Phng trnh Maxwell

James Clerk Maxwell

Cc phng trnh Maxwell bao gm bn phng trnh, ra bi James Clerk Maxwell,

dng m t trng in t cng nh nhng tng tc ca chng i vi vt cht. Bn

phng trnh Maxwell m t ln lt :

in tch to ra in trng nh th no (nh lut Gauss).

S khng tn ti ca vt cht t tch.

Dng in to ra t trng nh th no (nh lut Ampere).

V t trng to ra in trng nh th no (nh lut cm ng Faraday)

y cng chnh l ni dung ca thuyt in t hc Maxwell.

Lch s

Cc cng thc ca Maxwell vo nm 1865 bao gm 20 phng trnh vi 20 n s, nhiu

phng trnh trong c coi l ngun gc ca h phng trnh Maxwell ngy nay.

Cc phng trnh ca Maxwell tng qut ha cc nh lut thc nghim c nhng

ngi i trc pht hin ra: chnh sa nh lut Ampre (ba phng trnh cho ba chiu (x,

y, z)), nh lut Gauss cho in tch (mt phng trnh), mi quan h gia dng in tng

v dng in dch (ba phng trnh (x, y, z)), mi quan h gia t trng v th nng

vect (ba phng trnh (x, y, z), ch ra s khng tn ti ca t tch), mi quan h gia

in trng v th nng v hng cng nh th nng vect (ba phng trnh (x, y, z),

nh lut Faraday), mi quan h gia in trng v trng dch chuyn (ba phng trnh

(x, y, z)), nh lut Ohm v mt dng in v in trng (ba phng trnh (x, y, z)),

v phng trnh cho tnh lin tc (mt phng trnh). Cc phng trnh nguyn bn ca

Maxwell c vit li bi Oliver Heaviside v Willard Gibbs vo nm 1884 di dng

cc phng trnh vect. S thay i ny din t c tnh i xng ca cc trng trong

cch biu din ton hc. Nhng cng thc c tnh i xng ny l ngun gc hai bc

nhy ln trong vt l hin i l thuyt tng i hp v vt l lng t.

Tht vy, cc phng trnh ca Maxwell cho php on trc c s tn ti ca sng

in t, c ngha l khi c s thay i ca mt trong cc yu t nh cng dng in,

mt in tch... s sinh ra sng in t truyn i c trong khng gian. Vn tc ca

sng in t l c, c tnh bi phng trnh Maxwell, bng vi vn tc nh sng c

o trc bng thc nghim. iu ny cho php kt lun rng nh sng l sng in t.

Cc nghin cu v nh sng v sng in t, tiu biu l cc nghin cu ca Max Planck

v vt en v ca Heinrich Hertz v hin tng quang in cho ra i l thuyt lng

t.

S khng ph thuc ca vn tc nh sng vo chiu v h quy chiu - nhng kt lun

c rt ra t phng trnh Maxwell - l nn tng ca thuyt tng i. Ch rng khi ta

thay i h quy chiu, nhng bin i Galileo c in khng p dng c vo cc

phng trnh Maxwell m phi s dng mt bin i mi, l bin i Lorentz.

Einstein p dng bin i Lorentz vo c hc c in v cho ra i thuyt tng i

hp.

Tm tt

Bng sau y tm tt cc phng trnh v khi nim cho trng hp tng qut. K hiu

bng ch m l vect, trong khi nhng k hiu in nghing l v hng.

Tn Dng phng trnh vi

phnDng tch phn

nh lut

Gauss:

inh lut

Gauss cho

t trng

(s khng

tn ti ca

t tch):

nh lut

Faraday

cho t

trng:

nh lut

Ampere

(vi s b

sung ca

Maxwell):

Bng sau y lit k khi nim ca cc i lng trong h o lng SI :

K hiu ngha n v trong

h SI

Cng in trng volt / mt

Cng t trng ampere / mt

in thmcoulomb / mt

vung

Vect cm ng t

tesla,

weber / mt

vung

Mt in tch, coulomb / mt

khi

Mt dng in, ampere / mt

vung

Vect vi phn din tch A, c hng vung gc vi mt S mt vung

Vi phn ca th tch V c bao bc bi din tch S mt khi

Vect vi phn ca ng cong, tip tuyn vi ng knh

C bao quanh din tch S mt

(cn

gi l div)

ton t tnh sut tiu tn:

trn mt

(cn

gi l rot) ton t tnh xoy cun ca trng vect. trn mt

Cc i lng D v B lin h vi E v H bi :

trong :

e l h s cm ng in ca mi trng,

m l h s cm ng t ca mi trng,

l hng s in mi ca mi trng, v

l hng s t mi ca mi trng.

Khi hai hng s and ph thuc vo cng in trng v t trng, ta c hin

tng phi tuyn; xem thm trong cc bi hiu ng Kerr v hiu ng Pockels.)

Trong mi trng tuyn tnh

Trong mi trng tuyn tnh, vect phn cc in P (coulomb / mt vung) v vect

phn cc t M (ampere / mt) cho bi :

Trong mi trng khng tn sc (cc hng s khng ph thuc vo tn s ca sng in

t), v ng hng (khng bin i i vi php quay), v khng ph thuc vo thi

gian, phng trnh Maxwell tr thnh :

Trong mi trng ng u (khng bin i i vi php tnh tin), v khng i

theo khng gian, v c th c a ra ngoi cc php o hm theo khng gian.

Trong trng hp tng qut, v c th l tensor hng 2 m t mi trng lng chit.

V trong cc mi trng tn sc v/hoc ph thuc vo tn s nh sng (sng in

t), nhng s ph thuc ny tun theo mi lin h Kramers-Kronig.

Trong chn khng

Chn khng l mi trng tuyn tnh, ng ng (khng bin i theo php quay v php

tnh tin), khng tn sc, vi cc hng s 0 v 0 (hin tng phi tuyn trong chn khng

vn tn ti nhng ch quan st c khi cng nh sng vt qua mt ngng rt ln

so vi gii hn tuyn tnh trong mi trng vt cht).

ng thi trong chn khng khng tn ti in tch cng nh dng in, phng trnh

Maxwell tr thnh :