primena gausovog zakona zadaci

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  • 7/24/2019 Primena Gausovog Zakona Zadaci

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    1. a ab 2 , QQa QQb 2 ( 1).

    .

    r . rE r .

    1 1.1 1.2

    ________________________________________________________________________________

    .

    . .

    ( 1.1):

    I) a ( ar ),II) a b ( bra ),III) ( b , br ).

    r( ) :

    240cos rESEdSEdSEdSESdE

    SSSS

    ,

    .

    S dS S Sd

    , (

    ) , Sd

    .

    :

    0

    24

    QrE ,

    204 r

    QE

    .

    I) ar

    . , ( 1.2)

    S ,

    - .

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    ar ,

    , 0Q , :

    20

    20 4

    0

    4 rr

    QE

    ,

    0E .

    ar , :

    0)0( arE .

    II) bra

    a , QQa . :

    QQ ,

    , .

    , :

    204 r

    QE

    .

    ar br :

    204

    )0(a

    QarE

    ,

    )0(4

    1

    4)0(

    20

    arEb

    QbrE

    .

    III) br

    r, br , :

    QQQQ 32 .

    :

    204

    3

    r

    QE

    ,

    br , :

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    )0(4

    3

    4

    3)0(

    20

    arEb

    QbrE

    .

    ,

    ,

    .

    , , ,

    .

    III) br

    r

    Qdr

    r

    QdrE

    rr 02

    0 4

    3

    4

    3

    .

    ,

    ,

    rrr

    drEdrEdlE .

    br :

    a

    Q

    b

    Qbr

    00 8

    3

    4

    3)0(

    .

    II) bra

    ,

    :

    148

    311

    48

    3

    4 00002

    0 r

    a

    a

    Q

    a

    Q

    br

    Q

    a

    Qdr

    r

    QEdrEdrrdE

    b

    rb

    b

    rr

    .

    :

    a

    Q

    b

    a

    a

    Q

    br 00 8

    3

    14)0(

    ,

    a

    Qar

    02)0(

    .

    I) ar

    .

    , 2

    04 r

    QE

    ,

    204

    3

    r

    QE

    :

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    a

    Qardr

    r

    Qdr

    r

    QdrdrEdrEdrEdrE

    bb

    b

    a

    a

    r

    b

    a

    a

    rr 02

    02

    0 2

    )0(4

    3

    40

    .

    ( ar ) , .

    . ,

    , . ,

    .

    1.2 1.3,

    .

    1.2 1.3

    . ar ar , bra bra ,

    br br .

    ,

    .

    2. , a b

    QQa 2 QQb ( 2).

    r . .

    : cm4a , cm6b , nC6Q .

    2

    ________________________________________________________________________________

    .

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    r , .

    .

    :

    I) ar

    , , 0Q , 0E .

    II) bra

    QQ 2 ,

    0

    2 24

    QErEdSSdE

    SS

    ,

    0

    2 2

    4 Q

    Er ,

    :

    202 r

    QE

    .

    III) br

    QQQQ 2 ,

    0

    24

    QErEdSSdE

    SS

    ,

    204 r

    QE

    .

    : a

    b .

    , . :

    V90011

    211

    22 0

    02

    0

    ba

    Qkba

    Qdr

    r

    QEdrldEU

    b

    a

    b

    a

    b

    a

    ab

    ,

    gde jeF

    m109 90 k .

    3. , a ab 2 ,

    QQa QQb 5 ( 3). .

    .

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    3

    ________________________________________________________________________________

    :

    I) ar

    ,

    , 0Q , 0E , 00 arE .

    II) bra

    QQQ a ,

    0

    24

    Q

    rEdSEEdSSdE

    SSS

    ,

    204 r

    QE

    .

    (

    )

    :

    2

    040

    a

    QaE

    ,

    041

    1640

    20

    20

    aEa

    Q

    b

    QbE

    .

    III) br

    QQQQ ba 4 ,

    QrEdSEEdSSdE

    SSS

    44 2

    ,

    20rQE

    . 3.1

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    , .

    rEE

    . rE r .

    3.1

    .

    , , .

    ,

    .

    :

    2

    02

    0 40

    a

    Q

    b

    QbE

    .

    :

    III) br

    r

    QEdr

    r 0

    ,

    a

    Q

    b

    Qbr

    00 20

    .

