Elect Magnet 6

8/2/2019 Elect Magnet 6

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4. 4. 4.1. , 4.1. , 4.2. 4.2. 4.3. 4.3. 4.4. 4.4. 4.5. 4.5. 4.6. 4.6. 4.7.4.7. D D


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33

4.1. , 4.1. ,

(( )):: 11 =1=1 /B /B. ,. , , , R R ,, QQ

R=1 R=1 : : C=10C=10 --1212 ; R=6370; R=6370 :: C=0,7 C=0,7 1010 --33

V V , , 2 2 , , : : Q=CV Q=CV C C::

QQ V V Q/V Q/V

QC

V

0Q Q R

C 4 RV kQ / R k


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44

2 2

QQ 2 2 d d +Q+Q QQ s s =Q/A=Q/A , ,

QC V

,

0 0

Q E

As

0

Qd V Ed A

0

0

AQ QC C

V Qd / A d

d = A

+Q -Q

+++

++

+

- --

-

-

-

-

-

-


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55

23.2 23.2 ,, +Q+Q , , bb QQ . . l l . .

:: l>a, l>bl>a, l>b , ,

E E a


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66

, , V=V V=V aa--V V bb : : 2 2 : : , , (( , , , ..., ... ))

E E r r --

l l =Q/l =Q/l

b b

b a r a a

dr bV V E dr 2k 2k ln

r al l

Q Q l C

2kQ b bV ln 2k lnl a a

C 1bl

2kln a

+

+++ ++

+++Q

-Q


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23.3 23.3The spherical The spherical capacitor capacitor

A sphericalA sphericalcapacitorcapacitorconsists of aconsists of asphericalsphericalconducting shellconducting shellof radiusof radius

bbandand

chargecharge QQ thatthatis concentricis concentricwith a smallerwith a smallerconductingconductingsphere of radiussphere of radiusaa and chargeand charge +Q+Q(Fig.).(Fig.).Find itsFind itscapacitance.capacitance.

bb QQ

,, aa +Q+Q .. . .


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:: , , kQ/r kQ/r 22

:: bb a/k=4a/k=4 00aa ..

bb b

b a r 2a aa

dr 1 1 1V V E dr kQ kQ kQ ;r r b a

a b ( b a )V V kQ ab

Q abC

V k( b a )


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4.2. 4.2. 2 2

V V : : QQ11=C =C 11V, QV, Q 22=C =C 22V V Q=QQ=Q 11+Q+Q 22=(C =(C 11+C +C 22 )V )V

33-- C C eqeq=C =C 11+C +C 22+C +C 33++

1 2eq

( C C )V QC ;

V V

eq 1 2C C C

C eq=C 1+C 2

Q1 C 1

Q2 C 2

a b a ba bV

C 1

+ _

C 2

+ _


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. .

2 2

C C eqeq=Q/V =Q/V V=Q/C V=Q/C eqeq

33--

eq 1 2 3

1 1 1 1...

C C C C

eq 1 2

1 1 1

C C C

1 2eq

1 2

C C C

C C

1 21 2

Q QV ; V C C

1 21 2

1 1V V V Q

C C

ba

C 1

+Q -Q

C 2

+Q -Q

C 1 C 2a b

V 1 V 2

V


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1111

23.4 23.4Find the equivalent capacitance betweenFind the equivalent capacitance between aa andand bb forforthe combination of capacitors shown in Fig. Allthe combination of capacitors shown in Fig. Allcapacitances are incapacitances are in m m F. F. bb -- . . -- ..:: , ,

4 4

8 8

41

3

6

2

8a b a ab b

2

4

a b

6


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1212

4.3. 4.3.

(V=0)(V=0) QQ,, V=Q/C V=Q/C V=0V=0 , , V/2=Q/2C V/2=Q/2C QV/2=QQV/2=Q 22 /2C /2C qq,, V=q/C V=q/C dqdq qq +q+q

q=0q=0 q=Qq=Q

qdW Vdq dq

C

2Q

0

q 1 QW dqC 2 C


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~C ~C ~V ~V

V=Ed V=Ed

Ad Ad

C C --

21 QU ;2 C Q CV ;

2 2 200

A1 1U ( E d ) ( Ad )E

2 d 2

2

0

U 1u E Ad 2

21 1U QV CV 2 2


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, , 1414

23.5 23.5C C 11 C C 22 (( C C 11>C >C 22 )) V V 00 . . (( )) S S 11 , S , S 22 (( )) . .

11)) aa bb . . C C QQ11=C =C 11V V 00 QQ22==--C C 22V V 00

Q=QQ=Q 11+Q+Q 22=(C =(C 11--C C 22 )V )V 002 2

a b+

+

_

_

S 1 S 2

Q1 C 1

Q2 C 2( )

ab

+

+

_

_S 1 S 2

C 1

C 2

( )

1 2

01 2 1 2

C C Q

V V C C C C


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1515

22)) . .

2 2 2i 1 0 2 0 1 2 0

1 1 1U C V C V ( C C )V

2 2 2

2 2 2

f 1 2 1 2

1 1 1U C V C V ( C C )V 2 2 2

2 2

21 2 1 21 2 0 i

1 2 1 2

C C C C 1( C C ) V U2 C C C C

2

f 1 2

i 1 2

U C C U C C


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4.4. 4.4.

