Bai Giang LTT - Nguyen Viet Son - 22-11-2010

Gio vin: TS.

mn: v Tin cng

C1 - 108 - Bch Khoa H

- 2010 -

2

Ti tham :

1. l - Bnh Thnh , 1970.

2. Electromagnetics -John D. Krauss - 4th edition, McGraw-Hill, 1991

3. Electromagnetic fields and waves - Magdy F. Iskander, Prentice Hall, 1992.

4. Electromagnetics - E.J. Rothwell, M.J. Cloud CRC Press, 2001.

5. Engineering Electromagnetics - W.H. Hayt, J.A. Buck McGraw-Hill, 2007 (*).

6. Fundamentals of Engineering electromagnetics - R. Bansal - CRC Press, 2006 (*)

(*) http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/
http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/http://mica.edu.vn/perso/Nguyen-Viet-Son/Ly-Thuyet-Truong/

3

dung trnh:

1. tch vector

2.

3. Coulomb v

4. , Gauss, dive

5. v

6. - mi - dung

7. Cc trnh Poisson v Laplace.

8.

9. v

10. thin v trnh Maxwell

11. Sng

12. v tn sng

13. sng v

4

1: tch vector

I. V v vector.

II. Descartes.

III. Tch v - Tch c .

IV. .

V. .

5

I. V v Vector.

v : L cc 1 ( ,

m).

V : cch, gian, , , p , tch

K : t, m, E, P,

vector: L cc

m) v trong khng gian (2 3 .

V : gia

K : A, B, E, H, (c thay )

C 3 php m chnh xc 1 vector:

Descartes.

.

.

, , , ,...A B E H

6

3 vung gc nhau .

Cc theo quy .

A trong khng gian Descartes :

Giao 3 .

Xc xa, ya, za.

P l vi c cc vi phn kch

dx, dy, dz.

tch vi : dV = dxdydz

0

x

y

z

xa

za z = za

y = yax = xa

x

y

z

0

dy

dz

dx

dV = dxdydz

P

7

Xt vector r trong Descartes:

r = x + y + z

x, y, z r

Vector thnh x, y, z

vo vector r.

khng thay .

Phn tch theo cc vector .

x = xax ; y = yay ; z = zaz

r = xax + yay + zaz = rxax + ryay + rzaz

vector:

Vector theo B:

0

x

y

z

x

z

y

r

0

x

y

z

ax

az

ay2 2 2| | x y zB B BB

2 2 2 | |B

x y zB B B

B Ba

B

8

1. Tch v

A . B = |A| |B| cos AB

- |A|, |B|

- AB A v B

A . B = AxBx + AyBy + AzBz ; A . B = B . A

A . A = A2 = |A|2 ; aA . aA = 1

Xt vector B a B:

B . a = |B| |a| cos Ba = |B| cos Ba

(B . a) a B ln

a

a

B

Ba

B . a

B a

a

B

Ba

(B . a) a

B

a

9

1. Tch v

G = yax 2.5xay + 3az

.Tnh:

a. G

b. G aN

c. G aN

G(rQ) = 5ax 2,5.4.ay + 3az = 5ax 10ay + 3az

12 2

3N x y za a a a

1 1(5 10 3 ) (2 2 ) (10 10 6) 2

3 3N x y z x y zG a a a a a a a

1( ) ( 2) (2 2 ) 1.333 0.667 1.333

3N N x y z x y zG a a a a a a a a

10

2. Tch c

A x B = aN |A| |B| sin AB

N

A x B = - (B x A)

A

BAB

A B

a a a

A B

x y z

x y z

x y z

A A A

B B B

ax, ay, az x, y, z

A = 2ax - 3ay + az ; B = -4ax - 2ay + 5az

2 3 1 13 14 16

4 2 5

x y z

x y z

a a a

A B a a a

11

gc gc

hnh P ln .

z cao P so

gc.

P( , , z)

C P l :

z = const

cong = const.

t sinh = const.

Khng

12

IV. trn .

Vector trong trn: a , a , az

a : vector php = 1

a : vector php = 1

az : trong Descartes

Tnh :

a , a thay theo trong cc php

hm, tch phn theo , cc vector

a , a l hm .

a x a = azcos

sin

x

y

z z

2 2x y

yarctg

x

z z

13

IV. trn .

Xt vi c kch v cng c kch d , d , v dz

dV = d d dz

2 r.(h + r)

.r2.h

14

V.

Xy trn

Descartes: P trong khng gian xc

r cch P (tm .

gc z

P.

gc x

hnh P ln

.

C coi P trong khng gian l

giao 3 :

P(r, , )

15

V.

Xy trn

Descartes: P trong khng gian xc

:

r cch P (tm .

gc z

P.

gc x

hnh P ln

.

16

V.

Vector trong :

ar: vector php

P, c ra ngoi,

trn hnh nn = const, v

= const

a : vector php nn,

trong v

P.

a : trong trn.sin cos

sin sin

cos

x r

y r

z r

17

V.

Xt 1 vi c kch v cng :

dV = r2 sin dr d d

S = 4 .r2

V = 4/3. . r3

18

VI. cng tch vector

A A A

Grad Ax y z

x y za a a

x y z

Rotx y z

A A A

x y za a a

A A AA A

yx zAA A

divx y z

A A

- divergence)- gradient)

- rotationnel)

2 2 2

2 2 2divgrad

x y z

A A AA A

19

I.

III.

IV.

-

20

I. Khi

: l ,

c trong quy qun tnh trong chn khng, n

v qua tc khc l

mi mang .

Tnh : c tc ln cc

c phn trong khng

gian, .

Tnh : tc ln cc mi (vd:

lorenx) v lan tc .

21

I. Khi

Trong quy c qun tnh, c hai tc

( Lorentz) mang ty theo cch

trong .

FE: Thay theo tr khng vo

.

FM: tc khi .

F = FE + FM eBq

FM

v

eE FEq

cc Lorentz v

chng l khi do

xc trong quy .

22

II. Cc thng v mi mang

xy m hnh Mi mang xc

thng v m :

thi: v thi v qu trnh

tc cc thnh vin trong .

hnh vi: tnh quy cc hnh vi

trong qu trnh tc khc.

23


1. thi mang

thi mang l tch q mang .

tc tc .

v mang chia lm 2 :

mang tch m e = -1,6.10-19 (C).

mang tch .

v mang c tch khng n khng c

tc .

24


2. thi

a. Vector E:

Xt mang tch dq, trong quy c qun

tnh, dFE. Khi ta c ni ln mang c

.

Vector thi l thi v

tc Lorenx ln mang trong

: dFE = dqE

[ ][ ]

[ ]

F N Nm VE

q C Cm m

25


2. thi

b. Vector B:

Xt mang tch dq, trong quy c

qun tnh, dFM. Khi ta c ni ln mang c

.

dFM theo eF, vung gc v mang v

vung gc eB xc trong trong quy .

Ta c:

( )d dq dqvBM v BF v B e e

dldqv dq idl

dt

[T]d iBdlM v BF e e

26


3. Tnh E v B

E v B l trong

quy . thng qua E v B.

E v B theo

mang mang tnh .

Lorenz 2 thnh :

Khng :

vo quy :

( )qE MF F F E v B

qEF E

( )qMF v B

27


4. Quan tch q v Coulomb

Coulomb l tc cc mang :

tc 2 mang tch q1, q2, v

cch chng.

0

Q1, Q2

1 2

2

Q QF k

r

0

1

4k 12

0 7 2

18,854.10 /

4 10F m

c

28


4. Quan tch q v - Coulomb

Xt 2 tch cng Q1 v Q2 trong chn

khng, c xc vector r1 v r2.

