ELECTRODYNAMICS: PHYS 30441

Transcription

1 ELETRODYNAMIS: PHYS 44. Electomagnetic Field Equations. Maxwell s Equations Analysis in space (vacuum). oulomb Bon June 4, 76 Angoulême, Fance Died August 2, 86 Pais, Fance In 785 oulomb pesented his thee epots on Electicity and Magnetism: - Pemie Mémoie su l Electicité et le Magnétisme. In this publication oulomb descibes How to constuct and use an electic balance (tosion balance) based on the popety of the metal wies of having a eaction tosion foce popotional to the tosion angle. oulomb also expeimentally detemined the law that explains how two bodies electified of the same kind of Electicity exet on each othe. oulomb s Law qq Foce: F 2 = ; ε 2 = 8.854x F / m ( N m ) 4πε d Hee, we will use SI units thoughout. Electic Field F q q( ') E() = = e ˆ = q 4 πε 4 πε 2 ' test ' ' poduced fom a single point chage q Fo a distibution of chages with chage density ρ( ') : ρ(')( ') E() = d ' ; hee [ ρ]=m 4πε ' - ' d Since = (compae with = ) 2 ' ' dx x x then we have: ρ(') = φ E()= d ' (), 4πε ' (') whee φ ρ () = 4 is the scala static potential πε ' d ' Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste --

2 ELETRODYNAMIS: PHYS 642 Gauss Law qcosθ E.dS = E cos θ ds = ds 2 4πε S S S q = = ρ()d ε ε Also, EdS =.Ed S ρ.e = (M) - st Maxwell Equation, which also hold fo time vaying fields ε Magnetic Foce F π 2 2,2 μ II dx(d x ) = is the foce of the cuent loop 2 on (show that F = F ); 2 2 [I ] = A (Ampee) is the unit of cuent. F = I dxb(), 2 whee B( ) is the magnetic induced by 2 : μi2 d2xd2 B( ) = : 4π Biot - Savat Law. 2 Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste -2-

3 ELETRODYNAMIS: PHYS 44 No Fee Magnetic Monopoles.B = (M2) -2nd Maxwell Equation This implies that B= xa, whee A is a vecto potential. Thus,.B =.( xa) =, i.e. div-cul vanishes. A is not uniquely detemined, i.e. B emains invaiant unde the change: A(,t) A'(,t) + χ(,t) The above change on A(, t) is efeed to as a Gauge Tansfomation and B(, t) is gauge invaiant. Execises: Stating fom the Biot-Savat law show that: μ I d' μ J(') A() = = d ' 4 π - ' 4 -' π -2 whee J.dS=I, with [J]=Am ae the units of cuent density Execises: 2 Show that the vecto potential fo an homogeneous constant magnetic field B is given by: A=- xb. Is A unique? 2 Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste --

4 ELETRODYNAMIS: PHYS 44 Ampee s Law: Ande-Maie Ampee (Januay 2, 775 June, 86) The SI unit of measuement of electic cuent, the Ampee, is named afte him. B.d = μ I =μ J.dS tot S Using Stoke's theoem: B.d = xb.ds S xb =μ J: Ampee's ciuital law in cul fom (n.b. this only holds fo static B fields) hage onsevation Begin with the divegence theoem applied to J: =.Jd S =- qtot d J.dS =loss of chage/sec fom a closed suface = ρ hage consevation equies:.j(,t) + ρ (,t) = Since.( xb) =, Ampees cicuital law must now ead: E t xh = J +ε (M4) 4th Maxwell Equation hee ε E epesents a time vaying displacement cuent (absent in static field) t Execise: Show that M4 satisfies chage consevation. Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste -4-

5 ELETRODYNAMIS: PHYS 44 Faaday s Law Electomotive foce (EMF) = E.d = B.dS t S MagneticFlux Stokes (cul) theoem : E.d = xe.ds = B.dS t S S xe = B (M) d Maxwell Equation t Some vecto elations to commit to memoy ul of Gad is zeo : x φ= Div of ul vanishes:.( xa) = Stokes' (ul) theoem : E.d = xe.ds = Gauss' (Div) theoem :.Ed E.dS S Execise:. Using Maxwell s d equation (M) show that E can be witten in tems of a vecto and scala potential: E = φ A 2. Also show that E is gauge invaiant unde the gauge tansfomations of the potentials: φ(,t) φ '(,t) =φ(,t) χ(,t) A(,t) A'(,t) = A(,t) + χ(,t) Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste -5-

6 ELETRODYNAMIS: PHYS 44 Summay of Maxwell s Equations in a acuum ρ (M).E = :oulombs and Gauss' law ε (M2).B = :No monopoles (M) xe = B : Faaday's law t (M4) xh J E :Ampee's law and chage consevation t = +ε hage consevation:.j + ρ= E and B fields in tems of vecto and scala potentials: B= xa, E = - φ- A Loentz foce on a chage moving at v: F= q(e+ vxb) Gauge tansfomations: φ(,t) φ '(,t) =φ(,t) χ(,t) A(,t) A'(,t) = A(,t) + χ(,t) E and B emain unchanged oulomb gauge:.a = ---Othe gauges ae possible (eg φ = adiation gauge) Loentz gauge:.a + φ= 2 c Execise:. Show that in a egion of chage and cuent, the E and B fields satisfy the wave equations: E E =, B B = c c 2. Deive the potential field equations in the Loentz gauge: ρ φ φ=, A A=μJ c t c t ε What do these equations look like in othe gauges? Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste -6-

