dien_dong_luc_hoc

8/7/2019 Dien_dong_luc_hoc

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Bi tp in ng lc hc. Trng Vn Thanh. http://truongthanh85.violet.vn

1. Biu din rot, grad, div trong h ta cac v chng minh rng :

R=0

D Aur

=0

Bai gii: Ta c: Grad ur

=u u u

i j kx y z

+ +

urr ur

R Aur

= =

D Aur

= + +

1. R=

= iy

r

jz

r

+ kx

r

u u u

y z xi j k

z x y

urr ur

= 2 2 2 2 2 2. . . . . .u u u u u ui j k

y z z y z x x z x y y x

+ +

urr ur =0

2. D Aur

=A AA AA Ay yx xz z

x y z y z x z x y

+ +

=2 22 22 2

. . . . . .

A AA AA Ay yx xz z

x y x z y z y z z x z y

+ +

=0

2. Tnh div .I R r ur

? Trong Ir

l vector khng i, Rur

l bn knh vector.

Bai gii:

Ta c: x y zI iI jI kI = + +r r r r

v R xi y j zk = + +ur r r r


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. x y z

i j k

I R I I I

x y z

=

r r r

r ur

= ( ) ( ) ( )y z x z x yi I z I y j I z I x k I y I x + r r r

( ) ( ) ( ).

0

y z x z x y

x y z

div I R I z I y I z I x I y I xx y z

y z z x x yI I I

z y x z y x

= +

= + + =

r ur

3. Tnh .div I R M r ur uur

, Trong I va Mr uur

# l vector khng i, Rur

l bn knh vector.

Bai gii:

Ta c: x y zM iM jM kM = + +uur r r r


.M R uur ur

= ( ) ( ) ( )z y z x y xx y z

i j k

x y z i yZ zM j xZ zM k xM yM

M M Z

= +

r r r

r r r

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

. x y z

z y z x y x

y y x z z x

x y x z z y

x z x y z y

i j k

I R M I I I yZ zM xZ zM xM yM

i I xM yM I xZ zM

j I xM yM I yZ zM

k I xZ zM I yZ zM

=

=

+

r r r

r uuruur

r

r

r

( )

.

2

2

y y z z x x x x z z y y

x x y y z z

div I R M I M I M I M I M I M I M

I M I M I M

IM

= + + + + + = + +

=

r uur uur

W


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4. Tnh:

( ){ } .Rrot U R va rot I R ur r ur

#

Trong :

Ir

la vecor khng i va Rur

la bn knh vecor.

Bai gii:

Ta c: ( ) ( ) ( )R RU R U xi y j zk = + +ur r r r

( ) ( ) ( )R RU R U xi y j zk = + +ur r r r

( ){ }

( ) ( ) ( )

( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )

( ) ( )

( ) ( )

( )

R R

R R R

R R R

R R

i zU yU y zi j k

rot U R j zU xU x y z x z

xU yU zU k yU xU

x y

U R U RR Ri z y

R y R z

U R U RR Rj z x

R x R z

U R Rk y x

R x

= = +

=

+

rr r r

ur r

r

r

r

r ( )

( ) ( )1

U R R

R y

U R R R R R R Rz y i z x j y x k

R y z x z x y

= +

r r r

a. Mc khc, ta c:2 2 2 2

; ;R x R y R z

R x y z x R y R z R

= + + = = =


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( )2

R R y zz y z yy z R R

R R x z z x z x

x z R R

x yR R

y xy x R Rx y

=

T () v (), ta c: ( ){ } 0Rrot U R = ur

Ta c:

x y zI iI jI kI = + +r r r r


( ) ( ) ( ). x y z y z x z x yi j k

I R I I I i I z I y j I z I x k I y I x

x y z

= = +

r r r

r ur r r r

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

2

x y x z

x y y z

y z x z x y

x z y z

x x y y z z

x y

i I y I x I z I xy zi j k

rot I R j I y I x I z I yx y z x z

I z I y I z I x I y I xk I z I x I z I y

x y

i I I j I I k I I

iI jI

= +

+

= + + + + +

= +

rr r r

r ur r

r

r r r

r r

( ) 2zkI I+ = r r

W

5. Tnh:3

PRgrad

R

urur

Trong : , Pur

la vecor khng i , Rur

la bn knh vecor.

