KLASIKH HLEKTRODUNAMIKH II

FULLADIO SHMEIWSEWN 6

1

SQETIKISTIKH DIATUPWSH THS HLEKTRODUNAMIKHS

1. To Tetrnusma thc Puknthtac Hlektriko Rematoc.

Smfwna me sa anaptqjhkan sta prohgomena, kje fusik posthta

ja prpei na qei sugkekrimnh tautthta wc proc ton trpo metasqhmatismo

thc kat Lorentz. H tetrda twn posottwn kai ~J den mpore par naentssetai se na tetrnusma. Pollaplasizontac thn puknthta fortou me

c ste na qei tic diec diastseic me thn puknthta rematoc, orzoume toantallowto tetrnusma

J =( c, ~J

)=

J 0 = c

J i =J 1 = JxJ 2 = JyJ 3 = Jz

. (1)

O eidikc metasqhmatismc Lorentz pou sundei ta tetransmata thc pukn-thtac rematoc J kai J sta adraneiak sustmata anaforc kai , ektwn opown to detero kinetai me taqthta

~V = xV wc proc to prto, enai

= ( V

c2Jx

)

J x = (Jx V

c

)J y = Jy, J

z = Jz , (2)

pou = (1 V 2/c2)1/2.

PARATHRHSH.Mporome na epalhjesoume thn isq auto tou metasqhmatismo sthn perptwsh enc

shmeiako fortou. Prgmati, tte qoume ap thn prth exswsh

(~r ~r 0(t) ) = (~r ~r0(t) )

(1 V

c2vx

).

H sunrthsh dlta sto arister mloc metasqhmatzetai wc exc:d3r(~r. . .) =

d3r (~r . . .) = J (~r . . .) = (~r . . .) ,

pou J enai h IakwbianJ =

(xi x0i(t))xj

.2

Prpei epomnwc na apodeiqje ti

1 = J (

1 V vxc2

).

H Iakwbian upologzetai wc exc

J = xixj

x0i(t

)xj

= xixj

x0it

t

xj

= xixj

vit

xj

.Sugkekrimna, dedomnou ti t(t, x), qoume

J =

xx vx t

x

xy

xz

yx

yy

yz

zx

zy

zz

= + vx

Vc2

0 0

0 1 0

0 0 1

= (

1 +vxVc2

).

Epistrfontac sthn apodeikta sqsh qoume

1 =

(1 +

vxVc2

)

(1 V vx

c2

)

1 = 2

(1 +

(vx V )(1 vxV

c2

) Vc2

) (1 V vx

c2

)=

2

(1 V

2

c2

)= 1 .

Anloga apodeiknontai kai oi uploipec sqseic metasqhmatismo.

H exswsh sunqeiac mpore na graftei se sunallowth morf sunartsei tou

tetransmatoc J . 'Eqoume

t+Jixi

= 0

(c)

(ct)+Jixi

=J 0x0

+J ixi

= 0

J x

= 0 . (3)

To hlektrik forto enai posthta anexrthth ap to ssthma anaforc.

Aut mporome na to apodexoume wc exc:

'Estw Q to forto miac katanomc sto ssthma hremac ( ~J = 0). 'Estwkai na ssthma anaforc to opoo kinetai me taqthta

~V = xV . 'Eqoume

Q =

d3r =

d3r

( V

c2Jx

)=

d3r 1 = 1

d3r

xixj .3

H emfanizmenh Iakwbian enai

xx

xy

xz

yx

yy

yz

zx

zy

zz

=

0 00 1 00 0 1

= .

Sunepc, qoume

Q = 1d3r =

d3r = Q .

2. To Tetrnusma tou Dunamiko kai o Hlektromagnhti-

kc Tanustc.

H tetrda twn dunamikn kai ~A, efson aut apotelon fusikc pos-thtec, ja prpei na qei sugkekrimnh tautthta wc proc touc metasqhmati-

smoc Lorentz, dhlad na metasqhmatzetai me sugkekrimno trpo ap naadraneiak ssthma anaforc se na llo. Xekinntac me thn elogh up-

jesh ti ta dunamik antistoiqon se na antallowto tetrnusma tou qrou

Minkowski, grfoume

A =

A0

Ai =A1 cAxA2 cAyA3 cAz

(4)

To antstoiqo sunallowto tetrnusma ja enai A =(, c ~A

).

