Gauge Transformation, Lorenz Gauge, Coulomb Gauge

2y ago
33 Views
3 Downloads
291.07 KB
19 Pages
Last View : 2d ago
Last Download : 2m ago
Upload by : Wren Viola
Transcription

Gauge Transformation,Lorenz Gauge, CoulombGaugeHyun LimDepartment of Mathematics&Statistics,South Dakota State UniversityNov. 5. 20141 / 19

Overview- Gauge Transformation & invariance- Lorenz Gauge- Coulomb Gauge2 / 19

Invariance of theelectromagnetic field A B(1) must be unique, but many A s exist that correspond to anySince B . We want:given B A A 0B(2) Λ. Then,We choose A 0 A A 0 A Λ A B(3)3 / 19

Invariance of theelectromagnetic field Φ A tE(4) to A 0 A ΛIf we transform only A 0 Φ A Φ (A Λ) t t Λ A Φ 6 Φ t tE(5)(6)Need something more!4 / 19

Invariance of theelectromagnetic fieldConsider Φ0 Φ Λ t 0 A Λ 0 E Φ Φ (A Λ) t t t Λ A Λ Φ t t t A Φ t(7)(8)(9)5 / 19

Gauge Transformation A 0 A Λ ΛΦ Φ0 Φ tA(10)(11)The invariance of the elds under gauge transformation is calledgauge invariance.6 / 19

Gauge TransformationRevisit! Consider to the vacuum form of the Maxwell's Equation: ρ ·E 0 2 Φ ( · A ) ρ ·E t 0 ) ρ 2 Φ ( · A t 0(12)(13)(14)7 / 19

Gauge Transformation µ0 J µ0 0 E B t E Φ 2 A 2 t t t2 ) B ( A) A ( · A!2A Φ ( · A ) µ0 J µ0 0 2 A 2 t t 2 1 Φ µ0 J 1 A ·A 2 A222 tc tc(15)(16)(17)(18)(19)8 / 19

Gauge Transformation ) ρ 2 Φ ( · A t 0 2 1 Φ µ0 J 1 A ·A 2 A222 tc tc(20)(21)Maxwell equations which is written in terms of the potentials9 / 19

Lorenz Gauge , Φ)From the freedom of the transformation, a set of potentials (Asatisfy the Lorenz condition: 1 Φ 0 ·A2 tc(22)From Eqns. (20) and (21), it produces inhomogeneous waveequations: Φ t2 2 A 1 2Ac2 t 21 Φ2c t 1 2Φ 2 Φ (0) 2 A2c t 2 1 2Ac2 t 2ρ 0(23) µ0 J(24) 10 / 19

Lorenz GaugeWe can always nd the potentials that satisfy the Lorenz condition , Φ) satisfy the Maxwell equations, and it does notSuppose (Asatisfy the Lorenz condtion. Then, make the gauge transformed 0 , Φ0 ), and we demand that (A 0 , Φ0 ) satisfy the Lorenzpotential (Acondition:0 0 1 Φ 0 ·A2 tc(25)11 / 19

Lorenz GaugeFrom the gauge transformation: 1 Φ01 Λ0 ·A 2 · (A Λ) 2Φ c tc t t2 2 Λ 1 Φ 1 Λ 0 ·A2 t22cc t(26)(27)Thus, the gauge function Λ can be obtained to satisfy:2 Λ 1 2Λc2 t 2 1 Φ ·A2 tc (28)12 / 19

Lorenz GaugeWe make the transformation which always preserves the Lorenzcondition, called the restricted gauge transformation: A Λ ΛΦ Φ t2Λ1 2 Λ 2 2 0c tA(29) , Φ, and Λ in this restricted class are said to belongthe potentials Ato the Lorenz gauge13 / 19

Coulomb GaugeTheCoulomb, radiationortransverse gaugeis: 0 ·A(30)From Eqn. (20), we obtain the Poisson equation again: 2 Φ ρ (0) 2 Φ t 0(31)And, solution is very well-known:Φ( x , t ) 14π 0Zρ( x 0 , t ) 3 0d x x x 0 (32)14 / 19