Bai gii:

Ta c:( )

33 2 2 2 2

x y zPR xP yP zP

R x y z

= + + = + +

urur


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( ) ( ) ( )

( )

( )

( )

3 3 332 2 2 2 2 2 2 2 22 2 2

3

6

3

6

3

3.2

2

3.2

2

3.2

2

x y z x y z x y z

x x y z

y x y z

z x y z

xP yP zP xP yP zP xP yP zP PRgrad i j k

R x y z x y z x y z x y z

P R xP yP zP R x

i R

P R xP yP zP R yj

R

P R xP yP zP Rk

+ + + + + + = + + + + + + + +

+ +

+ +=

+ +

ururr r r

r

r

r

( ) ( ) ( )

( )

6

3 5

3 5

31

3

x y z

z

R

PR xi y j zk iP jP kP R R

PRPR

R R

+ += + +

=

urur r r r

r r r

urururur

W

6: Tnh thng lng ca bnknh vector R

urqua mt mt tr c

bn knh a v chiu cao h,t nh hnh v ( Tnh bng cngthc O G v bng phng phptrc tip).

Bai gii:

Tnh bng nh l O G:nh l O G:

S V

Rd S div RdV = ur ur ur

Ta bit:yx z

RR RR xi y j zk div R

x y z

= + + = + +

ur r r r ur

23 3

S V

Rd S dV a h = = ur ur

Tnh trc tip:


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( )1 2 3

1S S S S

Rd S Rd S Rd S Rd S = + + ur ur ur ur ur ur ur ur

( )

( )

1 1

2 2

3

3

2

1 1

1

2

2 2

2

cos

cos

0

2. 2

0

S S

S S

S

S

S

Rd S R dS

Rd S R dS

Rd S vi R S

hR dS hS h a

R

hR dS hS h a

R

=

= =

= == = =

ur ur

ur ur

ur ur ur ur#

T () v (), ta c:2

3S

Rd S a h= ur ur

7. Hai vng trn mng, bnknh cng bng , tch in u v


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xp t nh hnh v. in tch vnh

ngoi 1O l 1e in tch vnh ngoi

2O l 2e . Cng cn thit a

in tch e t v cc n 1O v 2O

ln lt l 1A

v 2A

. Tnh cc intch 1e v 2e

Bai gii:

- Xet vanh 1O :

Th v hng ti 1O gy bi 2 vanh in tch 1e v 2e vi khong cch v 2 2R a+

1

11

22

21 2 2

1

.4

14.

4

r r

e

Re drEdl

er

R a

== = = +

ur r

Theo nguyn l chng cht in trng ta c:

- Th vng hng ti O do1

e v2

e gy ra l:

1 21 11 212 2

1

4

e e

R R a

= + = + +

Vi 1 110 O = = = l hiu in th ti 1O

in th 1 bng cng dch chuyn in tch dng t 1O

( )1 1 21 2 21

14

A e e

e R R a

= = +

+

- Tng t i vi vanh2

O :

( )2 2 112 2

12

4

A e e

e R R a

= = +

+

T () v (), ta c:


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( )

( )

2 2 2 2

1 1 22

2 2 2 2

2 2 12

4

4

e R R a A R a A Rea

e R R a A R a A Rea

= + +

= + +

8. ng nh l O G tnh in trng trong v ngoi mt qu cu bn knh , tch inu vi mt in tch mt cont. Hng in mi trong v ngoi qu cu u bng

Bai gii:

- Theo nh l O G ta c:divD

divED E

= ==

urur

ur ur

- Xt trong h ta cu, do in tch trong qu cu phn b u: ( )E E r=ur ur

( ) ( )2 2 223

2

1

3.

3 3

r

divE r E r E r r r r

E rr

r E E r

= = = = = =

ur

uur r

- Xt nhng in tch ngoi qu cu:S

Dd S e=ur ur

D Dur

cng phng vi vector php tuyn ca mt cu, ta c:

2 3

3

33 3

2 2

44

3

3

3 3

S

N

Dd S D r RR

E rrR R

D Er r

= = = = =

ur ur

uuur r


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9. Tnh in dung ca mt t in cchiu di bng d v khong cch giahai bn cha hai in mi khc nhau.

Bai gii:

- Theo nh l O G ta c:

1 2 3S S S S

Dd S q Dd S Dd S Dd S = = + + ur ur ur ur ur ur ur ur

V D r=ur r

1 2 0S SDd S Dd S = =

ur ur ur ur

( )1 23

.2 .

2 . 2 . .