Ac jewrsoume tic exisseic pou ikanopoion ta dunamik kai ~A kai ac taantikatastsoume me tic antstoiqec tetranusmatikc sunistsec. Parnoume

2 ~ ~A = /0 = 2A0

xkxk

2Akx0xk

=J 0c0

2 ~A + ~(~ ~A) + 1c2~A + ~ = 0 ~J

2Ai

xkxk+

2Akxkxi

+2Aix0x0

+2A0xix0

= 0cJ i .

4

Suneqzontac, mporome na froume thn prth exswsh sthn morf

2A0xx

2A

xx0=J 0c0

. (5)

H deterh exswsh gnetai

2Aixx

2A

xxi= 0cJ i . (6)

O emfanizmenoc diaforikc telestc

2 2

xx=

2

x0x0

2

xkxk(7)

onomzetai telestc DAlembert kai enai nac bajmwtc telestc, dhlad, a-nallowtoc se metasqhmatismoc Lorentz. Enai faner ti oi exisseic (5, 6)apotelon to qronoeidc kai qwroeidc mroc miac tetranusmatikc exswshc

2Axx

2A

xx=J c0

. (8)

'Eqei anaferje se prohgomeno keflaio ti h sunjkh thc bajmdac

Lorentz enai anallowth kat Lorentz. Prgmati, qoume

~ ~A + 1c2

t= 0 = A

i

xi+A0x0

= 0

Ax

= 0 . (9)

Aut enai h sunjkh thc bajmdac Lorentz se sunallowth morf. Epibl-lontac thn sunjkh Lorentz, h exswsh tou dunamiko aplousteetai wc exc

2Axx

=J c0(10)

, pio suntomografik,

2A = J

c0. (11)

O eidikc metasqhmatismc Lorentz pou sundei ta dunamik metax doadraneiakn susthmtwn me sqetik taqthta

~V = xV enai

= ( V Ax)

Ax = (Ax Vc2

)Ay = Ay, Az = Az

(12)

5

se morf pnaka A = A

cAxcAycAz

=

V/c 0 0 V/c 0 0

0 0 1 00 0 0 1

cAxcAycAz

. (13)Ac epistryoume stic sqseic dunamikn kai pedwn kai ac tic gryoume

sunartsei tou tetrasnusmatiko dunamikoEi = i Ait = A

0

xi Ai

x0= Ai

x0 A0

xi

Bi =(~ ~A

)i

= 1c ijkAkxj

= 12cijk(Akxj Aj

xk

) (14)Parathrome ti kai stic do ekfrseic emfanzontai oi sunistsec tou diou

tanust deterhc txhc

F Ax

Ax

. (15)

O tanustc autc onomzetai Hlektromagnhtikc Tanustc kai enai ek ka-

taskeuc antisummetrikc, dhld isqei

F = F . (16)

Oi sqseic twn pedwn me tic sunistsec tou enai

Ei = F0i, Bi = 12cijkFjk . (17)

H teleutaa sqsh antistrfetai wc exc

Fij = cijkBk . (18)

O hlektromagnhtikc tanustc grfetai se morf pnaka wc exc

F =

0 Ex Ey Ez

Ex 0 cBz cBy

Ey cBz 0 cBx

Ez cBy cBx 0

. (19)

6

O antstoiqoc plrwc antallowtoc tanustc F enai

F =

0 Ex Ey Ez

Ex 0 cBz cBy

Ey cBz 0 cBx

Ez cBy cBx 0

. (20)

3. Sunallowth Diatpwsh twn Exissewn Maxwell.'Opwc edame sta prohgomena edfia oi do mh-omogenec exisseic Ma-

xwell grfontai sunartsei tou tetranusmatiko dunamiko wc

2Axx

2A

xx=Jc0

x

(Ax

Ax

)=Jc0

.