F2 trn tch Q2 c:

: Cng vector R12

Q1 v Q2.

R12 = r2 r1

: Cng vector R12.

1 2

2

0 124

Q Q

R2 12F a

a12 R12

| | | |

12 2 112

12 2 1

R r ra

R r r

29


4. Quan tch q v - Coulomb

V 1: Cho tch Q1 = 3.10-4 (C) A(1, 2, 3), v tch Q2 = -10

-4 (C)

B(2, 0, 5) trong chn khng. Tnh tc Q1 ln Q2.

1 2

2

0 124

Q Q

R2 12F a

2 1 (2 1) (0 2) (5 3) 2 2r r12 x y z x y zR a a a a a a

2 2 2

12

12

1 ( 2) 2 3

1 2 2

3 3 3

R

R

1212 x y z

Ra a a a

4 4

1 2

0 12 2

0 12

3.10 ( 10 ) 1 2 2( )

4 4 .8,854.10 .3 3 3 3

Q Q

R2 12 x y zF a a a a

10 20 20 ( )N2 x y zF a a a

1 2 230( )

3 3 3x y za a a

30

III. tch

Q1 Qt

Q1 Qt

1 1

2 2

0 1 0 14 4

t

t t t

Q Q Q

R Q R

tt 1t 1t

FF a a

nguyn: V/m

Vector: - R: Q

- aR R

2

04

Q

RRE a

31

III. tch

Q

r:

ar

Q

x, y, z)

2 2 2 2 2 2 2 2 2 2 2 204 ( )

Q x y z

x y z x y z x y z x y zx y zE a a a

2

04r

Q

rE a

33

III. tch

Q1 v Q2 trong chn khng.

P

21 0

( )4 | |

nk

k

Qk

k

E r ar r

Q1

Q2

z

x

y

r1

r - r1

r

r2 r r2P a1

a2

E1

E2E(r)

1 21 22 2

0 0 2

( )4 | | 4 | |

Q Q

1

E r a ar r r r

34

III. tch

V : Cho Q1 = 4.10-9C P1(3, -2, 1), Q2 = 3.10

-9C P2(1, 0, -2), Q3

= 2.10-9C P3(0, 2, 2), Q4 = 10-9C P4(-1, 0, 2). Tnh

P(1, 1, 1).

Trong :

31 2 41 2 3 42 2 2 2

0 0 2 0 3 0 4

( )4 | | 4 | | 4 | | 4 | |

QQ Q Q

1

E r a a a ar r r r r r r r

2 2

1 1 1

| | (2) 3 3,32

( ) ( ) ( ) 2 3 2 3

| | 3,32 3,32

x x y y z z1

1 x y z x y 11 x y

1

r r

r r a a a a a r ra a a

r r| | 3,16 0,32 0.95

| | 1,73 0,58 0,58 0,58

| | 2,45 0,82 0,41 0,41

2 2 y z

3 3 x y z

4 4 x y z

r r a a a

r r a a a a

r r a a a a24,66 9,99 32,4x y zE a a a

35

IV. tch lin

Xt vng khng gian cc mang .

V : Khng gian v cathode phng

trong tivi, mn hnh CRT...

C coi phn cc mang l lin v c m phn

hm tch (C/m3).

tch trong khng gian tch V l:

0limvv

Q

v

v

V

Q dv

36

IV. tch lin

V : Tnh tch chm hnh

tch

: p cng :

tch trn:

:

510 35 /zv e C m

56 105.10 . zvV V

Q dV e dV

dV d d dz

50,04 0,012

6 10

0,02 0 0

5.10 . zQ e d d dz

5 50,04 0,01 0,04 0,012

6 10 5 10

0,02 0 0 0,02 0

5.10 . 10z zQ d e d dz e d dz

5 5

0,040,01 0,04 0,01 0,01

5 10 10 10 10 4000 1000

0 0,02 0 00,02

10 10 10z zQ d e dz e d e e d

37

IV. tch lin

V : Tnh tch chm hnh

tch

:

tch c gi l:

510 25 /zv e C m

0,014000 2000

10

0

104000 2000

e eQ

10 1 110 0,07852000 4000 40

Q pC

38

IV. tch lin

r do tch Q gy ra tnh theo

cng :

Trong :

r: vector E

: vector tch ( )

Tch phn trn l tch phn 3 l trong Descartes

2 2 2

0 0 04 | | | | 4 | | | | 4 | | | |

v vQ Qr r' r r' r r'E(r)r r' r r' r r' r r' r r' r r'

3

0

( ) '( )

4 | |

v

V

dvr'E(r) r r'

r r'

39

V. tch

Xt tia trong phng cathode dy tch c bn

knh . :

Cc .

qua sinh ra cc .

Coi tia tch c tch L (C/m)

Xt dy tch di v trn z E.

ha tnh E tch :

thay E theo cc : , , z

E = E + E + Ez, thnh no tiu.

0limLL

Q

L

40

V. tch

thay E theo cc cc : , , z

z = const

= const

= var

z

x

y

L z = var

= const

= const

z

x

y

L z = const

= var

= const

var

const

z const constE var

const

z const

const

E

var

varz const

const

E

z

x

y

L

41

V. tch

E = E + E + Ez, thnh no tiu:

vi phn di tch ra E.

vi phn di tch ra thnh E , Ez, khng

ra thnh E (E = 0).

Thnh Ez hai vi phn di trn z c

nhau v thnh Ez tiu.

E = E ( )

42

V. tch

Xt dy di v ( L) trn

z . Tnh E P(0, y, 0).

dQ = L

r

dE

dEzdE

R = r

ar

z

x

y

L

P(0, y, 0)

Vi phn dE P

do vi phn tch dQ = L tnh

theo cng :

3

0

( )

4 | |

dQd

r r'E

r r'

Trong : r = yay = a ; = az

r = a - az

2 2 3/2

0

'( ' )

4 ( ' )

Ldz zd

z

a aE

E = 0

Ez2 2 3/2

0

'

4 ( ' )

L dzdEz

43

V. tch

qut: Tnh E P( , ,z) .

2 2 3/2 2 2 20 0 0

' 1 '

4 ( ' ) 4 2'

L L Ldz zEz z

(0, 0, )

dQ = L

r

R = r ar

z

x

y

L

P( , , z)

z

02

LE a

3

0

'( )

4 | |

v

V

dv r r'E

r r'

r = a + zaz

az2 2

2 2

( ')

( ') ( ')

( ')

R z z

z z z z

z zR

R r r' a a a aa

3/22 2

0

' ( ')

4 ( ')

Ldz z z

z z

a aE

3/2 3/22 2 2 2

0

' ( ') '

4 ( ') ( ')

Ldz z z dz

z z z z

za a

E

44

V. tch

Vector a , az lun const (gi v ) khi thay .

3/2 3/22 2 2 2

0

' ( ') '

4 ( ') ( ')

Ldz z z dz

z z z z

za a

E

a , az l hm

???

3/2 3/22 2 2 2

0

' ( ') '

4 ( ') ( ')

L dz z z dz

z z z zE a a

2 2 2 2 20

1 ( ') 1

4 ( ') ( ')

L z z

z z z zE a a

0

20

4

LE a a02

LE aVector

E tch

cch.

45

V. tch

V : Xt dy tch di v song song z, x = 6,

y = 8. Tnh vector E P(x, y, z).

