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Chng 9: IN TRNG TNH

189

Chng 9

IN TRNG TNH9.1 TNG TC IN NH LUT COULOMB1 in tch nh lut bo ton in tch: T xa xa, con ngi bit hin tng mt s vt sau khi c st th chng c th ht hoc y nhau v chng ht c cc vt nh. Ngi ta gi chng l cc vt nhim in v phn bit thnh hai loi nhim in dng v m. u th k XVII, ngi ta mi nghin cu lnh vc ny nh mt ngnh khoa hc. Cc vt nhim in c cha in tch. Trong t nhin, tn ti hai loi in tch: dng v m. in tch cha trong mt vt bt k lun bng s nguyn ln in tch nguyn t in tch c gi tr nh nht trong t nhin. n v o in tch l coulomb, k hiu l C. Gi tr tuyt i ca in tch c gi l in lng. in tch ca ht electron l in tch nguyn t m: e = 1,6.10 19 C. in tch ca ht proton l in tch nguyn t dng: +e = 1,6.10 19 C. in tch dng v in tch m c th trung ho ln nhau nhng tng i s cc in tch trong mt h c lp l khng i l ni dung ca nh lut bo ton in tch. 2 nh lut Coulomb: Cc in tch cng du th y nhau, tri du th ht nhau. Tng tc gia cc in tch c gi l tng tc in. Nm 1785, bng thc nghim, Coulomb (nh Bc hc ngi Php 1736 1806) xc lp c biu thc nh lng ca lc tng tc gia hai in tch c kch thc rt nh so vi khong cch gia chng gi l in tch im, t ng yn trong chn khng. Pht biu nh lut: Lc tng tc gia hai in tch im ng yn trong chn khng c phng nm trn ng thng ni hai in tch , c chiu y nhau nu chng cng du v ht nhau nu chng tri du, c ln t l thun vi tch ln ca hai in tch v t l nghch vi bnh phng khong cch gia chng. Biu thc:

Fo = k

q 1 .q 2 r2

=

1 q 1 .q 2 . 2 4 o r

(9.1)

Trong : k =

1 = 9.10 9 (Nm2/C2) l h s t l; 4. o

190o =

Giao Trnh Vat Ly ai Cng Tap I: C Nhiet - ien

1 = 8,85.10 12 (F/m) l hng s in. 9 36.10

Trong cht in mi ng nht v ng hng, lc tng tc gia cc in tch gim i ln so vi lc tng tc trong chn khng:

F=

q .q Fo 1 q1.q 2 = k 1 22 = r 4 o r 2

(9.2)

gi l h s in mi ca mi trng . l i lng khng th nguyn, c gi tr ty theo mi trng, nhng lun ln hn 1. Bng 9.1 cho bit h s in mi ca mt s cht thng dng. Bng 9.1: H s in mi ca mt s cht Vt liu Chn khng Khng kh Du ha (20 C) Du bin th Nc (20 C) Ebnto o

1 1,0006 2,2 4,5 80 2,7 2,9

Vt liu Ru tilic (20oC) Giy S Mica Gm titan Thy tinh

25 3,5 6,5 5,5 130 5 10

q1 +F21

r12

q2 + q2 +

F12

q1 +

r21

Hnh 9.1: Lc tng tc gia 2 in tch im Nu gi r12 l vect khong cch hng t q1 n q2 th lc do q1 tc dng ln q2 c vit l:

q .q r F12 = 1 2 2 . 12 4 o r r

(9.3)

Tng t, lc do q2 tc dng ln q1 l:

q .q r F21 = 1 2 2 . 21 4 o r r

(9.4)

Chng 9: IN TRNG TNH

191. 4o r 2 r qi q j rij

Tng qut, lc do in tch qi tc dng ln in tch qj l: Fij = trong rij l vect khong cch hng t qi n qj. 3 Nguyn l tng hp cc lc tnh in:

(9.5)

Gi F1 , F2 , ..., Fn ln lt l cc lc do in tch q1, q2, , qn tc dng ln qo. Khi lc tng hp tc dng ln qo s l:

F = F1 + F2 + ... + Fn = Fii =1

n

(9.6)

Da vo nguyn l ny, ngi ta chng minh c lc tng tc gia hai qu cu tch in u ging nhng tng tc gia hai in tch im t ti tm ca chng.

