"Electromagnetics"의 두 판 사이의 차이

2012년 6월 12일 (화) 10:36 판

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

vector potential \(\mathbf{A}(x,y,z,t)=(A_{x},A_{y},A_{z})\)
electrostatic potential \(\phi(x,y,z,t)\) (scalar)
electric field \(\mathbf{E}(x,y,z,t)\)
magnetic field \(\mathbf{B}(x,y,z,t)\)
charge density \({\rho} \) (for point charge, density will be a Dirac delta function)
current density \(\mathbf{J}=(J_x,J_y,J_z)\)
\(\mu_0\)
\(\varepsilon_0\)
\(c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}\)

맥스웰 방정식

전기장에 대한 가우스의 법칙
\(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\)
자기장에 대한 가우스의 법칙
\(\nabla \cdot \mathbf{B} = 0\)
패러데이의 법칙

\(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\)

앙페르-패러데이 법칙

\(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \)

electromagnetic field (four vector potential)

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

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
- \(dF=0\) (\(\nabla \cdot \mathbf{B} = 0\), \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\))
- \(d*F=J\) (\(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\), \(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \))
See Introduction to differential forms

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 \(e\mathbf{E}+e\mathbf{v}\times \mathbf{B}\)

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

encyclopedia

books

ELECTROMAGNETIC THEORY AND COMPUTATION

@@ 49번째 줄: / 49번째 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>
-*   <br>
 * [http://pythagoras0.springnote.com/pages/12166408 포벡터 포텐셜과 맥스웰 방정식]
 *  in covariant formulation, this is a '''1-form'''<br><math>A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}</math><br>
@@ 55번째 줄: / 54번째 줄: @@
-* <math>F=F_{01}dx^{0}\wedge dx^{1}+F_{02}dx^{0}\wedge dx^{2}+\cdots</math><br>