S

Dd S q D r d q R r R

q qD E

r d r d

= =

= =

ur urp p

Mt khc, ta c: E grad = 1

ln2

qr C

d

= +

Vi

1 111

1 1 11 1

1

1

ln2

ln2

ln2

R

R a

a

qR C

d q aR r a

q d Ra C

d

= + = = = +

p p

1 112 2

2 1 12

1

2

ln2ln

2ln

2

R

a R

a

q

R Cd Rqa r R

q d aa C

d

= + = =

= +

p p

Vy hiu in th gia hai bn t l:


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21 2

1 1 2

1 1ln ln

2

Rq a

d R a

= = +

in dung ca t l:

2

1 1 2

21 1

ln ln

q dCRa

R a

= = +

Bai gii:

Ta c:

10. Mt hnh cu bn knh , tchin u trn b mt ngoi vi mt bng v quay quanh trc ca n vivn tc . Tnh cm ng t bn tronghnh cu?


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( )

0 .0 04 . 1

4 4.

dSA i R d SImatr A dS RS

r rS Si V R

= = =

= =

uuuuuuuuurur r ur ur ur

ur ur

r ur ur ur

Theo nh l O G : . .n dS dV VS

= ur

( )21 4 43.

3 3 3

dS rdV r r

r r rVS = = =

uuur rr

T () v (), ta c:

40 0. .4 3 3

0 .3

RA R r r

RB rot A rot r

= =

= =

ur ur r ur r

ur ur ur r

( ) ( )

( )

( )

0 0

. . . . .

23 , . .

3. . 0, 0

rot r divr r r div r

divr r B R

r div r

= + = = = = =

ur r rur ur r r ur ur r

rur ur ur r ur uur

r ur ur r

11.Mt mch dao ng gm cun t cm L v mt t in phng c in tch mi bn bng Smi trng gia cc bn c dy bng d v hng in mi . Tnh chu k dao ng ca mc

, cho bit L . H S 5 2cm =1 02 = R=0

Bai gii:

- Cng in trng do mi bn gy ra: E

=

Ta c: E gradd

d Edxdx

= =

0 0 0

d d d

d Edx dx d

= = = =


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in dng ca t C l:

q s sC

dd

= = =

V dng in chy trong L, C, c , coi khng c th in ng ngoi lai : ( ) 0n =

Ta c:

( )

22

2

2

1 10

0

.sin .

d II if

dt LC LC

I I

I A t

+ = =

+ =

= +

Trong : Tn dng:

2 12 .

2 . 2 . .

fT LC

sT LC L

d

= = =

= =

Thay , ta c:

( )2

50

3 9

2 0,1.5.10 .22 . . 2 9, 4.10 sec

10 .4 .9.10

sT L

d

= = =

12. Chng minh rng vi cc ng phng n c, nu th vector Aur

tha mn phng trnh

( )0.expA A i kr t = ur ur r r

th cc vector in trng v t trng tha mn cc h thc:

.B i k A va E i A = =

ur r ur ur ur#

Bai gii:

- Ta c: B rot A=ur ur


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1

34 :

R re rE

r e dien tich hinh cau ban kinh R

=

rur

-

22 2 2

1 2

11 1 1

1 2

3 2

2 1

14 4 4

1 1

4

RR R R

R R

RR R R

R R

e r e eU Ed r d r d r r r r

eU

R R

= = = =

=

rur r r r

-

( )1 2

1 2

1 22 1

4

1 11 1

4

R R

e eC R R

U e

R RR R

= = =

pW

. 1 2,C C ; 1 2,e e ss

s

- N

2 2

1 11 2 2

1 1 1 21 22 2

1 22 22

2 2

1 1W

2 2 1W W W

21 1W

2 2

q e

C C e e

C Cq e

C C

= = = + = +

= =

- Nss

( )22

1 2 ' 1 2

1 2 1 2

1 1W2 2

C C C e eee e e C C C

= + + = == + +

-


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( )

( )

( )

2 2 2' 1 2 1 2

1 2 1 2

2

1 1 2 2

1 2 1 2

1 1W W W

2 2

10

2

e e e e

C C C C

e C e C

C C C C

+ = = + +

=

+

- Vytm

j

- s

-

1


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1

2

1 1 1 1 1. 22

C

jRH dl I j S RH j R H = = = =uur uur

2

D ( )1 ,H j ruur r r

1

1.

2H j r = uur r r

j

1O M r=uuuur r

'

2

1.

2H j r =

uruuur r

-

( )

( )

1 2

0 0 1

1. '

2

1. OO

2

H H H j r r

H j r r

= + =

= =

uur uur uuur r r r

uur r ur ur uuuur

1 3

.

4

j rIH dV

r

=

r ruur


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Theo nh l O-G:


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i u ki n lin t c:


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i u ki n lin t c:


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