9.2 IN TRNG1 Khi nim in trng: nh lut Coulomb th hin quan im tng tc xa, ngha l tng tc gia cc in tch xy ra tc thi, bt k khong cch gia chng l bao nhiu. Ni cch khc, vt tc truyn tng tc l v hn. Theo quan im tng tc gn, s d cc in tch tc dng lc ln nhau c l nh mt mi trng vt cht c bit bao quanh cc in tch l in trng. Tnh cht c bn ca in trng l tc dng lc ln cc in tch khc t trong n. Chnh nh vo tnh cht c bn ny m t bit c s cc mt ca in trng. Nh vy, theo quan im tng tc gn, hai in tch q1 v q2 khng trc tip tc dng ln nhau m in tch th nht gy ra xung quanh n mt in trng v chnh in trng mi tc dng lc ln in tch kia. Lc ny gi l lc in trng. Khoa hc hin i xc nhn s ng n ca thuyt tng tc gn v s tn ti ca in trng. in trng l mi trng vt cht c bit, tn ti xung quanh cc in tch v tc dng lc ln in tch khc t trong n. 2 Vect cng in trng: Xt im M bt k trong in trng, ln lt t ti M cc in tch im q1, q2, , qn (gi l cc in tch th), ri xc nh cc lc in trng F1 , F2 , , Fn tng ng. Kt qu thc nghim cho thy: t s gia lc tc dng ln mi in tch v tr s ca in tch l mt i lng khng ph thuc vo cc in tch th m ch ph thuc vo v tr ca im M trong in trng: F1 F2 F = = ... = n = const q1 q 2 qn

192

Giao Trnh Vat Ly ai Cng Tap I: C Nhiet - ien

Hng vect c trng cho in trng ti im M c v phng chiu v ln, c gi l vect cng in trng ti im M, k hiu l E . Vy:

E=

F q

(9.7)

Vect cng in trng ti mt im l i lng c trng cho in trng ti im v phng din tc dng lc, c gi tr (phng, chiu v ln) bng lc in trng tc dng ln mt n v in tch dng t ti im . n v o cng in trng l vn/mt (V/m). Nu E khng i (c v phng chiu ln ln) ti mi im trong in trng th ta c in trng u. Nu bit vect cng in trng ti mt im, ta s xc nh c lc in trng tc dng ln in tch q t ti im :

E

+ q>0

F

F

q 0 th F E ; Nu q < 0 th F E . 3 Vect cng in trng gy bi mt in tch im: Khi mt in tch im Q xut hin, n s gy ra xung quanh n mt in trng. xc nh vect cng in trng do in tch im Q gy ra ti im M cch n mt khong r, ta t ti M in tch th q. Khi in trng ca Q s tc

Qq r dng lc ln q mt lc F xc nh theo nh lut Coulomb: F = k 2 . . So snh r r

vi (9.7), suy ra vect cng in trng ti M do in tch im Q gy ra l:

Q r Q r E=k 2. = . 2 r r 4o r r

(9.9)

Trong , r l vect bn knh hng t Q n im M. Nhn xt: Vect E c: Phng: l ng thng ni in tch Q vi im kho st M Chiu: hng xa Q, nu Q > 0 v hng gn Q, nu Q < 0.

Q

+Q

r

M

-

EM

EM rM

Hnh 9.3: Cng in trng gy bi in tch im

Chng 9: IN TRNG TNH ln: E = k

193(9.10)

|Q| |Q| = 2 r 40 r 2

im t: ti im kho st M. Nu bao quanh in tch Q l mi trng in mi ng nht, ng hng, c h s in mi th cng in trng gim i ln so vi trong chn khng:

E ck Q r Q r E= . =k 2. = 2 r r 4 o r r

(9.11)

4 Nguyn l chng cht in trng: Nu cc in tch Q1, Q2, , Qn cng gy ra ti im M cc vect cng in trng E 1 , E 2 ,..., E n , th vect cng in trng tng hp ti M l:

E = E1 + E 2 + ... + E n = E ii =1

n

(9.12)

tnh cng in trng do mt h in tch phn b lin tc trn mt vt no gy ra ti im M, ta chia nh vt thnh nhiu phn t, sao cho mi phn t mang mt in tch dq coi nh mt in tch im. Khi phn t dq gy ra ti im M vect cng in trng:

dq r dq r . dE = k 2 . = 2 r r 4 o r r

(9.13)

v vect cng in trng do ton vt mang in gy ra ti M l:

E=

vat mang ien

dE

(9.14)

