gamoyenebiTi fizikis safuZvlebi III - ieeetsu

gamoyenebiTi fizikis safuZvlebi
III
leqciebis kursi
prof. roman jobava
q. Tbilisi 2010

Sesavali
Tanamedrove fizikaSi cnobilia sami tipis urTierTqmedeba: 1.
gravitaciuli; 2. Zlieri; 3. eleqtromagnituri. am urTierTqmedebis
gamovlena xdeba sxvadasxva sivrcul masStabebSi. kerZod, gravitaciuri
urTierTqmedeba SesamCnevia, Tu urTierTmoqmedi sxeylebidan erTi mainc
aris astronomiuli masStabebis. Zlieri urTierTqmedebis moqmedebis are
SemoisazRvreba atombiurTvis zomebiT (10 -13 sm). mxolod
eleqtromagnituri urTierTqmedeba vlindeba mTeli sisruliT
masStabebSi, romlebSic mimdinareobs Cveni ypveldRiuri cxovreba.
praqtikulad yvela Zala, romelic gansazRvravs fizikur movlenebs Cvens
irgvliv (garda mizidulbis Zalisa) gamowveulia eleqtromagnituri
urTierTqmedebiT.
1. eleqtromagnituri velis zogadi daxasiaTeba
fizikaSi nebismieri movlenis Seswavla emyareba bunebis
materialistur gagebas, romlis Tanaxmadac movlenebi da sagnebi
arseboben imis da miuxedavad, vcdilobT Tu ara Cven maT Seswavlas
movlenebis Seswavla mdgomareobs imaSi, rom SevusabamoT yovel
movlenas garkveuli cneba da davamyaroT sxvadasxva cnebebs Soris
kavSirebi, romlebic asaxaven kavSirebs Sesabamis movlenebs Soris.
eleqtromagnituri movlenebis Seswavlisas Cven vemyarebiT im
cnobebs, romlebsac gvaZlevs cda kerZod, imisaTvis rom SevamCnioT
romelime eleqtromagnituri movlena, Cven unda davafiqsiroT am
movlenis Sedegad miRebuli cvlilebebi, mag., sasinji sxeulebis
gadaadgilebebi maTze garkveuli Sesaswavl movlenasTan dakavSirebuli
Zalebis zemoqmedebis Sedegad.
vTqvaT, gvaqvs romelime А sxeuli, romlis eleqtrul Tvisebebsac
Cven vswavlobT da sasinji sxeulebi a, b, c, … cda gviCvenebs, rom Tu A
sxeulsa da sasinj sxeulebs Soris sruldeba eleqtruli xasiaTis
urTierTqmedeba (mas Cven advilad gavarCevT gravitaciuli an Zlieri
urTierTqmedebisagan), maSin SegviZlia davafiqsiroT sasinj sxeulebze A
sxeulis mxridan moqmedi Zalebi F r a
, F r
b
da a.S. Tu a da b sxeulebs
mimdevrobiT movaTavsebT erTsa da imave wertilSi, xolo Semdeg igive
cdas gavimeorebT sivrcis sxva wertilisaTvis, movnaxavT:
F
F
a
b
F ′
a
=
F′
b
2

