"Electromagnetics"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
| 53번째 줄: | 53번째 줄: | ||
* 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> | * 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> | ||
* gague field describing the photon | * gague field describing the photon | ||
| + | |||
| + | |||
| 68번째 줄: | 70번째 줄: | ||
* this is necessary for Maxwell equations with sources | * this is necessary for Maxwell equations with sources | ||
| − | * describes the distribution and | + | * describes the distribution and motion of charged particles |
* charge density <math>{\rho} </math> (for point charge, density will be a Dirac delta function) | * charge density <math>{\rho} </math> (for point charge, density will be a Dirac delta function) | ||
* current density <math>\mathbf{J}=(J_x,J_y,J_z)</math> | * current density <math>\mathbf{J}=(J_x,J_y,J_z)</math> | ||
2012년 1월 15일 (일) 12:29 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 \(\mathbf{J}=(J_x,J_y,J_z)\)
- \(\mu_0\)
- \(\varepsilon_0\)
- \(c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}\)
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. - in covariant formulation, this is a 1-form
\(A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}\) - gague field describing the photon
electromagnetic field strength
- in covariant formulation, this is a 2-form
- \(F=F_{01}dx^{0}\wedge dx^{1}+F_{02}dx^{0}\wedge dx^{2}+\cdots\)
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
\(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!\)
\(F_{\alpha \beta} = \left( \begin{matrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & -B_z & B_y \\ \frac{-E_y}{c} & B_z & 0 & -B_x \\ \frac{-E_z}{c} & -B_y & B_x & 0 \end{matrix} \right)\) - \(F_{01}=\partial_{0} A_{1} - \partial_{1} A_{0}=\partial_{t} A_{x} +\partial_{x} \phi=E_{x}\)
- \(F_{12}=-\partial_{1} A_{2} + \partial_{2} A_{1}=-\partial_{x} A_{y} + \partial_{y} A_{x}=-B_{z}\)
- In Gauge theory, we regard F as 2-form, A as 1-form
- \(A=A_{0}dx^{0}+A_{1}dx^{1}+A_{2}dx^{2}+A_{3}dx^{3}\)
- \(F=F_{01}dx^{0}\wedge dx^{1}+F_{02}dx^{0}\wedge dx^{2}+\cdots\)
- \(J=(-\rho,J_1,J_2,J_3)\)
- Maxwell's equation can be recast into
- \(dF=0\) (\(\nabla \cdot \mathbf{B} = 0\), \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\))
- \(d*F=J\) (\(\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}\), \(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ \))
- \(dF=0\) (\(\nabla \cdot \mathbf{B} = 0\), \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\))
- See Introduction to differential forms
- Maxwell's Equations in Terms of Dierential Forms
- Maxwell Theory and Differential Forms
- http://www.math.sunysb.edu/~brweber/401s09/coursefiles/ElectromagneticNotes.pdf
- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.7828&rep=rep1&type=pdf
- https://www.nottingham.ac.uk/ggiemr/downloads/GCEM.pdf
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
- 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
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