R

z

x

y

L

(6, 8, 0)

(6, 0, 0)

(0, 8, 0)

P(x, y, z)

(x, y, 0)

R

(6, 8, z)02

LE a

Thay

a = aR

2 2

02 ( 6) ( 8)

L

x yRE a

2 2

( 6) ( 8)

| | ( 6) ( 8)

x y

x y

x y

R

a aRa

R

2 2

0

( 6) ( 8)

2 ( 6) ( 8)

Lx y

x y

x ya a

E

46

VI. tch

tch l (vd: ) c tch phn

hm tch S (C/m2). z

x

y

0limSS

Q

S

Xt tch v c

tch S trn yOz.

S

Chia tch thnh cc tch di

v c ( 0).

Coi tch l tch .

'S SL

dS LdydQ

L L L'L Sdy

47

VI. tch

Xt P(x, 0, 0), p cng tnh E

tch :

z

x

y

2 20 0

'

2 2 '

SLdy

dR x y

RR

aE a E

S

2 2'R x yP(x, 0, 0)

2 2

0

'cos

2 '

Sx

dydE

x ydE

dEx

cosd dxE E

2 2cos

'

x

x y2 2

0

'

2 '

Sx

xdydE

x y

2 2

0 0 0

''

2 ' 2 2

S S Sx

x yE dy artg

x y x 02

SNE a

aN

48

VI. tch

02

SNE a E = E+ + E-

y

x

z

0a

S- S a

49

3

0

( ) '( )

4 | |

v

V

dvr'E r r'

r r'

02

SNE a

02

LE a

3

0

( )4 | |

QE r r'

r r'

50

VII. -

:

l hnh minh cch quan phn

trong khng gian.

Cc trong khng gian trng

vector .

pht mang v cng

mang m cho ta phn v .

V : Xt E dy di v :02

LE a

tch do trn

tch theo .

51

VII. -

:

Cho S v ln chu vi S

cc lm thnh hnh bao khng

gian, l .

S3

S2S1

cho ln v phn

E theo

.

52

- - Dive

I.

III. Dive.

IV.

53

I.

- - Dive

Th M. Faraday (1937):

khng gian (2cm) 2 dung

mi.

Hai kim tm, ngoi 2

bn c nhau.

ngoi, +Q cho trong.

ngoi v mi 2 .

ngoi.

tch trn ngoi -Q.

: tch ngoi c

tch vo trong, khng vo mi 2 .

: ( ) trong ra ngoi:

= Q

54

I.

ra trn ton tch

:

D l dng .

-Q

+Q

- - Dive

2 24 ( )aS a m

cho

ra khi vector

D [C/m2]:

24r aQ

arD a 24r b

Q

brD a24

Q

rrD a

D

D cho gi trung bnh qua

vung gc .

55

I.

Trong chn khng:

tch :

- - Dive

tch :

2

0

2

0

4

4

Q

r

Q

r

r

r

D a

D E

E a

2

04

v

V

dv

RrE a 24

v

V

dv

RrD a

56

II. Gauss

1. Pht : Thng ra kn S tch do bao

trong kn .

- - Dive

QSP

DS

DS,

S

Xt nhm cc tch bao

kn hnh dng .

tch S kn, c thng DS

qua (vector DS thay v

tr S).

l thng qua S: = DS, php cos S = DS. S

thng qua kn l (cng Gauss):

.Sd dD S = tch trong kn = Q

57

II. Gauss

1. Pht

- - Dive

S

.Sd dD S = tch trong kn = Q

nQ Q

LQ dL

S

S

Q dSV

V

Q dv

.S VS V

d dvD S

58

II. Gauss

1. Pht

Xt tch Q tm

bn knh a

- - Dive

Trn bn knh a:

Khi :2

04

Q

rrE a 0 24

Q

rrD E a

24

Q

arD a

cong dS trn c tch:

2 2sin sindS r d d a d d

thng qua :2

2

2

0 0

. sin . sin sin4 4 4

S

S S S

Q Q Qd a d d d d d d

aR RD S a a

59

II. Gauss

1. Pht

- - Dive

:

thng qua kn

tch bn trong .

2 2 2

0 0 0 00

. sin ( cos )4 4 2

S

S

Q Q Qd d d d d QD S

Th M. Faraday

Gauss..

gp to Gauss khng l

pht m l tm ra cng ton

cho ny.

60

II. Gauss

1. Pht

- - Dive

V : Tnh thng qua hnh 6

x, y, z = 5, phn tch trong hnh l:

Hai tch Q1 = 0,1 C A(1, -2, 3), v Q2 = 0,14 C B(-1, 2, -2).

p cng : .SS

d d QD S

thng qua hnh l : = Q = 0,1 + 0,14 = 0,24 C

tch L = C/m x = -2 v y = 35

5

5

5

10 31,4L LQ dz z C

61

II. Gauss

2. Gauss

D(E Q

- - Dive

.SS

Q dD S

tnh D(E) kn mn 2

( Gauss):

DS vung

.0

S

S

D dSdD S

DS = const t DS.dS

S S

S S

Q D dS D dS

62

II. Gauss

2. Gauss

Q

E.

- - Dive

kn bao quanh tch Q mn 2 trn Gauss) l

cc bn knh r, c tm trng tr tch

2 2

04 4R r

Q Q

r rD a E a

2

2 2

0 0

. sin 4S S S SQ d D dS D r d d D rD S

gi r, vector DS lun qua theo php ta c

63

II. Gauss

2. Gauss

V 2: Xt dy di v trn z

. Tnh vector E.

- - Dive

xt: D = D a v D = f( )

02 2

L LD E

. 0 0S SQ d D dS dS dSD S

2

0 0

2

z L

S S

z

Q D d dz D L2 2 2

L LS

LQD

L L

ta c:

L

L

Gauss l bao kn dy tch

64

II. Gauss

2. Gauss

V 3: Xt hai trn di v

(cp ). Bn knh trong l a, bn knh

ngoi l b. tch trong l S.

- - Dive

khc:

ab

Gauss: trn di L, bn knh a < < b, khi ta c:

2SQ D L

tch trn = a, di z = L l:

2

0 0

2 ( )

z L

S SS S S

z

a aQ ad dz aL D a bD a

1 12L S SL m L mQ S a 2

LS

a

ta c:2

LD a

65

II. Gauss

2. Gauss

- - Dive

ab

li hnh trn bn

trong ra ngoi v tch m

trong hnh trn ngoi. Do tch

trn ngoi l:

Q Q

M

,2 SQ aL

M

,2 SQ bL

, ,S S

a

b

66

II. Gauss

2. Gauss

- - Dive

ab

Gauss l hnh trn cp

c bn knh > b ( < a), ta c:

2 0b r a SD L Q Q

0 ( )SD b a

xt:

cp khng bn ngoi v bn trong

dy .

L L >> b

67

II. Gauss

2. Gauss

- - Dive

V 4: Xt cp c: L = 0,5m, bn knh li 1mm, bn knh 4mm.

li v l khng kh. tch li: 30nC. Tnh tch

trn li, ; Tnh E, D.9

2

, 3

30(10 )9,55 /

2 2 (10 )(0,5)S

QC m

aLtch :

92

, 3

30 102,39 /

2 2 (4 10 )(0,5)S

QC m

bL

Tnh vector E v vector D:

3 3

3 6, 2

10 4.10

10 (9,55 10 ) 9,55/

SaD nC m

3 3

9

1210 4.100

9,55 10 1079/

8,854 10

DE V m

68

II. Gauss

2. Gauss

- - Dive

V 5: c: tch Q = 0,25 C tm ; 2

tch tr: (r1 = 1cm, S = 2mC/m2) v (r2 = 1,8cm, S = -0,6mC/m

2). Tnh

D : r3 = 0,5cm ; r4 = 1,5cm ; r5 = 2,5cm. Tnh tch tr r6 =

3cm c D = 0 tr r7 = 3,5cm.