Aut grfetai sunartsei tou hlektromagnhtiko tanust wc

Fx

=Jc0

. (21)

Oi omogenec exisseic Maxwell mporon epshc na grafton sunartseitou hlektromagnhtiko tanust. Ac arqsoume ap thn posthta

Fx

=

0nrs Frsxn = nrs

Frsxn

iFx = i0rs

Frsx0

+ inFxn = irs Frsx0 + ins F0sxn inr Fr0xn

=

2c~ ~B

2Bit + 2(~ ~E)iQrhsimopoisame to gegonc ti 0ijk = ijk. Epomnwc, qoume

~ ~B = 0

~ ~E + ~Bt = 0= F

x= 0 (22)

7

To zegoc twn tetranusmatikn exissewn

Fx

=Jc0

, Fx

= 0 (23)

apotelon kai thn sunallowth diatpwsh twn exissewn touMaxwell. Enaifaner ti h omogenc exswsh ikanopoietai tautotik en qrhsimopoisoume

thn kfrash pou orzei to hlektromagnhtik tanust sunartsei tou tetra-

nusmatiko dunamiko. H exswsh aut, qrhsimopointac tic idithtec tou

plrwc antisummetriko sumblou kai to gegonc ti o hlektromagnhtikc

tanustc enai antisummetrikc, grfetai wc

Fx

+Fx

+Fx

= 0 (24)

kai onomzetai Tautthta Jacobi .O tanustc

F = 12F (25)o opooc emfanzetai stic exisseic Maxwell, onomzetai Duikc Hlektroma-gnhtikc Tanustc. Oi sunistsec tou tanust auto enai

1

F0i = 120ijkFjk = 1

2ijk (cjk`B`) = c

22i`B` = cBi

kai

Fij = 12ijF = ij0kF0k = ijk Ek .Sunepc, qoume

F =

0 cBx cBy cBz

cBx 0 Ez Ey

cBy Ez 0 Ex

cBx Ey Ex 0

. (26)

Ac shmeiwje ti

F F

~E c ~B

c ~B ~E(27)

1

Isqoun oi tautthtec ijk`mk = i`jm jmi` kai ijk`jk = 2i` .

8

Sqseic Sunallowtwn Posottwn

J 0 = c , J i = Ji,

A0 = , Ai = cAi

F0i = Ei, Fij = c ijk Bk,

Bi = 12cijk Fjk

SHMEIWSH: Sunallowth Morf twn Kajusterhmnwn Dunamikn.

Oi do ekfrseic twn kajusterhmnwn dunamikn grfontai se enopoihmnh morf wc

A(x) = 14pi0c

d3r J(~r , tK)|~r ~r| .

Gia na proqwrsoume proc mia plrwc sunallowth kfrash grfoume to anwtrw dunamik wc exc:

A(x) = 14pi0c

d3r 1|~r ~r|

dx0 (x

0 x0 + |~r ~r|)J(x)

A(x) = 14pi0c

d4x (x

0 x0 + |~r ~r|)|~r ~r| J

(x) .

Akolojwc anatrqoume sthn idithta thc dlta sunrthshc

(f(x)) =

a

(x a)|f (a)| ,

pou f(a) = 0. 'Eqoume

((x x)2) = ((x0 x0)2 (~r ~r)2) =

(x0 x0 + |~r ~r|)|2(x

0 x0)|

+(x0 x0 |~r ~r|)

|2(x0 x0)|

9

=1

2|~r ~r|((x0 x0 + |~r ~r|) + (x0 x0 |~r ~r|)

)kai

2 (x0 x0) ((x x)2) =(x0 x0 + |~r ~r|)

|~r ~r| .

Telik, qoume

A(x) = 12pi0c

d4x(x0 x0) ((x x)2)J(x) .

H kfrash aut enai plrwc sunallowth.

