)
Lagrangian formulation
action
\(S=-\frac{1}{4}\int F^{\alpha\beta}F_{\alpha\beta}\,d^{4}x\)
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
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- 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
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