amis garda
r r
F a | F
|
b da
a
Fb
r r
F ′ | | ′ . maSasadame, F
a
Fb
sidide koordinatebisagan
damoukidebelia da ganisazRvreba mxolod sasinji sxeulebis TvisebebiT.
sasinj sxeulze moqmedi Zala F r
ki damokidebulia koordinatebze. am
moTxovnebis dakmayofileba SesaZlebelia mxolod maSin, Tu
aq
r r
r
= e E( x,
y z)
, F e E(
x , y z)
F
a a
,
r
= da a.S.
b b
,
e , e a b
, … koordinatebisagan damoukidebelia da axasiaTeben sasinj
r
E x,
y,
z aris Sesaswavli A sxeulis Tvisebebis aRmweri
sxeulebs, xolo ( )
funqcia, misi arseboba dakavSirebulia mxolod A sxeulis arsebobasTan.
E r -veqtoris gaaCnia garkveuli mniSvneloba sivrcis (x,y,z) wertilSi
imisda miuxedavad, moTavsebulia Tu ara iq sasinji sxeuli. amboben rom
A sxeuli Tavis irgvliv qmnis E r -vels. es veli aris obieqturi realoba.
mas eleqtruli veli ewodeba. xolo E r
veqtori warmoadgens eleqtruli
velis daZabulobis veqtors. is ganisazRvreba rogorc erTeulovan
muxtze moqmedi Zala:
r
E
( x,
y,
z)
=
r
F
( x,
y,
z)
e
analogiurad SemoaqvT magnituri velisa da misi daZabulobis
r
H x,
y,
z -is cneba.
veqtoris ( )
eleqtruli da magnituri velebi warmoadgenen erTi movlenis,
eleqtromagnituri velis sxvadasxva saxeebs. es faqti iyo damtkicebuli
eqsperimentalurad. eleqtruli da magnituri velebis arsebobis faqtic
warmoadgens cdis Sedegs. misi dadgena ar moiTxovs eleqtruli Tu
magnituri muxtebis erTmaneTTan urTierTqmedebis konkretuli kanonis
codnas.
2. eleqtruli muxtebis mikroskopuli matareblebi
muxti warmoadgens eleqtromagnituri velis wyaros erTis mxriv da
misi zemoqmedebis obieqts meores mxriv.
muxtebis mikroskopuli matareblebi aris damuxtuli nawilakebi da
ionebi. elementul nawilakebs Soris mxolod ramdenimes gaaCnia
sakmarisad didi sicocxlis xangrZlivoba. esenia eleqtroni, protoni da
maTi antinawilakebi.
eleqtroni warmoadgens elementaruli uaryofiTi muxtis
materialur matarebels
3

−19
e = 1,6021892 ⋅10
kl.
me
= 9,1 ⋅10
−31
kg.
protoni aris dadebiTi elementaruli muxtis materialuri
matarebeli.
klasikur fizikaSi eleqtronsa da protons Tvlian materialur
wertilebad. magram es modeli Sedis winaaRmdegobaSi TviT klasikuri
fizikis kanonebTan. dReisaTvis cnobilia, rom protoni ara marto
moculobiTi obieqtia, aramed mis SigniT eleqtroba aris ganawilebuli
garkveuli kanonis mixedviT. eqsperimentalurad iqna Seswavlili
eleqtronebis gabneva protonebze. gabneuli eleqtronebis traeqtoriebi
gansxavdebian im traeqtoriebisagan, romlebsac Cven davinaxavdiT,
protoni rom sferulad simetriulad yofiliyo damuxtuli. muxtis
simkvrivis ganawileba protonis SigniT naCvenebia nax. 1-ze. analogiuri
eqsperimentebidan neitronebisaTvis iqna dadgenili, rom maT gaaCniaT
Sinagani struqtura (nax.2).
4πr 2 ρ
4πr 2 ρ
0 0.5 1 1.5 r⋅10 -15 m
0 0.5 1 1.5 r⋅10 -15 m
nax. 1 nax. 2
neitronis sruli muxti nulis tolia, magram mis SigniT
eleqtroba ganawilebulia araerTgvarovnad.
am Sedegebis axsna Tanamedrove fizikaSi xdeba Semdegnairad:
elementaruli nawilakebi Sedgebian kidev ufro mcire nawilakebisagan –
kvarkebisagan. kvarkebs gaaCniaT danawevrebuli muxtebi. kerZod, protoni
2 2 1
1 1
Sedgeba sami kvarkisagan: + e , + e , − e , neitroni: − e , − e ,
3 3 3
3 3
2
+ e . kvarkebi nawilakebis SigniT moZraoben. am moZraobis Sedegs Cven
3
aRviqvamT rogorc eleqtronis uwyvet da araerTgarovan ganawilebas.
Tavisufal mdgomareobaSi kvarkebi jer-jerobiT ar aris aRmoCenili.
protonis SigniT kvarkebis arseboba pirdapiri eqsperimentebiT ar aris
4