:

2

3 2 2

0.25( 0,5 ) 796 /

4 4 0,005

Qr cm C m

ar r rD a a a

r3 r3 Q

r4 r4 Q

r1 = 1cm, S = 2mC/m2

6 2 32

3 2 2

0,25.10 4 .0,01 .2.10( 1,5 ) 977,3 /

4 4 .0,015

Qr cm C m

ar r r

D a a a

69

II. Gauss

2. Gauss

- - Dive

V 5: c: tch Q = 0,25 C tm ; 2


2). Tnh


3cm c D = 0 tr r7 = 3,5cm.

:

r5 r5 Q

3 2( 1,5 )

4

Qr cm

arD a

6 2 3 2 32

2

0,25.10 4 .0,01 .2.10 4 .0,018 .( 0,6.10 )40,79 /

4 .0,025C mr rD a a

70

II. Gauss

2. Gauss

- - Dive

V 5: c: tch Q = 0,25 C tm ; 2


2). Tnh


3cm c D = 0 tr r7 = 3,5cm.

:

D r6 r6

6 2 3 2 30,25.10 4 .0,01 .2.10 4 .0,018 .( 0,6.10 ) 320,37Q nC

r7 = 3,5cm l

2

2 2

320,3728,33 /

4 4 0,03S

QC m

r

71

II. Gauss

2. Gauss

- - Dive

p Gauss (tnh D, E) theo cch trn tm Gauss

mn 2 : DS vung gc DS = const trn kn)

0 0 0 0

( , , )

x x y y z z

P x y z

D D DD D a a a

z

yx

z

y

x

Trong kh tm Gauss, php l kn sao

cho DS const trn kn .

Xt P(x, y, z) trong khng gian Descartes:

0 0 0 0x x y y z zD D DD D a a a

kn hnh ( x, y, z) c tm

l P: D const trn .

.S sau

Q dD S

72

II. Gauss

2. Gauss

- - Dive

0 0 0 0

( , , )

x x y y z z

P x y z

D D DD D a a a

z

yx

z

y

x

Xt :

Do P l tm hnh cch P l x/2

trong Dx0 l gi Dx P

.S sau

Q dD S

,. . x xD y z D y zD S a

, 0 0( )2 2

xx x x x

Dx xD D D theo x D

x

02

xx

DxD y z

xta c:

73

II. Gauss

2. Gauss

- - Dive

xt sau c: ,. .( )sau sau sau x x sausau

D y z D y zD S a

, 0 0( )2 2

xx sau x x x

Dx xD D D theo x D

x

02

xx

sau

DxD y z

x

Khi ta c:x

sau

Dx y z

x

xt - tri), (trn - :

yDx y z

yzD x y z

z

74

II. Gauss

2. Gauss

- - Dive

Tm :

.yx z

S

DD DQ d x y z

x y zD S

yx zv

DD DQ v

x y z

75

II. Gauss

2. Gauss

- - Dive

V 6: Xc tch tch 10-9m3

vector :

sinxxD

e yx

92 2.10 2yx z

v

DD DQ v v nC

x y z

2sin cos 2 ( / )x xx y ze y e y z C mD a a a

thin D theo cc x, y, z l:

siny x

De y

y2z

D

z

ta c

sin 0xxD

e yx

sin 0y x

De y

y2z

D

z

tch 10-9m3 l:

76

II. Gauss

2. Gauss

- - Dive

4 2 4 2 3 28 4 16 ( / )x y zxyz x z x yz pC mD a a a

:

a. Thng qua z = 2, 0 < x < 2, 1 < y < 3 theo az l:2 3 2 3

2 32 3 3 2

0 10 1 0 1

1 116 (2) 16 1365,33

3 2dxdy x y dxdy x y pCzD

V 7: Trong chn khng :

a. Tm thng qua : z = 2, 0 < x < 2, 1 < y < 3 theo az.

b. Tnh E P(2, -1, 3)

c. Tnh tch c tch 10-12m3 P(2, -1, 3).

b. E P(2, -1, 3)

4 2 4 2 3 12

12

0

8.2( 1)3 4.2 .3 16.2 ( 1)3 .10

8,85.10

146,44 146,4 195,2 /

x y z

V mx y z

a a aDE

E a a a

77

II. Gauss

2. Gauss

- - Dive

4 2 4 2 3 28 4 16 ( / )x y zxyz x z x yz pC mD a a a

:

c. tch c tch 10-12m3 P(2, -1, 3).

V 7: Trong chn khng :

a. Tm thng qua : z = 2, 0 < x < 2, 1 < y < 3 theo az.

b. Tnh E P(2, -1, 3)

c. Tnh tch c tch 10-12m3 P(2, -1, 3).

12 12 4 2 2

(2, 1,3)

(2, 1,3)

10 10 .(8 48 )yx z

P

P

DD DQ yz x yz

x y z

212,376.10Q C

78

III. Dive

pht cng :

- - Dive

.yx z

v

S

DD Dd Q v

x y zD S

.yx Sz

dDD D Q

x y z v v

D S

0 0

.

lim limyx Sz

vv v

dDD D Q

x y z v v

D S

0

.

limyx Sz

v

dAA A

x y z v

A S

0

.

lim Sv

d

Dive divv

A S

A ACng :

79

III. Dive

- - Dive

0

.

lim Sv

d

Dive divv

A S

A A

yx zDD D

divx y z

D

1 1( ) z

D Ddiv D

zD

2

2

1 1 1( ) (sin )

sin sinr

Ddiv r D D

r r r rD

80

III. Dive

- - Dive

0

.

lim Sv

d

Dive divv

A S

A A

divA ( thng vector A) l thng

ra kn tch c tch zero.

Dive l php ton c l vector, l

gi v .

Dive cho l c bao nhiu thng (trn

tch) ra kn (dive khng cho thng tin

thng .

81

III. Dive

- - Dive

V 8: Tm divD D = e-xsinyax e-xcosyay + 2zaz

:

p cng tnh div:

Gi div D = 2 = const m khng vo tr tnh.

sin sin 2 2y x xx z

DD Ddiv e y e y

x y zD

D l C/m2, khi divD l C/m3

tch .

82

III. Dive

- - Dive

V 9: Tm divD :

2 2 0 10yx z

DD Ddiv yz x

x y zD

2 2 2 2) (2 ) ( 2 ) / (2,3, 1)Aa xyz y x z xy x y C m Px y zD a a a

2 2 2 2 2 2

0

) 2 sin sin 2 2 sin /

( 2, 110 , 1)B

b z z z C m

P z

D a a a

1 1( ) z

D Ddiv D

zD

2 2 2 2 24 sin 2 cos2 2 sin 9div z zD

p cng tnh div trong trn:

p cng tnh div trong Descartes:

83

III. Dive

- - Dive

V 9: Tm divD :

2

0 0

) 2 sin cos cos cos sin /

( 1.5, 30 , 50 )C

c r r r C m

P r

rD a a a

2

2

1 1 1( ) (sin )

sin sinr

Ddiv r D D

r r r rD

p cng tnh div trong :

cos cos 2 cos6sin cos 2,57

sin sindivD

84

IV. trnh Maxwell 1 trong

- - Dive

0

.

lim Sv

d

divv

D S

Dcng div c:

vdivD

v

khc, theo Gauss c: .S

d QD S

.S

dQ

v v

D S

v

Xt vi c tch zero:0 0

.

lim limSv v

dQ

v v

D S

85


- - Dive

Cng Maxwell 1 p cho v

vdivD

Pht : Thng trn tch ra vi

gi tch

trnh Maxwell 1 coi l vi phn Gauss v:

Gauss lin gi thng tch mang

ra kn bao quanh.

trnh Maxwell 1 pht thng trn tch

ra vi (coi 1 tch .