    II) bra

    aar

    Q

    a

    Q

    br

    QEdrEdr

    b

    b

    r

    2

    2

    11

    42

    11

    4 000 ,

    ar

    Q

    2

    51

    4 0 .

    022

    5

    2

    1

    40

    00

    br

    a

    Q

    aa

    Qbr

    ,

    a

    Q

    aa

    Qar

    00

    8

    3

    2

    51

    40

    .

    I) ar

    a

    Qardr

    r

    Qdr

    r

    QdrdrEdrEdrEdrE

    bb

    b

    a

    a

    r

    b

    a

    a

    rr 02

    02

    0 8

    3)0()(

    40

    .

    3.2 3.3, .

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    3.2 3.3

    4. , a b ( a

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    , .

    ( ar , bra ,

    br ).

    I) ar

    , 0Q , 0E (

    ).

    II) bra

    22

    0

    2

    0

    24 aradrrr

    adVQ

    r

    aV

    ,

    24rESdE

    S

    ,

    2

    0

    220

    0

    2202

    2

    24

    r

    araE

    araEr

    .

    III) br

    22

    0

    2

    0 24 abadrrr

    a

    dVQ

    b

    aV

    ,

    2

    0

    220

    0

    2202

    2

    24

    r

    abaE

    abaEr

    .

    2

    1

    bar

    2

    0

    220

    2 r

    araE

    .

    r 1r :

    0

    0210

    2210

    194

    21

    araraEE rr .

    bar 2 ,

    20

    220

    2 r

    abaE

    , r 2r

    :

    0

    0220

    220

    23

    4

    22

    a

    r

    abaEE rr

    .

    5. , cm4a cm1b , m3d ( bad , ) (5).

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    ?

    nC10Q .

    5 5.1

    ________________________________________________________________________________

    ,

    Q . . , a a

    aQ , b bQ .

    .

    ,

    :

    QQQ ba .

    bad , ,

    .

    .

    (.. ). .

    ( 0E (A B), a b ,

    0B

    Edr ):

    ba ,b

    Qk

    a

    Qk ba 00 .

    :

    bba QQb

    aQ 4 , ba QQQ , bQQ 5 ,

    nC25

    Q

    Qb , nC85

    4

    QQa .

    , :

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    20x

    QkE aa ; 20 )( xd

    QkE bb

    .

    ( ba EE ):

    2020 )(4

    xdQk

    xQk bb

    ,

    0483 22 ddxx ,

    :

    m2

    3

    2

    dx . ( dx 2 ).

    6. , cm3a , cm1b cm4c , ( 6).

    c nC40Q , a , :

    ) ;

    ) .

    6

    ________________________________________________________________________________

    a) , , . Q ,

    :

    cba QQQQ .

    ,

    cba ;c

    Qk

    b

    Qk

    a

    Qk cba 000 ;

    c

    Q

    b

    Q

    a

    Q cba ,

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    ( 2: 2Srang ).

    bQ cQ aQ ,

    :

    ab Qa

    bQ , ac Qa

    cQ , aQa

    c

    a

    bQ

    1 ,

    :

    nC15aQ ; nC5bQ ; nC20cQ .

    b)

    , :

    m

    kV15020 a

    QkE aa ; m

    kV45020 b

    QkE bb ; m

    kV5.11220 c

    QkE cc .

    7. , cm1a , cm4b cm5c , nC100Q .

    ( 7). a

    (), :

    ) ;

    ) :

    ) .

    7

    ________________________________________________________________________________

    .

    ) nC50cQ ; nC10aQ ; nC40bQ .

    )m

    MV9.0

    20 a

    QkE aa ;

    m

    MV225.0

    20 b

    QkE bb ;

    m

    MV18.0

    20 c

    QkE cc .

    ) kV90 aEaQk aa

    cba .

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    8. a , Q ,

    b bc ( 8). . ,

    r .

    rE r . : .

    8 8.1

    ________________________________________________________________________________

    ar0 crb . ,

    ( 8.1).

    , , ,

    . , .

    .

    0

    QSdE

    S

    ,

    2

    4rEdSEdSESdESSS

    ,

    0

    24

    QrE .

    ar , 0Q , , , 0E .

    ( ).

    :

    bra : QQ 204 r

    QE

    ;

    crb : 0 QQQ 0E ;

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    cr : QQQQQ 204 r

    QE

    .