7 ELETRODYNAMIS: PHYS 44 Maxwell s Equations in Mateials Dielectics (Mateials in E field) ρ.e = = ρ +ρ ε ε ( ) ind If ρ ind =.P (whee P is the polaisation).( ε E + P) =ρ (M).D =ρ with D= ε E + P, and P =ε χ E with χ = electical susceptibility (may be a tenso depending on medium) E E Then, D=(+ χ ) ε E =ε ε E, whee ε = +χ is the elative pemittivity fo a linea media E E (M2)=(M2); (M)=(M) emain unchanged in mateial media Diamagnetics (Mateials in B field, e.g. in a solenoid) As in dielectics, fo E/ = : xb =μ J =μ (J + J ) (M4) ind ind with J = xm (M = magnetisation) x(b μ M) =μ J E D o xh = J (M4)' = = whee H= B M is the magnetic intensity μ M =χ H; H χ = magnetic susceptibility (may be a tenso) H B =μ (H+M) =μ ( +χ )H =μ μh; with μ = elative pemeability H E D Fo xh=j + (M4) Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste -7-

8 ELETRODYNAMIS: PHYS 44 Behaviou of Fields Acoss Mateial Boundaies Bounday conditions fo E and D: (Dielectics) =ρ = ρ (M) :.D d.d d Δ Δ Div theoem dsn.d ˆ = dz dsρ =δsρ ΔS Δ s whee ρ is the suface density of chage [ ρ ]=m s Since ds n.d ˆ =δsn.(d ˆ D ) = δsρ ΔS 2 s We have n.(d ˆ D ) =ρ. Futhe, if ρ = then D is continuous. 2 s s s -2 Fo the E-field, we have (fo a suface loop ΔS) (M2): xe = B ds. xe = ds. B (as ΔS ) ΔS ul theoem d.e = E ΔS E2 = δ E is always continuous Execise:. Fo a diamagnetic bounday show that: B B =, i.e. B is continuous 2 (Hint: conside (M2).B = in a pillbox) 2. Pove that the diamagnetic bounday condition fo B is: ˆnx(H H ) = J, J is the suface cuent pe unit width [J ]=Am 2 s s s Show that if J s =, H is continuous - (Hint: conside (M4) on a suface loop) Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste -8-

9 ELETRODYNAMIS: PHYS 44 Point hage and Diac s Delta-Function An electon can be consideed as a point chage, without stuctue (exp < -2 m) We intoduce the Diac delta function to descibe chage density: +, x=x' δ(x x') =, x x' + + and dx δ(x x ') = o dx f (x) δ(x x ') = f (x ') In -D: ( ') (x x ') (y y ') (z z ') δ =δ δ δ δ = v d f() ( ') f(') Theefoe, it is natual to define a point chage density: ρ()=q δ (- ) situated at q onside the static scala potential: = ; ( = ). 4πε ρ() Then, E=- and.e= (M) ε ρ ε 2 2 o equivalently = - =δ () 4π In spheical coodinates one can show that: 2 2 =, fo > 2 = In geneal we have: 2 - =δ ( ') 4π ' Also, note that G(,') =δ ( ') 2 2 G(, ') = + f(, ') with f(, ') =. 4π ' Hee, G is the Geen's function without bounday conditions. Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste -9-

10 Execise on Diac Delta Function In the following execises pove vaious elations elating to Diac's delta function. The following epesentation is used (although othe epesentations ae of couse equally valid): (x) = p lim exp( a! a x2 =a 2 ). Z + dx(x) = 2. Z + dx(x)f(x) = f(). (cx) = (x); c 6= jcj 4. x(x) = 5. Z + dx (x)f(x) = f () 6. (x 2 c 2 ) = 2jcj ((x c) + (x + c)) 7. (f(x)) = X (x x jdf=dxj i ) ;whee x i ae the oots of the function f(x i )= x=xi i within the integation intevals. 8. Z +x dy(y c) = (x c) = fo x<c fo x>c d 9. (x c) = (x c) dx In veifying these elations you may nd the following elations useful. Z+. Gaussian integal: dx exp( x 2 =a 2 ) = p a 2. Taylo expansion: f(x) = f() + f ()x + 2 f (x) + ::: Electodynamics: PHYS 642

11 ELETRODYNAMIS: PHYS 44 Refeence on Diac Delta and Geen s functions Baton, G., Elements of Geen s functions and popagation (Oxfod Univesity Pess, 25) Electodynamics PHYS 44, EM FIELD EQS PART, R.M. Jones, Univesity of Mancheste --