Gauss xem l tch phn trnh Maxwell 1

86


- - Dive

V 1: Tnh tch v trong khng gian xung quanh

tch Q .

:

Vector thng D tch Q :

p cng tnh divD trong :

24

Q

rrD a

2

2

1 1 1( ) (sin )

sin sinr

Ddiv r D D

r r r rD

2

2 2

1( ) 0

4

d Qdiv r

r dr rD (r 0) 0v

v Q

87

- - Dive

x y zx y za a a

ton vector nabla l ton del)

Xt: . . x y zD D Dx y z

x y z x y zD a a a a a a

.yx z

DD Ddiv

x y zD D

88

- - Dive

pht Gauss, c:

ta c:

. vD

.S

d QD S

khc: vQ dv trong

. .S

d dvD S D

Pht : thnh php vector c

hm ring trn kn dive vector trong

khng gian trong kn.

89

- - Dive

V 1: l D = 2xyax + x2ay C/m

2 v hnh

x = 0, x = 1 ; y = 0 , y = 2 ; v z = 0, z = 3

tri:

:

. .S

d dvD S D

.S sau

dD S

2 2 23 3 3

1 1

0 0 0 0 0 0

( ) .( ) .( ) 2

y y yz z z

x x x

z y z y z y

dydz D dydz ydydzxD a

3

0

4 12

z

z

dz C

90

- - Dive


2 v hnh

x = 0, x = 1 ; y = 0 , y = 2 ; v z = 0, z = 3

tri:

:

. .S

d dvD S D

.S sau

dD S

2 23 3

0 0

0 0 0 0

( ) .( ) .( ) 0

y yz z

x x x

sau z y z y

dydz D dydzxD a

3 2 3 2 3 2

2

2 20 0 0 0 0 0

( ) .( ) .( ) ( )

z x z x z x

y y y yz x z x z x

dxdz D dxdz x dxdzD a

91

- - Dive


2 v hnh

x = 0, x = 1 ; y = 0 , y = 2 ; v z = 0, z = 3

tri:

:

. .S

d dvD S D

.S sau

dD S

3 2 3 2

0 00 0 0 0

( ) .( ) .( )

z x z x

y y y yz x z x

dxdz D dxdzD a

3 2

2

0 0

( )

z x

z x

x dxdz

92

- - Dive


2 v hnh

x = 0, x = 1 ; y = 0 , y = 2 ; v z = 0, z = 3

tri:

:

. .S

d dvD S D

.S sau

dD S

V D = 2xyax + x2ay , khng vo z D song song trn v

D.dS = 0

0

93

- - Dive


2 v hnh

x = 0, x = 1 ; y = 0 , y = 2 ; v z = 0, z = 3

tri:

:

. .S

d dvD S D

.S sau

dD S

3 2 3 2

2 2

0 0 0 0

. 12 0 ( ) ( ) 0 0

z x z x

S z x z x

d x dxdz x dxdzD S

. 12S

d CD S

94

- - Dive


2 v hnh

x = 0, x = 1 ; y = 0 , y = 2 ; v z = 0, z = 3

:

:

. .S

d dvD S D

.V

dVD

2. 2 0 2yx z

DD Dxy x y

x y z x y zD

2 23 1 3 3

0 0 0 0 0 0

. 2 2 2 4 12

y yz x z z

V V z y x z y z

dV ydV ydxdydz ydydz dz CD

95

- - Dive


2 v hnh

x = 0, x = 1 ; y = 0 , y = 2 ; v z = 0, z = 3

:

. .S

d dvD S D

. .S V

d dVD S D =12C = Qxt:

C l Dive tnh thng ra kn

tnh tch bn trong bao kn.

C 2 cch tnh:

Gauss

Dive

96

- - Dive

V 2: l D = 6 sin0,5 a + 1,5 cos0,5 a C/m2

v cong = 2, = 0 ; = , v z = 0, z = 5

:. .

S

d dvD S D

225

97

-

I.

III. -

IV.

98

-

Xt tch Q dL tc

E. Khi do tc ln tch: FE = QE

Thnh theo dL: FEL = F.aL = QE.aL

. .LdW Q dL Q dEa E L

tc tch: Ftd = -QE.aL

cng sinh ra tch Q trong

dL l:

99

-

Cng tch Q tiu :

.dW Q dE L

Q = 0, E = 0, L = 0

E vung gc dL

Cng tch Q trong qung :

.W Q dE L

Xt tch Q yn trong khng gian c E.

100

-

V : Xt khng gian c . Tnh vi phn cng

tch 6nC qung di 2 m P(2, -2, 3) theo

:

vi phn cng tch l:

:

2 2

2

18 4 4 /xyz x z x y V m

zx y zE a a a

6 3 2

7 7 7L x y za a a a

2 2

2

(2, 2,3)

18 4 4 10,67 5,33 3,56 /P

P

xyz x z x y V mz

x y z x y zE a a a a a a

6

2 2 2

6 3 2

12 6 47 7 72.107 7 76 3 2

7 7 7

Ld dL mx y z

x y z

a a a

L a a a a

9 12 6 4. 6.10 ( 10,67 5,33 3,56 )( ) 149,377 7 7

PdW Q d Jx y z x y zE L a a a a a a

101

-

1 1 2 2 6 6( ... )L L LW Q E L E L E L

Xt cng tch Q B

A trong khng gian c E.

B

A

E

E

EE

E

E

L6

L5

L4

L3L2

L1

EL6

EL5

EL4EL3

EL2

EL1

Chia B-A thnh 6 : L1, L2,

L3, L4, L5, L6

c: EL1, EL2, EL3,

EL4, EL5, EL6

Cng tch Q B A tnh theo cng :

( ... ) ( ... )W Q Q1 1 2 2 6 6 1 2 6E

BAW QE L

102

-

B

A

E

E

EE

E

E

L6

L5

L4

L3L2

L1

EL6

EL5

EL4EL3

EL2

EL1

xt: Cng tch

:

Gi tch Q

E v khng

onsA Ado c t

BA

B B

W Q d Q d QE

E L E L E L

cch v LBA (khng vo

2 B, A).

103

-

V 1: Cho khng gian vector E = yax + xay + 2az.

Xc cng tch Q = 2C B(1, 0, 1)

A(0,8 ; 0,6 ; 1) theo cong: x2 + y2 = 1, z = 1.

:A

B

W Q dE Lp cng : trong :2y xx y zE a a a

d dx dy dzx y zL a a a

2 ( 2 ) ( )

A A

B B

W Q d y x dx dy dzx y z x y zE L a a a a a a

0,8 0,6 1

1 0 1

2 2 4W ydx xdy dz

0,8 0,6

2 2

1 0

2 1 2 1 0W x dx y dy

0,8 0,62 1 2 1

1 0

1 sin 1 sin 0,96W x x x y y y J

104

-

V 1: Cho khng gian vector E = yax + xay + 2az.

Xc cng tch Q = 2C B(1, 0, 1)

A(0,8 ; 0,6 ; 1) theo cong: x2 + y2 = 1, z = 1.

:A

B

W Q dE Lp cng : trong :

2y xx y zE a a a


0,8 0,6 1

1 0 1

2 2 4

A

B

W Q d ydx xdy dzE L

0,8 0,6

1 0

6 ( 1) 2 1 0,963

yW x dx dy J

2 B A c trnh:

( )A BB BA B

y yy y x x

x x3( 1)y x

105

-

Descartes:


trn:

d d d dz zL a a a

:

sinrd dr rd r dL a a a

106

-

V 2: Xt tch L trn z trong chn khng. Tnh cng di

tch Q trn trn bn knh , tm trn z v trn

song song Oxy.