4. Metasqhmatismo Lorentz HM Pedwn.Wc tanustc deterhc txhc, o hlektromagnhtikc tanustc metasqhmat-

zetai kat Lorentz wc exc

F = F (28)

, se morf pinkwn,

F = F . (29)Gia thn perptwsh tou eidiko metasqhmatismo Lorentz h anwtrw sqshenai

0 Ex Ey Ez

Ex 0 cBz cBy

Ey cBz 0 cBx

Ez cBy cBx 0

=

V/c 0 0

V/c 0 0

0 0 1 0

0 0 0 1

0 Ex Ey EzEx 0 cBz cByEy cBz 0 cBxEz cBy cBx 0

V/c 0 0

V/c 0 0

0 0 1 0

0 0 0 1

=

0 Ex

(Ey V Bz

)(Ez + V By

)Ex 0 c

(Bz V

c2Ey

)c(By +

Vc2Ez

)(Ey V Bz

)c(Bz V

c2Ey

)0 cBx

(Ez + V By

)c(By +

Vc2Ez

)cBx 0

.

Ap ta anwtrw sumperanoume tic sqseic metasqhmatismo

10

Ex = Ex, Ey = (Ey V Bz) , Ez = (Ez + V By)

Bx = Bx, By = (By +

Vc2Ez), Bz =

(Bz Vc2Ey

)

~E || = ~E||, ~E =

(~E ~V ~B

)~B || = ~B||, ~B

=

(~B +

~V ~Ec2

)

Metasqhmatismo Lorentz

Enai endiafron ti, pwc problpoun oi parapnw sqseic, en se na a-

draneiak ssthma qoume mhdenik magnhtik pedo (

~B = 0), se na llossthma, kinomeno me taqthta

~V wc proc aut, ja emfanisje magnhtikpedo

~B =

c2~V ~E . (30)Ep plon, gia to hlektrik pedo sto kinomeno ssthma qoume

V ~E = V ~E, V ~E = V ~E . (31)

Antstrofa, en se na ssthma uprqei mno magnhtik pedo, sto kinomeno

ssthma ja emfanisje kai hlektrik pedo

~E = ~V ~B . (32)

PARADEIGMA: Ta peda enc omal kinoumnou fortou. Ac jewrsoume na shmeiak forto Q,

to opoo kinetai me stajer taqthta

~V = V x wc proc na {aknhto} adraneiak ssthma . Epshc, stw to ssthma {hremac} tou swmatidou. Wc proc to ssthma aut qoume apl mno to pedo Coulomb

~E

=Q

4pi0

~r

r3

11

kai mhdenik magnhtik pedo

~B = 0 . Ta peda sto aknhto ssthma ja enai

~E = Ex, Ey = E

y, Ez = E

z

Bx = 0, By = V

c2Ez , B

z =

V

c2Ey .

Sugkekrimna, qoume

~E =Q

4pi0 r3(xx + y

y + z

z)

~B =Q

4pi0 r3V

c2

(yz zy

).

Ekfrzontac ta peda wc proc tic suntetagmnec tou aknhtou sustmatoc parnoume

~E =Q

4pi02

(~r ~V t)[(x V t)2 + 2(y2 + z2)

]3/2~B =

1

c2~V ~E .

To hlektrik pedo mpore na grafte kai wc exc:

~E =Q

4pi0

~R

R3

(1 V 2

c2

)[1 (R~Vc2

)2]3/2 , (33)pou

~R = ~r ~V t .

12

5. H Dnamh Lorentz.Poi enai h Drsh enc fortismnou swmatidou? Isodnama mporome na

rwtsoume, poi enai h exswsh knhshc enc swmatidou up thn epdrash

enc hlektromagnhtiko pedou? O aplosteroc roc pou mporome na pro-

sjsoume sthn Drsh tou eleujrou swmatidou (S0 =ds) enai nac rocgrammikc wc proc to hlektromagnhtik dunamik A. Dedomnou mwc tinac ttoioc roc ja prpei na enai na bajmwt kat Lorentz, ja prpei toantallowto tetrnusma tou dunamiko na pollaplasiaste me na sunallow-

to tetrnusma. To mno ttoio pou diajtoume enai to tetrnusma thc jshc

dx. 'Ara, h aplosterh Drsh, me grammik allhlepdrash metax dx kaiA, enai h

S = mcds q

c

dxA(x) , (34)

pou ds = (dxdx)1/2enai to diaforik tou anallowtou mkouc. q enaito hlektrik forto kai h stajer c qei emfanisje ste na qei o rocallhlepdrashc tic swstc diastseic.

