bai giang ltt - nguyen viet son - 22-11-2010
DESCRIPTION
Lý thuyết trường điện từTRANSCRIPT
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Gio vin: TS.
mn: v Tin cng
C1 - 108 - Bch Khoa H
- 2010 -
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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
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4
1: tch vector
I. V v vector.
II. Descartes.
III. Tch v - Tch c .
IV. .
V. .
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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
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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
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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
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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
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11
gc gc
hnh P ln .
z cao P so
gc.
P( , , z)
C P l :
z = const
cong = const.
t sinh = const.
Khng
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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
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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, , )
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15
V.
Xy trn
Descartes: P trong khng gian xc
:
r cch P (tm .
gc z
P.
gc x
hnh P ln
.
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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
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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
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19
I.
III.
IV.
-
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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 .
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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 .
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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.
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23
II. Cc thng v mi mang
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 .
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24
II. Cc thng v mi mang
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
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25
II. Cc thng v mi mang
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
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26
II. Cc thng v mi mang
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
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27
II. Cc thng v mi mang
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
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28
II. Cc thng v mi mang
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
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29
II. Cc thng v mi mang
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
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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
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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
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32
III. tch
Q
).
P(x, y, z)
3/22 3 2 2 20 0 0
( ') ( ') ( ')( )
4 | | | | 4 | | 4 ( ') ( ') ( ')
y zQ x x y y z zQ Q
x x y y z z
xa a ar r' r r'E
r r' r r' r r'
| |
'
| |
R
R
r r'
R r r r r'a
r r'
-
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
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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
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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
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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
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.
:
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
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.
:
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
IV. trnh Maxwell 1 trong
- - 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
IV. trnh Maxwell 1 trong
- - 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
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 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
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
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
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
V D = 2xyax + x2ay , khng vo z D song song trn v
D.dS = 0
0
-
93
- - 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
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
V 1: l D = 2xyax + x2ay C/m
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
V 1: l D = 2xyax + x2ay C/m
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
d dx dy dzx y zL 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:
d dx dy dzx y zL a a a
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