QQ00 , ,

d d

QQ00 , C , C 00

V V 00=Q=Q 00 /C /C 00 , , , ,

,, V=V V=V 00 / /

0 0 0

0 0

Q Q QC V V / V 0C C

0 AC d


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d d --

::, , , , E, E, / /

11,000 ,0001,000591,0005980803,7 3,7

3,783,785,6 5,6 233233

33 10106 6

--16 16 10106 6

88 10106 6 1414 10106 6

88 10106 6


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23.6 23.6 22 3 3 . . 1 1 . . ))

. . =3,7 =3,7 ..

)) ? ? E=16 E=16 10106 6 / /

2 4 212 120

2 3

A 6 10 C 3,7 8,85 10 19,6 10 19,6

d 1 10

6 3 3max max

V E d (16 10 ) 1 10 16 10

12 3 6 max maxQ CV (19,6 10 )(16 10 ) 0,31 10 0,31

2 12 3 2 3

max max

1 1U CV (19,6 10 )(16 10 ) 2,5 10

2 2


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, , 1919

23.7 23.7

:: V V 00=Q=Q 00 /C /C 00

C C

C=C= C C 00

>1>1

QQ00 . . . . . .V 0

C 0 Q0+ -

20 0 0

1U C V

2200

0

QU 2C

20Q

U 2C 20 0

0

Q U U

2 C

C


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2020

. . . .

++ -- Fasin Fasin q q, , asinasin q q : : t t =2Fasin=2Fasin q q . F=. F=qE qE, p=2aq , p=2aq t t =2aqEsin=2aqEsin q q ==pEsin pEsin q q .. 2 2

? ? , , 22, 2 , 2 :: p=2aq p=2aq ,, -- ++ E E

q q 2 2 , , F=qE F=qE

--q

+

+q

2a

p

- -q

++q

E -F

+F

q O

p E t


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2121

++ d d q q dW=dW= t t d d q q t t =pEsin=pEsin q q , , U U . . q q 0 0 -- q q

coscosq q 00 q q 00=90=90 cos90cos90 =0=0

. .

0 0 0U d pE sin d pE sin d

q q q

q q q t q q q q q 0 0 pE cos pE(cos cos )q

q q q q

U pE cos q U p E

p E


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2222

, , , , (( ))

:: HClHCl


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2323

23.8 23.8TheThe HCl HCl molecule has amolecule has adipole moment of dipole moment of 3,43,4 1010 --3030C C mm.. A sample containsA sample contains 10102121

such molecules, whosesuch molecules, whosedipole moments are alldipole moments are alloriented in the direction of oriented in the direction of an electric field of an electric field of 2,52,5 101055

N/C N/C .. How much work isHow much work is

required to rotate therequired to rotate thedipoles from this orientationdipoles from this orientation(( q q =0=0 ) to one in which all of ) to one in which all of the moments arethe moments areperpendicular to the fieldperpendicular to the field(( q q =90=90 ))??

HCl HCl 3,43,4 1010 --3030 . . 10102121

2,52,5 101055 / / . .

(( q q =0=0 )) (( q q =90=90 )) ??

: 90 90 0

W=U 90-U 0=(-pEcos90 )-(-pEcos0 )=pE=8,5 )=pE=8,5 1010 --2525 10102121

W t =1021

8,58,5 1010--2525

=8,5=8,5 1010--44

4 6 4 6


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4.6. 4.6.

, ,

: : . : . :

0 E E


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: : E E 00== s/ s/ 0 0 : : E E ii== s s i i / / 0 0 E=E E=E 00 / / == s/ s/ 0 0

>1>1 s s i i


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23.9 23.9 C C 00 . . d/3d/3

. . . .:: 2 2 d/3d/3 ,,

2d/32d/3 , ,

2 2

0 01 2

A AC ; C

d / 3 2d / 3

1 2 0 0

1 1 1 d / 3 2d / 3C C C A A

0 A3C 2 1 d

00

AC

d

0

3C C

2 1

d

13 d 23 d

C 1

C 2

13 d

23 d

23 10 23 10


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23.10 23.10

:: 2 2

, , 2 2 (d (d--a)/2a)/2

aa d d C C .. ??

d d ,, A A. . aa

. . . .

0 01 2

1 1 1 1 1 A AC C C

( d a ) / 2 ( d a ) / 2

0 AC d a

d a

++ + ++ s

-- - -- s ++ + ++-- - --

s s

C 1

C 2


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4.7.4.7. D D

:: 2 : 2 :

, , ::

E E : :c c -- , , ,, c >1 c >1

q , , s

0 E E E

0 E E

ii

1 P p

V

0 P E c

0 P E c


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P P P P P P : , : , P P E E (( ))E E

D D D D 2 2 , ,

in P dA q

0 in E dA ( q q )

0 in E dA P dA q

0 D E P

in D dA q


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-- 1 c 1 c >1 >1 ,, 1 1 ,, 81 81 D D , , , , D D : : D D

0 D E P,

in D dA q

0 0 0 0 D E P E E (1 )E; c c

0 D E

D

in D dA q

Elect Magnet 6

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