Ac jewrsoume tra dunatc metabolc x, oi opoec na mhdenzontai sta kra. Gia ttoiec, kajar sunar-thsiakc, dunatc metabolc isqei (dx) = d(x). H metabol tou diaforiko ds enai

(ds) = (

dxdx) =1

2

2(dx)dxdxdx

=d(x)dx

ds= (dx

)dx

ds=

1

c(dx

)dx

d=

1

c(dx

)u .

H metabol thc allhlepdrashc enai

(dxA) = (dx)A + dxA = (dx)A + dx (x)Ax

.

H antstoiqh metabol thc Drshc enai

S = m

(dx

)u q

c

((dx)A dx (x)

Ax

)

S = m

d(x

)u q

c

(d(x)A dx (x)

Ax

)= m

d(x

u) + m

xdu

q

c

d(xA) +

q

c

x dA

q

c

dx

Ax

x.

O prtoc kai o trtoc roc mhdenzontai, dedomnou ti h metabol x mhdenzetai sta kra thc oloklrwshc.Suneqzontac, qoume

S =

d x

(mdu

d+q

c

dAd

qc

dx

d

Ax

)

S =

d x

(mdu

d+q

c

(Ax

Ax

)u

)(35)

13

S =

d x

(mdu

d+q

cFu

). (36)

H Arq thc Elqisthc Drshc upagoreei

S = 0 = mdud

=q

cFu . (37)

Oi exisseic knhshc enai

md2xd2

=q

c

dx

dF . (38)

To dexi mloc thc anwtrw exswshc apotele thn sqetikistik genkeush

thc dnamhc

f =q

cu F . (39)H qwrik sunistsa thc dnamhc enai

f i =q

c( cF0i + vj Fji ) = q

(Ei +

(~v ~B

)i

). (40)

Me exaresh to pargonta , o opooc, gia v

Ac shmeiwje ti h posthta Ekin = mc2 tautzetai me thn qronik sunist-sa thc tetraormc diairemnh di c, gia na elejero swmatdio. Mporome nathn onomsoume {kinhtik enrgeia}.

H Drsh gia na fortismno swmatdio, thn opoa qrhsimopoisame sta

amswc prohgomena

S = mc2d q

c

duAgrfetai kai wc

S =

dt

{mc2 q

cu0A0 q

cuiAi

}. (44)

Ap thn teleutaa kfrash sumperanoume ti h sunrthsh Lagrange tousustmatoc enai

L = mc2

1 v2

c2 q + q(~v ~A) . (45)

Ap thn sunrthsh Lagrange mporome na paraggoume kateujean tic exi-sseic knhshc wc

d

dt

(L

xi

)=

L

xi.

Mpore na deiqje ti h exswsh knhshc qei thn morf

md~v

dt= q

(~E + ~v ~B ~v

c2

(~E ~v

)). (46)

Ap ton orism thc kanonikc ormc pou isqei sta plasia thc Klassikc

Mhqanikc, qoume

pi Lvi

=mvi1 v2

c2

+ q Ai = mui + qAi . (47)

Parathrome ti h sqsh taqthtac kai ormc perilambnei kai ton ro q ~A,o opooc ofeletai sto hlektromagnhtik pedo. H sqsh aut dnei tic qw-

roeidec sunistsec thc ormc gia to tetrnusma

p = mu + qA . (48)To qronoeidc mroc tou tetransmatoc thc ormc dnei thn enrgeia tou sw-

matidou diairemnh me c

p0 =Ec

= mc + q

c. (49)

15

H sunrthsh Hamilton tou swmatidou upologzetai ap ton genik thcorism sunartsei thc sunrthshc Lagrange

H(pi, xj) = pivi L .

Dedomnou mwc ti h sunrthshHamilton ja prpei na ekfrasje sunartseithc ormc, mac qreizetai kai h antstrofh sqsh taqthtac/ormc. Aut

exgetai wc exc:

m(v)~v = ~p+ q ~A =

v2 = c2 (~pq

~A)2

[m2c2+(~pq ~A)2]

1 =

mcm2c2+(~pq ~A)2

= ~v = c(~p q~A)

m2c2 + (~p q ~A)2.