:

W Q dE L

02

LEE a a

2

002

LW Q da a2

00

02

LQ d a a

L

z

y

x

dLp cng tnh cng:

0

0

d d d dz

d

dz

zL a a a

Q

107

-

V 3: Xt tch L trn z trong chn khng. Tnh cng di

tch Q = a = b.

:

W Q dE L

02

LEE a a

0 0 0

ln2 2 2

b b

L L L

a a

Qd bW Q d Q

aa a

L

z

y

x

dL

p cng tnh cng:

0

0

d d d dz

d

dz

zL a a a

Q

a

b

108

-

-

: 2 A v B (VAB) l cng

tch 1C trong E B A.

A

AB

B

JV d V

CE L

Trong coi 1 trong c 0

( tham th cc

khc so tham chnh l ( ) chng.

VA, VB 2 A, B (chung tham ) th

A v B (VAB) tnh theo cng :

AB A BV V V

109

-

-

2

04r

QE

rE a a

V 1: Tnh 2 A, B cng trn 1 xuyn tm c

cch rA, rB trong tch Q.

c tm trng tr tch Q

Vector do Q ra:

2

0 0

1 1

4 4

A

B

rA

AB

A BB r

Q QV d dr

r r rE L

VAB l:

110

-

-

V 2: Trong khng gian c E = 6x2ax + 6yay + 4az V/m.

a. Tnh VMN M(2, 6, -1), N(-3, -3, 2)

2(6 6 4 ) ( )

M M

MN

N N

V d x y dx dy dzx y z x y zE L a a a a a a

2 6 1

2

3 3 2

6 6 4 139MNV x dx ydy dz V

b. Tnh VN P(1, 2, -4) c VP = 2

3 3 2

2

1 2 4

2 2 6 6 4 19

N

N NP P

P

V V V d x dx ydy dz VE L

111

-

minh 2 A, B cng trn 1

xuyn tm c cch rA, rB trong tch Q

tnh theo cng :

0

1 1

4AB

A B

QV

r r

B(rB, B, B)

A(rA, A, A)

E = Er.ar

dL = drar + rd a + rsin d a

rA

rB

r

Q

2 A, B

di tch Q B

A tnh theo cng :

2

0 0

1 1

4 4

A A

B B

r r

AB r

A Br r

Q QV E dr dr

r r r

112

-

B(rB, B, B)

A(rA, A, A)

E = Er.ar

dL = drar + rd a + rsin d a

rA

rB

r

Q

2 A, B

di tch Q B

A tnh theo cng :

2

0 0

1 1

4 4

A

B

r

AB

A Br

Q QV dr

r r r

2 trong tch

vo cch 2 tch

m khng vo qung 2 .

Coi rB = v VB = 0:

04

QV

r

113

-

tch cho ta cng di 1 tch 1C

tr xa v cng tham V = 0) 1 cch tch Q

r.

04

QV

r

tch l v khng c vector .

l cc c cng v do

cng tch trn lun khng.

tch l cc tm, c tm trng

tr tch .

1

114

-

V : Cho tch Q = 15nC . Tnh VP P(-2, 3, -1) v:

a. V = 0 A(6, 5, 4)

9

0 0

1 1 15.10 1 120,68

4 4 4 9 1 36 25 16PA

P A

QV V

r r

b. V = 0 v cng9

0 0

15.1036,1

4 4 4 9 1PA

P

QV V

r

c. V = 5 B(2, 0, 4)

9

0

15.10 1 15 10,89

4 4 9 1 4 0 16P PB BV V V V

115

-

Xt khng gian, 1 tch Q1. Khi

A tnh theo cng :

Q1

r1

r

r - r1

r2 r r2

A

Q2

1

0

( )4 | |

QV

1

rr r

khng gian c n tch Q1, Q2, , Qn, A l:

1 0

( )4 | |

nk

k

QV

k

rr r

Coi Qk l phn tch lin V vm:

1 0

( )( )

4 | |

nv k

k

vV k

k

rr

r r0

( ') '( )

4 | |

nv

V

dvV

rr

r r'

116

-

C tch L (dy mang di v :

C tch S tch v

0

( ') '( )

4 | |

v

V

dvV

rr

r r'

mang :

C tch V:

0

( ') '( )

4 | |

L dLVr

rr r'

0

( ') '( )

4 | |

S

S

dSV

rr

r r'

117

-

Ta c cng :

V 1: Tnh 1 trn z trong

dy trn L, bn knh a, trn z = 0

0

( ') '( )

4 | |

L dLVr

rr r'

(0, 0, z)

y

x

z

r

L

= a

2 2| | a zr r'

trong :' ' ; ; dL ad z azr a r' a

2 2| | a zr r'

2

2 2 2 20 0 0

'

4 2

L Lad aVa z a z

xt:

1 l cng sinh ra 1 tch v cng m

khng vo chng.

tch l cc do

tch nn.

118

-

khc, A tnh theo cng :

A

AV dE L

2 A, B khng vo A v BA

AB A B

B

V V V dE L

(vector khng thay

v theo gian t):

0dE L

119

-

V 2: Trong chn khng, coi v cng c 0, tnh

A(0, 0, 2) gy ra mang :

a. tch L = 12nC/m, = 2,5m, z = 0

b. tch Q = 18nC B(1, 2, -1)

9

2 2 2 2

0 0

12.10 .2,5529,4

2 2 2,5 2

LA

aV V

a z

9

1

0 0

18.1043,26

4 | | 4 1 4 9A

B

QV V

r r

120

-

C 2 cch xc gy ra mang :

Thng qua vector E (tch phn

Thng qua hm phn tch (tch phn

Tuy nhin gi vector v hm phn

tch .

Trong ta hai . Khi

xc E phn tch cc

.

php gradient

121

-

pht cng :

Xt L sao cho E = const:

V dE L

cosV E LE L

cosdV

EdL

Xt vi phn qung L:

1ax

(cos )m

dVE

dL

E gi

thin theo cch.

Gi vi phn cch

E E nhanh .

+50

+40

+60

+70

+80

V = +90

+30

+20+10

E

LP

122

-

aN l vector php cc

v c pha cc c

cao. Khi

axm

dV

dLNE a

Do dV/dL max khi dL cng aN

+50

+40

+60

+70

+80

V = +90

+30

+20+10

E

LP

aN

axm

dV dV

dL dN

dV

dNNE a

123

-

ton gradient (grad) vector T :

Gradient of T = grad TdT

dNNa

ta c:

V V VdV dx dy dz

x y z

grad VE

khc ta c: V = V(x, y, z)

x y zdV d E dx E dy E dzE L

x

y

z

VE

x

VE

y

VE

zSuy ra:

V V V

x y zx y zE a a a

V V Vgrad V

x y zx y za a a

aN l vector php cc

c theo

vector T

124

-

khc ta c

ta c:

VE

V V V

grad Vx y z

x y za a a

x y zx y za a a

T T TT

x y zx y za a a

TT grad

VgradEkhc:

Suy ra quan vector v :

125

-

VE

Descartes:

V V VV

x y zx y za a a

trn:

1V V VV

za a a

:

1 1

sin

V V VV

r r rra a a

126

-

Ch phn 2 ton

V V VV

x y zx y za a a

Gradient:

Dive:

.yx z

DD D

x y zD

Gradient v l vector

Dive vector cho ta gi v .

127

-

V 1: Xt V = 2x2y - 5z v P(-4, 3, 6). Hy tnh

E, hm D, v hm phn

tch V P.