    :

    cr :r

    Q

    r

    Qdr

    r

    QdrE

    rrr 002

    0 4

    1

    44

    ;c

    Qc

    04 .

    crb : cc

    c

    rr

    drr

    QdrrdE

    2

    040

    ;c

    Qcb

    04 .

    bra : drr

    Qdrdr

    r

    QdrEdrEdrEdrE

    c

    c

    b

    b

    rc

    c

    b

    b

    rr

    2

    02

    0 40

    4 ,

    cbr

    Q

    c

    Q

    br

    Q 111

    44

    11

    4 000 ;

    cba

    Qa

    111

    4 0 .

    ar : drr

    Qdrdr

    r

    QdrdrEdrEdrEdrEdrE

    c

    c

    b

    b

    a

    a

    rc

    c

    b

    b

    ar

    a

    r

    2

    02

    0 40

    40

    ,

    cba

    Qa

    111

    4 0 .

    8.2 8.3.

    8.2 8.3

    9. , a ab 2 , , qq 1 qq 32 ,( 9). r )(rE )(r .

    arp 4 .

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    9

    9.1

    ________________________________________________________________________________

    , .

    I) ar0 , , , 0E , , 0)0( arE .

    II) bra , , r L ( 9.1).

    ( S

    SdE

    ) - .

    ( GBS

    DBS

    )

    2

    . ,

    .

    .

    :

    LrEdSEdSEdSESdESdESdESdE

    SSSS SSS

    20cos2

    cos2

    cos

    0DBGBGB 0DB

    0

    0

    DB

    0

    GB

    .

    :

    LqLqdlqQ

    L

    ''' 1 .

    :0

    2

    Lq

    ErL

    ,

    r( ):

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    r

    q

    r

    qE

    00

    1

    22

    .

    , :

    a

    q

    arE02

    )0(

    , a

    q

    b

    q

    brE00 42

    )0(

    .

    III) br , LqLqqdlqQL

    '2)''(' 21 ,

    :

    r

    q

    r

    qqE

    00

    21

    2

    .

    :

    a

    qbrE

    02)0(

    .

    : pp r

    r

    r

    r

    rErE dd

    .

    III) br , :

    r

    aq

    r

    rqr

    r

    qrE

    p

    r

    r

    r

    r

    pp

    4lnlndd

    000

    ,

    ,

    2ln)0(0

    q

    br

    .

    II) bra , :

    pr

    b

    b

    rrr

    q

    rr

    q

    dd2 00 ,

    :

    2ln2

    ln2 00

    q

    r

    aq

    .

    , ,

    :

    )0(2ln)0(0

    brq

    br

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    :

    2ln2

    2ln2ln2

    )0(000

    qqq

    ar

    .

    I) ar0 , :

    pr

    b

    b

    a

    a

    r

    rr

    qr

    r

    qr dd

    2d0

    00 ,

    2ln2

    )0()0(0

    q

    arar

    .

    ,

    9.2 9.3:

    9.2

    9.3

    10. a , q , b

    bc ( 10). . r .

    : cm20a , cm60b , cm65c , m/nC100q .

    10

    ________________________________________________________________________________

    ,

    ( q ),

    ( q ). ,

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    , ,

    .

    , r

    h , ,

    .

    , :

    I) ar , 0Q , , . 0E .

    II) , bra ,

    hqQ , :

    hrEESSdE oSo

    2

    ,

    :

    0

    2

    hq

    hrE

    ,

    r

    qE

    02

    .

    III) , crb , 0 hqhqQ , 0E .

    IV) , cr , hqhqhqhqQ , :

    r

    qE

    02

    .

    , ,

    10.1.

    10.1

    11. , ,

    ( L ) , a

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    ( 11).

    ( r), 2

    2

    0a

    r ,

    0 . arp 2 .

    11 11.1________________________________________________________________________________

    ,

    L , r . ,

    .

    , ,

    .

    ,

    .

    I) ar0 , , :

    VS

    dVSdE 0

    1;

    r

    SSS

    drrLa

    rEdSEdSEdS

    ODBGB 0

    2

    2

    0

    0

    00

    21

    0cos2

    cos2

    cos

    ;

    r

    drra

    LLrE

    0

    3

    20

    022

    ;

    4

    4

    20

    0 r

    aEr

    ;

    2

    0

    30

    4 a

    rrE

    .

    II) , ar , :

    VS

    dVSdE 0

    1;

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    a

    SSS

    drrLa

    rEdSEdSEdS

    ODBGB 0

    2

    2

    0

    0

    00

    21

    0cos2

    cos2

    cos

    ;

    a

    drra

    LLrE

    0

    3

    2

    0

    022

    ;

    4

    4

    20

    0 a

    aEr

    ;

    r

    aE

    0

    20

    4

    .