H sunrthsh Hamilton enai

H = c

(~p q ~A)2 +m2c2 + q . (50)

Ac shmeiwje ti sto mh-sqetikistik rio, mporome na anaptxoume thn k-

frash aut se dunmeic tou 1/c kai na proume thn gnrimh sunrthsh Ha-milton

H mc2 + 12m

(~p q ~A

)2+ q . (51)

Tloc, Exisseic Hamilton enai

xi =H

pi, pi = H

xi.

H prth dnai aplc thn sqsh taqtrhtac/ormc pou brkame pio pnw. H

deterh enai

pi =qc(pk qAk)m2c2 + (~p q ~A)2

Akxi

q xi(52)

kai mpore ekola na deiqje ti tautzetai me thn exswsh knhshcsthn opoa

katalxame prohgoumnwc.

PARADEIGMA: Stajer Hlektrik Pedo. H perptwsh stajero kai omogenoc hlektriko pedou

~E = . enai idiatera ekolh kai analsimh. Sthn perptwsh aut qoume mno to bajmwt dunamik

= ~r ~E = ~ = ~E .

H sunrthsh Hamilton enai

H = c

m2c2 + p2 q ~r ~E

16

kai dnei tic exisseic Hamilton

~v ~r = H~p

=c~p

m2c2 + p2

kai

~p = H~r

= q ~E .

Epilgontac to ssthma suntetagmnwn tsi ste

~E = E x parnoume

px = q E, py = pz = 0 .

Qwrc blbh thc genikthtac mporome na epilxoume tic arqikc sunjkec

px(0) = pz(0) = 0, py(0) = p0 .

Tte qoume

px(t) = q E t, py(t) = p0, pz(t) = 0 .

Ap thn llh exswsh Hamilton qoume

x = vx(t) =c q E t

m2c2 + p20

+ (qEt)2, y = vy(t) =

c p0m2c2 + p2

0+ (qEt)2

.

H knhsh gnetai apokleistik sto eppedo x, y mia kai vz(t) = vz(0) = 0 z(t) = z(0). Proqwrmeepilgontac arqikc sunjkec x(0) = y(0) = 0. Ap thn prth parnoume

x(t) = c

t0

dtqEt

m2c2 + p20

+ (qEt)2=

c

2qE

m2c2+p20+(Eqt)2

m2c2+p20

d

x(t) =c

qE

(m2c2 + p2

0+ (qEt)2

m2c2 + p2

0

).

Ap thn llh qoume

y(t) =p0c

qE

qEt/m2c2+p20

0

d1 + 2

=p0c

qEln

(qEt

m2c2 + p20

+

1 +

(qEt)2

m2c2 + p20

).

Den enai dskolo na antistryoume aut thn sqsh, opte qoume

cosh

(qE

p0cy

)=

1 +

(qEt)2

m2c2 + p20

.

Sunduzontac aut thn sqsh me thn kfrash gia to x(t) kai apalefontac to qrno, parnoume thn troqi(Alussoeidc)

x =c

p20

+m2c2

qE

(cosh

(qE

p0cy

) 1).

Sto mh-sqetikistik rio mporome na anaptxoume se dunmeic tou 1/c, opte qoume

cosh

(qE

p0cy

)=

1

2

(e

qEp0c

y+ e qEp0c

y) 1 + y

2

2c2

(qE

p0

)2+ . . . .

kai

x (qEm

2p20

)y2,

pou enai mia parabol.

17

PARADEIGMA: Stajer Magnhtik Pedo. Sthn perptwsh pou to swmatdio kinetai up thn

epdrash enc stajero kai omogenoc magnhtiko pedou

~B h knhs tou enai idiatera apl kai epilsimh.Mia epilog gia to dianusmatik dunamik pou odhge se stajer kai omogenc magnhtik pedo enai h

~A =1

2

(~B ~r

)= ~ ~A = ~B .

H sunrthsh Lagrange tou sustmatoc enai

L = mc2

1 v2

c2+q

2~v (~B ~r

)kai odhge sthn exswsh knhshc

md ( ~v)

dt= q(~v ~B

).