2

( 4,3,6) ( 4,3,6)4 2 5 48 32 5 /P P P

V xy x V mx y z x y zE a a a a a a

:

P: VP = 2(-4)2.3 5.6 = 66V

Vector E P:

Hm D P:

2 3

0 35,4 17,71 44,3 / xy x pC mx y zD E a a a

Hm phn tch V:335,4 / =-106,2 V y pC mD

128

-

nghin cho php ta phn tch cc qu trnh

trong cc mi khi chng trong E.

( ) l khi 2 tch tri c

cng nhau sao cho cch chng so

cch P xt EP hay VP)

r R2

R1P

z

y

x

+Q

-Q

d

2 1

0 1 2 0 1 2

1 1

4 4

R RQ QV

R R R R

P(r, , =-900):

z = 0 R1 = R2 V = 0

1 2R R 0PV

129

-

r

R2

R1

z

y

x

+Q

-Q

d

2

0

cos

4

QdV

r

R1 R2

E

R2 R1 = dcos

2 1 cosR R d

1 1

sin

V V VV

r r rrE a a a 3 3

0 0

2 cos sin

4 4

Qd Qd

r rr

a a

3

0

2cos sin4

Qd

rrE a a

130

-

0

2

14

cos

Qd

Vr3

0

2cos sin4

Qd

rrE a a2

0

cos

4

QdV

r

0,2

0,4

0,6

0,8

1

-0,2

-0,4

-0,6

-0,8

-1

z

00

E

131

-

Momen : p = Qd [C.m]r R2

R1P

z

y

x

+Q

-Q

d

2

0

.

4V

r

rp a

r P

ar

2

0

cos

4

QdV

r

. cosdrd a

2

0

1 '.

4 | ' | | ' | V

r rp

r r r r

xt:

V do gy ra bnh

cch.

E do gy ra

cch ba.

132

-

V 1: trong chn khng, c momen

p = 3ax 2ay + az nC.m.

a. Tnh V A(2, 3, 4)

b. Tnh V B(r = 2,5 ; = 300 ; = 400)

2

0

.

4V

r

rp a

p cng :

2 2 2

2 3 4

2 3 4

x y z

r

a a aa 2 2 22 3 4 29r

3

0

(3 2 ).(2 3 4 )

4 29

x y z x y za a a a a a0,23

2

0

.

4V

r

rp a3

0

(3 2 ).(0,96 0,8 2,17 )

4 2,5

x y z x y za a a a a a1,985V

2 2 2

0,96 0,8 2,17

0,96 0,8 2,17

x y z

r

a a aa 2 2 20,96 0,8 2,17 2,5r

B(0,96 ; 0,8 ; 2,17)

133

-

V 2: trong chn khng, c momen

p = 6az nC.m. Tnh E A(r = 4 ; = 200 ; = 0)

/ : 1,584 0,288 / D S V mrE a a

134

-

di 1 tch Q2 xa v cng vo khng gian c

gy ra tch Q1 , ta cng.

Q2 nguyn: Q2 c

Q2 do:

Q2 ra xa Q1

Q2 tch trong qu trnh .

xc tch .

135

-

Xt tch Q2 trong khng gian c Q1

Q2 = Q2V2,1

cng di =

V2,1 l tr Q2 do Q1 ra

Khng gian c tch Q3

Q3= Q3V3,1 + Q3V3,2

Khng gian c tch Q4

Q4= Q4V4,1 + Q4V4,2 + Q4V4,3

WE = Q2V2,1 + Q3V3,1 + Q3V3,2 + Q4V4,1 + Q4V4,2 + Q4V4,3

136

-

ta c:

khc:

WE = Q2V2,1 + Q3V3,1 + Q3V3,2 + Q4V4,1 + Q4V4,2 + Q4V4,3

13 3,1 3 3

0 13 3 3,1 1 1 1,3

0 31

13 31

44

QQ V Q Q

R Q V Q QVR

R R

WE = Q1V1,2 + Q1V1,3 + Q1V1,4 + Q2V2,3 + Q2V2,4 + Q3V3,4

2WE = Q1(V1,2 + V1,3 + V1,4 + ...) +

Q2 (V2,1 + V2,3 + V2,4 + ...) +

Q3 (V3,1 + V3,2 + V3,4

V1,2 + V1,3 + V1,4 + ... = V1

V3,1 + V3,2 + V3,4 3

V2,1 + V2,3 + V2,4 + ... = V2

1 1 2 2 3 3

1

1 1...

2 2

N

E k k

k

W QV Q V Q V Q V

k VQ dv

1

2WE V

V

Vdv

tnh 1 mang , coi:

137

-

Cng cho php tnh tch ,

mang c hm phn tch V

1

2WE V

V

Vdv

tch

1

1

2W

N

E k k

k

Q V

Cng tnh mang c hm phn tch

V c coi l cng tnh qut cho cc mang

khc nhau:

tch

tch

138

-

Xt cng :1

2WE V

V

Vdv

p trnh Maxwell 1: V D

1( )

2WE

V

VdvD

khc: ( ) ( ) ( )V V VD D D

1( ) ( )

2WE

V

V V dvD D

1 1( ) ( )

2 2WE

V V

V dv V dvD D

p l Dive: .S V

d dvD S D

ta c cng :1 1

( ) ( )2 2

WES V

V d V dvD S D

139

-

Ta c:

ta c:

1 1( ) ( )

2 2WE

S V

V d V dvD S D

0

1

4

QV

r r

2 2

1

4

Q

r rrD a

2S

1( ) 0

2S

V dD S

1( )

2WE

V

V dvD

VETheo cng gradient :

2

0

1 1

2 2 WE

V V

dv E dvD E

140

-

V 1: Tnh cp di L,

c phn trong cp S

p cng :

abCch 1:

2

0

1

2WE

V

E dv

trong : ( )Sa

D a b0

SaE a

2 2 2 2 2

0 2 2

0 00 0

1ln

2W

L b

S SE

a

a La bd d dz

a

141

-

V 1: Tnh cp di L,

c phn trong cp S

p cng :

abCch 2:

1

2WE V

V

Vdv

0 0

ln

0

a

a aab S S

ab

b b

b

V d a a bV E d d

aV

E L

Coi cc trn ngoi cp l tham (V = 0). cc

trn trong cp l:

142

-

V 1: Tnh cp di L,

c phn trong cp S

abCch 2:

0

1ln

2W VE V

V

a bdv

a

, ,2 2

SV

t ta a t a

t

2 2 22

0 00 0

2

1ln ln

2W

ta

z L

S S SE

tza

Lab ba d d dz

t a a

Ch :

2 SQ aLtch li cp:

li cp:

0

lnSaa b

Va

2 2

0

1ln

2W SE a

La bQV

a

143

-

V 2: Tnh WE mang 2mm < r < 3mm, 0 < < 900,

0 < < 900 trong chn khng, V:

a.200

Vr

b.2

300 oscV

r

144

- -

I. -

145

I. Dng - dng

5: - Mi - Dung

Dng l dng c cc mang

thin tch theo gian qua 1 cho .

[A]dQ

Idt

dng J [A/m2] phn dng trn tch.

Dng ra S vung gc dng tnh

theo cng : I = JN S

S khng vung gc dng : I = J. S

dng qua S c dng J tnh theo cng :

S

I dJ S

146

I. Dng - dng

5: - Mi - Dung

Xt mang c hm tch V

V VQ V S L

VJ v

z

y

x

S

VQ V

L

z

y

x

S

VQ V

L

x

ha: Coi mang song

song x: x trong gian t

VQ S x

trong t, dng I qua

vung gc x l:

V V x x

Q xI S I Sv J S

t t

ta c:

147

I. Dng - dng

5: - Mi - Dung

V : Cho vector dng Tnh

dng qua trn = 3, 0 < < 2 , 1 < z < 2

2 2 210 4 cos / z A mJ a a

:

2 2 2

3 10.3 4.3cos 90 12cosz zJ a a a a

p cng :3

S S

I d dJ S J S

Ta c:3d d dz d dzS a a

Suy ra:22 2

1 0 1

270 270 2 .270 2,54

z z

S S z z

I d zd dz zd dz zdz AJ S

148

I. Dng - dng

5: - Mi - Dung

Theo : Dng ra kn cc

mang tch ln cc mang tch m).