damtkicebuli. amitomac kvarkebis arseboba dReisaTvis warmoadgens
hipoTezas, romlis meSveobiTac SegviZlia garkveuli faqtebis axsna.
3. elementaruli muxti da misi invariantuloba
eleqtruli muxtis diskretulobis Sesaxeb daskvnis gakeTeba
SesaZlebelia m. faradeis mier 1834 w. aRmoCenili eleqtrolizis
kanonebis safuZvelze. magram aseTi daskvna realurad gakeTebuli iyo
mxolod 1881w. helmholcisa da stoneis mier. 1895 wels h. lorencma
SeimuSava eleqtromagnetizmis Teoria, romelic dafuZnebuli iyo
eleqtronebis arsebobaze. elementaruli muxtis ricxviTi mniSvneloba
eleqtrolis kanonebis safuZvelze iyo gamoTvlili Teoriulad, xolo
misi pirdapiri eqsperimentaluri gazomva 1909 w. moaxdina milikenma.
milikenis cdebi. am cdebis sqema mocemulia naxazze. mcire
sferuli nawilakebi moZraoben blant siTxeSi erTgvarovan eleqtrul
velSi. TiToeul nawilakze moqmedebs arqimedis amwevi Zala F r a
, xaxunis
Zala ( F r b, romelic gamoiTvleba stoqsis formuliT), simZimis Zala mg
r
da eleqtruli Zala qE
r , sadac E r
cnobili sididea. yvela Zalis
gamosaxuleba CvenTvis cnobilia, amitomac nawilakis moZraobis kanonebis
safuZvelze Cven davadgenT qE
r Zalis sdides, saidanac gavigebT
nawilakis q muxts.
F r a
qE
r
E r v r
mg
r
nax. 1
F r b
droTa ganmavlobaSi muxti icvleba, rac gavlenas axdens mis
moZraobaze. Tu ganvsazRvravT nawilakis q
1
da q
2
muxtebs drois
sxvadasxva momentebSi, movnaxavT sxvaobas ∆ q = q 2
− q1
. milikenma SeamCnia,
rom ∆q yovelTvis aris erTi da igive sididis e -s jeradi: ∆ q = n e ;
n=±1, ±2,…,
−19
e = 1,6 ⋅10
k.
dadebiTi da uaryofiTiel ementaruli muxtebis toloba.
dReisaTvis didi didi sizustiT cnobilia rom dadebiTi da uaryofiTi
elemntaruli muxtebi erTmaneTis tolia, kerod
5

e
+
− e
e
±
−
≤
10 −21
.
elmentaruli muxtis invariantoba. elementaruli muxtis
invariantoba niSnavs mis damoukidebloba siCqarisagan. sxvadaxva
atomebSi sxvadasxva wreebze moZrav eleqtronebs gaaCniaT sxvadasxva
siCqareebi. am siCqareebze rom yofiliyo damokidebuli eleqtronebis
muxtis sidide. atomebis neitraloba aucileblad dairRveoda.
maqsimaluri siCqareebi, romlebsac aRwevs eleqtronebi atomebSi, 0,5c
tolia, maSasadame swored am siCqarebamde mainc elementaruli muxti
SeiZleba CaiTvalos invariantulad; Tumca ar arsebobs faqtebi, romelTa
mixedviTac muxtis invariantoba dairRveoda ufro maRali
siCqarebisaTvisac.
4. muxtis Senaxvis kanoni
muxtis Senaxvis kanoni: izolirebul sistemaSi sruli eleqtruli muxti,
e.i. dadebiTi da uaryofiTi muxtebis
algebrauli jami mudmivia.
es kanoni damtkicebuli aris mravali eqsperimentiT. amasTan misi
samarTlianoba damtkicebulia ara marto stabiluri nawilakebisaTvis
rogoricaa eleqtronebi da protonebi, aramed Zalze mcire sicocxlis
xangrZlivobis mqone nawilakebisaTvis. garda amisa fotonebis
zemoqmedebis Sedegad vakuumSi SesaZlebelia nawilakebis vakuumidan
“amogleja”, magram amasTan maTi sruli eleqtruli muxti aucileblad
nulis toli iqneba.
e -
fotoni
nax. 1
e +
movaxdinoT muxtis Senaxvis kanonis maTematikuri Cawera:
Caketil sistemaSi
Q = const
(1)
movaxdinoT am eqsperimentaluri faqtis ganzogadeba da davakavSiroT is
muxtis moZraobasTan. Tu sruldeba (1), maSin araCaketil sistemaSi
6