Pollaplasizontac eswterik me thn taqthta parnoume

~v d ( ~v)dt

= 0 = dv2

dt= 0 = = 0

11 v2(0)

c2

.

Epilgontac to ssthma anaforc ste

~B = Bz parnoume tic exisseic

vz = 0 z(t) = z(0) + vz(0) t

vx = vy

vy vx,

pou

qBm0

.

Paragwgzontac llh mia for tic exisseic autc parnoume

vx = 2 vx, vy = 2 vy

me profanec lseic tic

vx(t) = vx(0) cos(t) +vx(0)

sin(t) = vx(0) cos(t) + vy(0) sin(t)

vy(t) = vy(0) cos(t) +vy(0)

sin(t) = vy(0) cos(t) vx(0) sin(t) .

Oloklhrnontac, parnoume

x(t) = x(0) +vx(0)

sin(t) vy(0)

(cos(t) 1)

y(t) = y(0) +vy(0)

sin(t) +

vx(0)

(cos(t) 1) .

Ap tic sqseic autc sumperanoume thn troqi{ (x(t) x(0) vy(0)

)2+(y(t) y(0) + vx(0)

)2=

v2x(0)+v2y(0)

2

z(t) = z(0) + vz(0) t

pou enai mia lika.

PARADEIGMA: Orjognia omogen kai qronik stajer hlektrik kai magnhtik pe-

da. Ja jewrsoume thn perptwsh enc omogenoc kai qronik stajero hlektriko pedou

~E, to opoo enai

kjeto proc na omogenc kai qronik stajer magnhtik pedo

~B, kai ja meletsoume thn knhsh enc for-tismnou swmatidou mzac m kai hlektriko fortou q. 'Estw to {aknhto} ssthma anaforc wc proc to

18

opoo to swmatdio qei taqthta ~v kai ta peda enai ~E kai ~B. Ex upojsewc ~E ~B = 0 . Ac shmeiwje tieswterik aut ginmeno enai anallowto se metasqhmatismoc Lorentz kai, epomnwc, ta peda exakoloujonna enai orjognia se kje ssthma anaforc. stw na ttoio ssthma anaforc , to opoo kinetai metaqthta

~V wc proc to . Ta peda wc proc aut to ssthma ja enai

~E|| = ~E||, ~B

|| = ~B|| ,

~E = (V )

(~E ~V ~B

), ~B

= (V )

(~B +

~V ~Ec2

).

Enai dunatn h taqthta

~V na epilege tsi ste sto ssthma na mhdenzetai to hlektrik pedo? Prgmati,en qoume E < Bc, h epilog

~V = ~E ~BB2

dnei, ek kataskeuc,

~E = 0

kai

~E ~V = 0 = ~E|| = 0 = ~E || = 0 .

Dedomnou ti ta

~E kai ~B enai kjeta, qoume

V =E

B= c

(E

Bc

)< c .

Ac shmeiwje epshc ti, lgw thc orjogwnithtac

~V ~B = 0, qoume ~B|| = 0 kai ~B || = 0. Telik, sto ja

qoume

~E

= ~B|| = 0, ~B

= (V )

(~B

1

c2B2

(~E ~B

) ~E)

=1

1 E2c2B2

(~B

E2

c2B2~B

)=

~B(V )

.

H exswsh thc knhshc sto enai

md~v

dt=

q

(v)

(~v ~B

).

Enai amswc faner ti

dv2

dt= 0 = (v) = 0

11 v2(0)

c2

.

Epilgontac to ssthma anaforc ste

~B = Bz, qoume

vz(t) = 0 = z(t) = z(0) + v z (0) t

vx(t) = v

y(t)

vy(t) = v x(t) ,

pou

qBm0

1 E

2

c2B2.

H troqi sto enai

x(t) = x

(0) +

vx(0)

sin(t)

vy(0)

(cos(t

) 1

)y(t) = y

(0) +

vy(0)

sin(t

) +

vx(0)

(cos(t

) 1

).