Coulomb GaugeFrom Eqn. (21), the vector potential satis es the inhomogeneouswave equation: 2 A 1 2Ac2 t 2 2 A 2c t 2 µ0 J(33)1 Φ µ0 J 2c t(34) 0 1 2A1 Φc2 tCurrent density J can be decomposed into two pieces: alongitudinal(irrotational), J , and transverse(rotation), J :tl J J J lt(35)with J 0 · J 0lt(36)(37)15 / 19

Coulomb GaugeThe vector identity: ( J ) ( · J ) 2 J(38)From this identity, the current density splits into two parts: 2 (J J ) ( J ) ( · J ) 2 J ( J )ltt 2 J ( · J )l(39)(40)(41)Since, Eqns. (40) and (41) are Poisson equation, and we have theform 2 (1/ x x 0 ) 4πδ( x x 0 )16 / 19

Coulomb Gauge 1 J4πt 1 4πJlZ J x x 0 Z 0 · J x x 0 3 0(42)3 0(43)d xd xThen:1Z Φ1 1 ρ( x 0 , t ) 3 0 2 d xc tc 4π 0 t x x 0 Z1 ρ( x 0 , t )13 0 µ0 0 d x4π 0 t x x 0 2(44)(45)17 / 19

Coulomb GaugeUsing the continuity equation · J ρ t 0:Z Φ1 0 · J 3 0 µ d x02c t4π x x 0 !Z1 0 · J 3 0 µ0 d x4π x x 0 1(46)(47)(48) µ0 J lThus, Eqn. (34) becomes: 2 A 1 2Ac2 t 2 µ0 J µ0 J µ0 J lt(49)18 / 19

Coulomb GaugeThe Coulomb or transverse gauge is often used when no sources arepresent, i.e. Φ 0 A t A BE(50)(51)19 / 19

Gauge Transformation, Lorenz Gauge, Coulomb Gauge Hyun Lim Department of Mathematics & Statistics, South Dakota State University Nov. 5. 2014 1/19

Related Documents:

experiment, you are going to verify the Coulomb’s law by using a Coulomb balance. Apparatus High Voltage Power Source (0- 6 kV), PASCO Coulomb Balance Description of Apparatus You will be using a Pasco Coulomb balance in this lab which is shown in Fig. 1. The Coulomb Balance is a delicate a

Coulomb branches and clusters Conjecture (Gaiotto) K-theoretic Coulomb branches are cluster varieties. In physics, the Coulomb branch M C Spec(A G;N) is the Coulomb branch of moduli of vacua in a 4d N 2 G-gauge theory on R3 S1. BPS quiver of the theory !quiver of the cluster variety. Theorem (Schrader-S.’in progress)

In this lesson we cover Coulomb’s law in words and mathematically. We also solve problems using Coulomb’s law to calculate the force exerted on a charge by one or more charges in 1D. 3. Coulomb’s Law in 2D This video looks at solving problems using Coulomb’s law to calculate the f

Fall 2010 Volume XXIV, No. 3 AFRP 10-1 Senior Leader Perspectives Lorenz on Leadership 5 Part 3 Gen Stephen R. Lorenz, USAF Reenabling Air Force Command and

Owner’s and Parts Manual LORENZ 6610 & 7810 SNOW BLOWER May 2010 LORENZ MFG. CO. 185 30

This notebook contains all of the material given in class on the Lorenz equations, and it constitutes section 2.5 of the class notes. The Lorenz equations are given by x s(y - x) , y rx - y-xz, z xy - bz. (1) These equations contain three parameters: s, r and b. In what follows, w

THE LORENZ SYSTEM 1 FORMULATION 1 Formulation The Lorenz system was initially derived from a Oberbeck-Boussinesq approximation. This approximation is a coupling of the Navier-Stokes equations with thermal convection. The original problem was a 2D problem considering the thermal convection between two parallel horizontal plates.

This textbook is designed for use on ten- or twelve-week introductory courses on English phonology of the sort taught in the first year of many English Language and Linguistics degrees, in British and American universities. Students on such courses can struggle with phonetics and phonology; it is sometimes difficult to see past the new .