muxtis cvlileba dakavSirebuli iqneba am sistemidan muxtebis
gasvlasTan (an piriqiT, SemosavlasTan). muxtebis mimarTul moZraobas
Cven dens vuwodebT. SemovitanoT muxtisa da denis simkvriveebis cnebebi:
sivrculi muxtis simkvrive:
1 ∆Q
ρ = ∑ei
=
(2)
∆V V ∆V
∆
aq e i
aris elementaruli muxtebi
moculobaSi sruli muxti, xolo
zedapirze, maSin
zedapiruli muxtis simkivrive:
∆ V moculobaSi. sadac ∆ Q aris V
ρ = ρ ρ
+ +
−
. Tu muxti ganawilebulia S
σ
1
= ∑e
∆
i
S ∆ S
∆Q
=
∆S
(4)
da
∫
Q = σdS
S
(5)
eleqtruli denis simkvrive:
r 1
j =
∆V
∑
∆V
e v r
i i
(6)
sadac v r
i
aris e i
nawilakis siCqare.
r
r
r r
j = ρ ρ v anu j = r
j + r
j + −
+
v+
+
−
−
r r
j = ρv
(7)
Tu raime S zedapirSi gadis j r
maSin sruli deni tolia
simkvrivis mqone deni yovel dS ubanSi,
I
=
∫
S
r r
jdS
(8)
7

jdS
warmoadgens r j veqtoris gegmils dS
r mimarTulebaze. am
mimarTulebas Cven ganvsazRvravT rogorc dS ubnis normals.
S
dS
dS
r
r
j
Semotanili cnebebis meSveobiT Cven SeiZleba ganvazogadoT muxtis
Senaxvis kanoni SemTxvevisaTvis, roca S zedapiriT SemosazRvrul V
moculobaSi xdeba sruli muxtis cvlileba. Tu
dS
r -is mimarTulebs Cven
ganvsazRvravT rogorc zedapiris dadebiT normals, maSin zedapirSi
Semavali denis mniSvneloba iqneba uaryofiTi. es deni gamoiwvevs muxtis
raodenobis zrdas V moculobaSi, maSasadame, dQ dt > 0 . aqedan
vRebulobT:
dQ
dt
d
r r
= − I ; ∫ ρ dV = −∫
jdS
dt
V
S
an
∂ρ
∂t
∫ dV = −∫
V
S
r r
jdS
es aris muxtis Senaxvis kanonis integraluri formulireba. imisaTvis,
rom CavweroT es kanoni diferencialur formaSi, saWiroa SemovatanoT
axali cneba.
divergencia
r
divA
=
lim
∆V
→ 0
∫
∆S
r r
A⋅
dS
∆V
∆ S aris usasrulod mcire Caketili zedapiri, romelic moicavs
usasrulod mcire ∆ V moculobas.
veqtorul analizSi mtkicdeba, rom:
r ∂Ax
divA =
∂x
∂Ay
+
∂y
∂Az
+
∂z
8

gausis formula:
∫
V
r r
divAdV =
∫
S
r r
A ⋅ dS
muxtis Senaxvis forma. mextis Senaxvis kanonis integralur formaSi
r r
gausis Teoremis Tanaxmad SevcvaloT ∫ jdS = ∫ divjdV . maSin
S
V
∫
V
∂ρ
dV
∂t
= −
∫
V
r
divjdV
aqedan
∫
V
⎛ ∂ρ
⎜
⎝ ∂t
r⎞
+ divj ⎟dV
⎠
da
∂ρ
r
+ divj = 0
∂t
9