* Trng hp in tch ca vt phn b theo chiu di L, ta gi =

dq (9.15) d

l mt in tch di (in tch cha trn mt n v chiu di). Suy ra, in tch cha trn yu t chiu di d l dq = .d v cng in trng do vt gy ra l:

E = dE =L

1 d .r 4o r 3 L

(9.16)

* Trng hp in tch ca vt phn b trn b mt S, ta gi =

dq dS

(9.17)

l mt in tch mt (in tch cha trn mt n v din tch). Suy ra, in tch cha trn yu t din tch dS l dq = dS v cng in trng do vt gy ra l:

194

Giao Trnh Vat Ly ai Cng Tap I: C Nhiet - ien

E = dE =(S)

1 4 o

dS r 3 . r (S)

(9.18)

* Trng hp in tch ca vt phn b trong min khng gian c th tch , ta gi

=

dq d

(9.19)

l mt in tch khi (in tch cha trong mt n v th tch). Suy ra, in tch cha trong yu t th tch d l dq = .d v cng in trng do vt gy ra l:

E = dE =( )

1 4 o

d ) r 3 . r (

(9.20)

T nguyn l chng cht in trng, ta chng minh c vect cng in trng do mt qu cu tch in u gy ra ti nhng im bn ngoi qu cu cng c xc nh bi (9.9), song phi coi in tch trn qu cu nh mt in tch im t ti tm ca n. 5 Mt s v d v xc nh vect cng in trng: V d 9.1: Xc nh vect cng in trng do h hai in tch im Q1 = Q2 = Q, t cch nhau mt on 2a trong khng kh gy ra ti im M trn trung trc ca on thng ni Q1, Q2 , cch on thng y mt khong x. Tm x cng in trng c gi tr ln nht. Gii Vect cng in trng ti M l E = E1 + E 2 , vi E1 , E 2 l cc vect cng in trng do Q1, Q2 gy ra ti M. Do Q1 = Q2 v M cch u Q1, Q2 nn t (9.10) suy ra: E1 = E2 = k Do :

|Q| |Q| =k . 2 r (x 2 + a 2 )k|Q| x k|Q|x . = 2 2 2 2 (x + a ) x + a (x 2 + a 2 )3/ 2

E = 2E1cos =

(9.21)

T qui tc hnh bnh hnh suy ra E nm trn trung trc ca on thng ni Q1, Q2 v hng ra xa on thng nu Q > 0 (hnh 9.4), hng li gn nu Q < 0. tm c gi tr ln nht ca E, ta c th ly o hm (9.21) theo x ri lp bng bin thin ca E(x), t suy ra gi tr ln nht. Hoc c th dng bt ng thc

1 2 1 2 a4 3 x2. Cauchy nh sau: x + a = x + a + a 3. 2 2 42 2 2

Chng 9: IN TRNG TNH

195

(x + a )2

2 3/ 2

4 2 a 27x . 4

3/ 2

a2 = 3 3 .x 2

E E2M r Q1 x a a Q2

E=Vy:

k|Q|x 2k | Q | = const 2 2 3/ 2 (x + a ) 3 3a 2 E max = 2k | Q | 3 3a 2(9.22)

E1

khi x 2 =

1 2 a a x= 2 2

V d 9.2: Xc nh vect cng in trng do mt vng dy trn, bn knh a, tch in u vi in tch tng cng Q, gy ra ti im M nm trn trc ca vng dy, cch tm vng dy mt on l x. T kt qu hy suy ra cng in trng ti tm vng dy v tm x cng in trng l ln nht. Gii Ta chia nh vng dy thnh nhng phn t rt nh sao cho in tch dq ca mi phn t y c coi l in tch im v n gy ra ti M vect cng in k.dq trng c ln: dE = . Vect d E c phn r 2

+

+

Hnh 9.4

d En

dE

M

d Et

r xdqa O

tch thnh 2 thnh phn: thnh phn php tuyn d E n song song vi trc vng dy v thnh phn tip tuyn

d E t vung gc vi trc vng dy.Cng in trng tng hp ti M l: E = d E = d E t + d E nL L L

Hnh 9.5

V ng vi mt phn t dq, ta lun tm c phn t dq i xng vi dq qua tm O ca vng dy v do lun tn ti d E' i xng vi d E qua trc ca vng dy. Tng cp d E v d E' ny c cc thnh phn tip tuyn trit tiu nhau. Do : d E t = 0 v E = d E n = n o . dE n = n o . dE.cos = n o .L L L L

r

kdq x . 2 r L(9.23)

E =