    :

    ar :

    r

    radr

    r

    a pr

    r

    p

    ln44 0

    20

    0

    20

    ;

    r

    aa 2ln

    4 0

    20

    ;

    2ln4 0

    20

    aar .

    ar0 :

    pr

    a

    a

    r

    drr

    adrr

    a 0

    203

    20

    0

    44

    ;

    a

    rara

    a

    pln

    444 0

    20

    44

    20

    0

    ;

    2ln41

    16 4

    4

    0

    20

    a

    ra

    ;

    2ln41160 0

    2

    0

    ar ;

    2ln4 0

    20

    aar .

    11.2 11.3

    .

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    11.2 11.3

    12. , mm1a

    mm718.2b ,

    4r .

    )

    ?

    )

    kV2U ?________________________________________________________________________________

    ) , ,

    .

    .

    ,

    ( 1C ), ( 2C ), :

    a

    bC

    ln

    2 r01

    ;

    a

    bC

    ln

    2 02

    .

    :

    pF/m8.1663 212 CCCC .

    , .

    .

    ) kV2U , .

    1q 2q , .

    :

    UCq 11 , UCq 22 ;

    nC/m6.3333 212 qqqq .

    :

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    211

    2

    1UCW , 222

    2

    1UCW ,

    J/m106.333 6

    12 WWW .

    13. , C100Q ,

    5r . , -.

    mMV20krE .

    ________________________________________________________________________________

    ,

    .

    5r . ,

    , 0 , , 0r .

    :

    ar

    r

    Q

    ar

    E

    r

    ,4

    ,0

    20

    .

    . ,

    , ar :

    2

    0

    max4 a

    QEarE

    r

    a

    .

    .

    , .

    .

    ,

    .

    .

    mMV3Vkr E , , , ,

    ...

    ,

    , (

    ).

    ,

    .

    , . . ,

    .

    , .

    .

  • 7/24/2019 Primena Gausovog Zakona Zadaci

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    , :

    kr

    r

    Ea

    QE

    20

    max4

    ; krr

    Ea

    Q

    2min04

    ; mm868,944 0

    min krrE

    Qa

    .

    14. ( a b ) r ( 14).

    U . :

    ) ;

    ) , q .

    : mm8a , mm16b , 3.2r , kV35U .

    14

    ________________________________________________________________________________

    :

    br

    bra

    r

    r

    qE

    0

    ,

    ,0

    2

    ,0

    r0;

    abq

    rr

    qrErEU

    b

    a

    b

    a

    b

    a

    ln2

    d2

    ddr0r0

    m

    C452.6

    ln

    2 r0 U

    a

    bq

    m

    MV156.3

    ln2 r0

    min

    a

    bb

    U

    b

    qEE b

    ,

    m

    MV312.6

    ln2 r0

    max

    a

    ba

    U

    a

    qEE a

    .

    15. , a c ,

    q .

    b d(

    15).

    ) r ;

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    ) cm1a , cm2.1b cm2c , ,

    ( 0C ) (C )

    5.00

    C

    C.

    15

    ________________________________________________________________________________

    15.1 15.2

    ) ,

    ( q ),

    , q ( 15.1).

    . :

    hErESSdESdESdESdE oSSSS BBo

    2

    00

    21

    ;

    ar : 00 EQ ;

    bra :r

    qEhqQ

    02

    ;

    dbrb : 00 EhqhqQ ;

    crdb :r

    qEhqhqhqhqQ

    02

    ;

    cr : 00 EhqhqhqhqQ .

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    b) :

    a

    cC

    ln

    2 00

    ,

    :

    I) :

    dbacbq

    db

    c

    a

    bqr

    r

    qrr

    r

    qrErEU

    c

    db

    db

    b

    b

    a

    c

    a

    c

    a

    ln2

    lnln2

    d2

    d0d2

    dd0000

    ,

    :

    dbacb

    C

    ln

    2 0 .

    II)

    ( 15.2): ,

    a

    bC

    ln

    2 01

    ,

    ,

    db

    cC

    ln

    2 02

    . :

    dbacb

    CCC

    CCC

    CCC

    ln

    2111 0

    21

    21

    21

    .

    CC 2

    10 , :

    cm5.01

    a

    cbd .