19

Oi lsh aut enai mia lika me kuklik probol sto eppedo x, y. Sto aknhto ssthma ektc ap thnanwtrw elikoeid knhsh grw ap ton xona B, to swmatdio metqei kai se omal metaforik knhsh procthn die;ujunsh E B.T sumbanei tan E > cB? Sthn perptwsh aut enai dunatn na brome na adraneiak ssthma , toopoo na kinetai wc proc to me katllhlh taqthta ste na mhdenzetai to magnhtik pedo ~B . H taqthtaaut enai

~V = c~E ~BE2

.

Sto ssthma ta peda enai~E|| = 0, ~B

|| = 0

~E =

~E(V )

, ~B = (V )

(~B ~V ~E

),

pou

(V ) =1

1 c2B2E2

.

H exswsh knhshc tou swmatidou sthn perptwsh aut enai

md~v

dt=

q

(v)

(~E ~v

c2

(~v ~E

))

md~v

dt=

q

(v)(V )

(~E ~v

c2

(~v ~E))

.

Aut h exswsh enai tautsimh me thn exswsh knhshc parousa mno hlektriko pedou me mnh diafor

ton stajer pargonta 1(V ). Sunepc h knhsh sto enai mia epitaqunmenh knhsh me alussoeid

(kat prosggish parabolik) troqi. H knhsh tou swmatidou sto prokptei ap thn snjesh autc thc

epitaqunmenhc knhshc me mia omal knhsh kat thn diejunsh E B.

20

6. Sunallowth Morf twn Jewrhmtwn Diatrhshc.

Jewreste thn tetrda twn exissewn diatrhshc twn prohgoumnwn e-

dafwn

U

t+ ~ ~S + ~J ~E = 0 (53)

~ ~(i) 1c2Sit

= Ei +(~J ~B

)i(54)

Qrhsimopointac touc orismoc J = (c, ~J), Ei = F0i kai ijkBk = 1cFij ,h prth ap autc thc exisseic (Jerma Poynting) grfetai

T 0x

=1

cJ F0 , (55)

pou

T 0 =(U,

~S

c

). (56)

H deterh exswsh grfetai

ijxj

x0

(Sic

)=

1

c

(J 0F0i + J jFji

)

T ix

=1

cJ Fi , (57)pou

T i =T 0i = Sic

T ji = ij. (58)

Oi exisseic autc sumptssontai se na eniao {Jerhma Poynting} sthnmorf miac tetranusmatikc exswshc

T x

=1

cJ F , (59)

pou

T =

U Sx/c Sy/c Sz/c

Sx/c xx xy xz

Sy/c yx yy yz

Sz/c zx zy zz

. (60)

21

To tanustik mgejoc (60) onomzetai Hlektromagnhtikc Tanustc Ormc-

Enrgeiac. Ekola fanetai ti o summetrikc tanustc T enai kai iqnoc,dhlad

T = 0 . (61)'Etsi pwc enai grammnoc o pnakac T , sunartsei mh-summetablhtnposottwn, den qei mia ekpefrasmnh tautthta kat Lorentz. Ja proqwr-soume grfontc ton sunartsei sunallowtwn megejn ste na enai faner

h tauttht tou wc antallowtou tanust deterhc txhc. H pio genik k-

frash pou mpore na grafte, h opoa efenc na enai digrammik wc proc ta

peda kai afetrou na qei do elejerouc summetrikoc dektec enai h

T = C FF + Dg FF ,

pou C kai D prosdioristaoi suntelestc. Efarmzontac aut to ansatzsthn perptwsh = = 0, qoume

T 00 = U = 02E2 +

B2

20= C F0jF j0 + D

(2F0jF0j + F ijFij

)= CE2 + D

(2E2 + c2ijkij`BkB`

)= E2(C + 2D) + 2c2DB2

pou, an sugkrnoume, afenc me thn {00} sunistsa thc (60) kai afetrou me

thn gnwst kfrash thc puknthtac enrgeiac, sunepgetai

C = 0, D = 04.

Epomnwc, o tanustc ormc-enrgeiac qei thn akloujh sunallowth k-

frash

T = 0FF + 0g

4FF . (62)Kai saut thn morf enai mesa diapistsimo ti o tanustc autc enai

iqnoc. Prgmati

T = 0FF + 0FF = 0 .

22