Qi l cc mang trong kn.

l Dive:

S

I dJ S

i

S

dQI d

dtJ S

( )S V

d dvJ S J

Xt kn S:

i V

V

Q dv

( ) VVV V V

ddv dv dv

dt tJ

( ) Vv vt

J V

tJ

149

I. Dng - dng

5: - Mi - Dung

t = 1s, dng ra kn bn knh

tch :

2 A/m

te

rrJ a

1 21 4 5 23,15

ArI J S e

V : st dng :

1 21 4 6 27,76

ArI J S e

2

2 2

1 1 1 1t t tV e r e et r r r r r

rJ a

2 2

1 1( ) ( )t tV e dt K r e K r

r r

3

20

1 r C/m

V

khit

V er

Bn knh r = 5m:

Bn knh r = 6m:

tch:

2

1

1 m/s

t

rV r

tV

eJ rv r

er

J v

150

5: - Mi - Dung

nguyn :

nhn mang tch .

Cc electron mang tch m xung quanh.

Electron c nhn (v

.

Khi electron ny sang khc th

n pht) ra .

Cc electron ha c cao kch thch, thot

ra thi cn v thnh cc electron do (dng cc

electron do).

151

5: - Mi - Dung

Xt electron do trong trong E

F = - eE

Chn khng: electron lin

: electron gi trung bnh

v

d e e e

J vv E J E e [m

2/Vs]

152

5: - Mi - Dung

thay theo (VD:

nhm thay 0,4% khi 10K).

thnh siu 0) khi

00K (VD: Nhm siu t0 ~1,140K).

e e

e eJ E e: do (lun m)

Trong cc kim ta c quan :

J E [S/m]:

153

5: - Mi - Dung

Xt dy hnh c J v E

S

I d JSJ S

J = constS

E = const

L

Ta c:

b b

ab ba ab

a a

V d d V ELE L E L E L E L

IJ E

S

Suy ra:I V

S L

LV I

S

L

S

V RI

dy c tnh theo cng :

b

ab a

S

dV

RI d

E L

E S

154

5: - Mi - Dung

Xt : cc electron bn trong .

cc electron lm chng ra

v c xu tch nhau.

tch bn trong khng,

tch .

trong dng khng

trong khng (theo Ohm)

-

155

5: - Mi - Dung

Xt : Phn cch

v chn khng.

a b

cd

w

w

hS

E

Et

END

Dt

DN

Chn khng

Vector : E = EN + Et ; D = DN + Dt

Ta c: 0dE L

0

b c d a

a b c d

Trong : E = 0

, , 02 2

wt N Nh h

E E E

hh

0h

0

0

t

t

E

D

p Gauss:S

Q dD S

0 0 ; ; Nxungquanh

D S

N SD S Q S

0N S ND E

-

156

5: - Mi - Dung

a b

cd

w

w

hS

E

Et

END

Dt

DN

Chn khng

Tnh trong

hh

0

0

0

t t

N N S

y

xy

x

E D

D E

V dE L

bn trong khng.

trn vector

lun vung gc .

c tnh .

-

157

5: - Mi - Dung

V : Cho V = 100(x2 y2) v P(2, -1, 3) trn bin

v chn khng. Tnh V, E, D, S P v trnh .

P:2 2100 2 ( 1) 300PV V

Do l trn c V=300V

tch cc c V = 300V = 100(x2 y2) x2 y2 = 3

Tnh2 2100 ( ) 200 200V x y x yx yE a a

400 200 V/mP x yE a a2

0 3,54 1,771 nC/mP P x yD E a a

2

, 3,96 /N P PD nC mD

2

, , 3,96 /S P N PD nC m

-

158

5: - Mi - Dung

quan l

lun c khng c c

v v dy khng.

C thay tch v m

khng lm thay cc trn .

+Q

-Q

V = 0

+Q

V = 0

-Q

V = 0

L

L

- L

159

5: - Mi - Dung

V : Tnh tch S P(2, 5, 0)

trn z = 0 c tch

L = 30nC/m x = 0 v z = 3

30nC/m

x

z

P(2, 5, 0)

y

30nC/m

x

z

P y

-30nC/m

R+

R-

p php soi .

2 3+ x zR a a 2 3- x zR a a

9

0 0

2 330.10

2 2 13 13

L

R

x y

+ R+

a aE a

9

0 0

2 330.10

2 2 13 13

L

R

x y

R

a aE a

9

0

180.10249

2 (13) V/mz+ - z

aE E E a

2

0 2,20 /S NE nC m

160

5: - Mi - Dung

Trong cc bn c 2 mang : Electron, v

bn khi kim

IV. Bn

e e h h

J, E

Trong cc bn cc

do electron ( ) di

electron).

Cc electron vng ha

kch thch qua vng

vng .

bn :

bn ln khi c (n-type, p-type)

161

5: - Mi - Dung

Cc mi cc phn trong chn

khng.

1. Khi

Cc phn khng phn qu trnh kim

hay bn do chng tc nguyn v phn .

Khi c tc ngoi, cc phn

theo ra .

Tnh : Cc mi c tch

.

thi bnh cc phn xoay theo cc khc

nhau.

162

5: - Mi - Dung

p l vector momen : p = Qd [Cm]

1. Khi

thi nhin, cc pi nhin p khng.

pi cng tc ngoi) p kh .

Qd

E

vi phn tch v c n p momen :

1

n v

i

i

p p

Vector phn P cho momen trn tch

2

01

1lim [C/m ]

n v

iv

ivP p

163

5: - Mi - Dung

Xt mi c P = 0

1. Khi

Xt vi phn tch S tc E

tc E, phn mi c : p = Qd

ES

S

S

+

-+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

d

1cos

2d

1cos

2d

tch theo S dcos

tch cng S

tch m S

164

5: - Mi - Dung

: n phn / m31. Khi

phn theo S trong vi phn tch:

bQ P S

p Gauss cho kn:

cosbQ nQd nQS d S

Q nQp d P db

S

Q dP S

0 b

S

Q d Q QE S

0b

S

Q Q Q dE P S

0 D E P

Theo l Dive:

S V

d dvD S D

V

V

Q dv

VD

165

5: - Mi - Dung

Trong E lun cng P, khng

.

1. Khi

e : phn mi (kp)0eP E

Ta c:0 0 0 0(1 )e eD E P E E E

: phn1r e

: l mi0 rD E E 0 r

Trong E khng cng P

0

x xx x xy y xz z

y yx x yy y yz z

R

z zx x zy y zz z

D E E E

D E E E

D E E E

D E

166

5: - Mi - Dung

w

Ett1S

2

1

h

Ett2

DN2

DN1

Xt phn cch 2 mi

0dE L

1 2 0tt ttE w E w02 2

wt N Nh h

E E E

0h 1 2 tt ttE E

1 21 2

1 2

tt tttt tt

D DE Edng D: 1 1

2 2

tt

tt

D

D

Xt EN : 1 2N N SD S D S Q S

1 2 N N SD D0

1 2 S

N ND D

1 1 2 2

1 2

2 1

N N

N

N

E E

E

E

Documents

Bai Giang LTT - Nguyen Viet Son - 22-11-2010