課程電介質(zhì)物理第二章極化

電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化本章重點(diǎn)第一節(jié)電介質(zhì)的分類第二節(jié)電介質(zhì)的極化第四節(jié)電介質(zhì)的宏觀參數(shù)及其與微觀參數(shù)的關(guān)系第三節(jié)電介質(zhì)的極化強(qiáng)度與介電常數(shù)第五節(jié)電介質(zhì)的極化機(jī)理與極化率第六節(jié)鐵電體電介質(zhì)的極化電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理本章重點(diǎn)電介質(zhì)的分類及極化類型電介質(zhì)不同極化類型的理論推導(dǎo)

電介質(zhì)的宏觀參數(shù)及其與微觀參數(shù)的關(guān)系有效電場(chǎng)及介電常數(shù)電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化第一節(jié)電介質(zhì)的分類一.電偶極矩電偶極子：由等量正負(fù)電荷不對(duì)稱分布而形成的電偶極矩的分子。電偶極矩：電荷q與矢徑l的乘積，用一矢量μ表示，m=

ql方向：負(fù)電荷指向正電荷矢徑l大小：正負(fù)電荷中心的距離電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化非極性電介質(zhì)單原子分子

(He,Ne,Ar等)相同原子組成的分子(H2,N2,Cl2等)對(duì)稱結(jié)構(gòu)的多原子分子

(CO2,CCl4,CnH2n+2等)弱極性電介質(zhì),μ0≤0.5D中極性電介質(zhì),0.5D<μ0<1.5D0強(qiáng)極性電介質(zhì),μ

>1.5D石英，云母，金紅石型離子晶體玻璃陶瓷其他無(wú)機(jī)電介質(zhì)一般具有對(duì)稱的化學(xué)結(jié)構(gòu)，介電常數(shù)

εr=2～5,體電阻率ρv=1014～1016Ω·m化學(xué)惰性，性能穩(wěn)定r化學(xué)結(jié)構(gòu)不對(duì)稱，介電常數(shù)ε

=2.6～80,體電阻率高于非極性電介質(zhì)介電常數(shù)較大，較高的機(jī)械強(qiáng)度電介質(zhì)按正負(fù)電荷和分布特性可分為無(wú)外電場(chǎng)作用時(shí)，由正：負(fù)電荷中心重合，電偶極矩為零的分子組成極性電介質(zhì):無(wú)外電場(chǎng)作用時(shí)，由正負(fù)電荷中心不重合，具有固有偶極矩的分子組成離子性電介質(zhì):

通常由正負(fù)離子組成電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化xyoθ+q-qlcosθPrprrn圖1計(jì)算電偶極子周圍的電場(chǎng)真空中任意一點(diǎn)P的電位為：qp4pe0

rp

rnf

=(

1

-

1

)近似作以下的假設(shè)：rp=rn=r;rp-rn=-lcosθ；μ=ql;（2-1）電介質(zhì)物理第二節(jié)電介質(zhì)的極化一.電偶極子周圍的電場(chǎng)（圖1）電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理則式1可寫為：000pr

2f4pe4pe

r

2

4pe

r

2=

q

rn

-

rp=

q

l

cosq

=

m

cosq向量形式為：0p4pe

r3f

=

m

r

根據(jù)電場(chǎng)強(qiáng)度與電位的關(guān)系，P點(diǎn)的電場(chǎng)強(qiáng)度為：p0000r3

r3r4r5pr4pe

r3

4pe=m

r(

-3

r

)

+

m

=1

m

r

r

-

m

4pe

r3

4pe

r3

mr

-1

1

1

E

=

-Gradf

=

-Grad

=

(m

r)Grad

+

Grad(

-1

3

（2-2）（2-3）m

r)（2-4）電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化p

m,

rE

,

Ep

,

m,

r將

寫成向量形式：E

=

iE

+

jE

+

kE

p

px

py

pz

m

=

i

mx

+

j

m

y

+

k

mz

r

=

ix

+

jy

+

kz

（2-5）則有：：5555pxpypz3xy3xzr5rr53zyrrr4pe4pe4pe1

3x2

-

r

2E

=mx

+

r5my

+

r5mz

0

1

3xy3yz

3y2

-

r

2E

=mx

+my

+

r5mz

0

1

3xz3z2

-

r

2E

=mx

+

my

+mz

0

（2-6）電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化k如果m

=m，則(6)式成為00pxpyr

5r

54

peE=

1

3

xz

m

r

5z

E

1

3

yz=4

pem

z

1

3

z

2

-

r

2

E

pz=4

pem

z

0

（2-7）電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化圖2正方體六個(gè)面心上點(diǎn)偶極子的分布例1：試求位于正方體的六個(gè)面心上六個(gè)點(diǎn)偶極子在正立方體中心產(chǎn)生的電場(chǎng)強(qiáng)度。解：以正立方體中心為原點(diǎn)，則六個(gè)點(diǎn)偶極子的坐標(biāo)分別為：m1(0,0,-a);m2(0,0,a);m3(0,-a,0);m4(0,a,0);m5(-a,0,0);m6(a,0,0);電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理假設(shè)正方體的連長(zhǎng)為2a,所有偶極矩為μ,方向與z軸平行，根據(jù)7式可以得到六個(gè)點(diǎn)偶極矩在中心點(diǎn)產(chǎn)生的場(chǎng)強(qiáng)三個(gè)分量分別為：30000000;2

x2

y2

z3

x3

y3

z4

x4

y4

z5

x5

y5

z6

x6

y6

za

52m4pe

a4pe

a

34pe

a

34pe

a

34pe

a

34pe

a

3

1 3a

2

-

a

2=

0,

E1

y

=

0,

E1

z

=

4pem

=

E1

xE

=

0,

E

=

0,

E=

2m

;E=

0,

E

=

0,

E

=

-

m

;

E=

0,

E

=

0,

E=

-m

;

E=

0,

E

=

0,

E=

-m

;

E=

0,

E

=

0,

E=

-m

;:電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理于是

E

=

iEx

+

jEy

+

kEz

=

0注意：同樣可以證明正立方體八個(gè)頂點(diǎn)的八個(gè)點(diǎn)偶極子和在每邊中點(diǎn)十二個(gè)點(diǎn)偶極子在中心點(diǎn)產(chǎn)生的電場(chǎng)強(qiáng)度也分別等于零。電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理例2：在一介電常數(shù)為εr的無(wú)限均勻介質(zhì)中，有一半徑為a的真空小圓球，圓球中心有一電偶極矩μ，試求圖3圓球內(nèi)外介質(zhì)情況其電場(chǎng)分布。解：由于ρ=0，因此小圓球內(nèi)外任何一點(diǎn)的電位滿足Laplace方程，選用球形坐標(biāo)系，Laplace方程表示為：D

f

=

?

?f

+

?

q

?f

??q

?q

?q

?

f+=q

?j2222211nisnis202n1

is2rrrrrr(1)電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理取圓球中心為原點(diǎn)，z軸方向與μ的方向一致，根據(jù)對(duì)稱關(guān)系，式1可以寫成:2211nis0nis2rrrrr+q

?

f

=?

?

?

??q

?q

?q

(2)Rf

=

QR用分離變量法解此微分方程，令f

=，Q

R=R(r)，Q=Q為(q)的函數(shù)，代入上式兩端除以，乘以r2得，22222112gtcRRrrRrr?

?

?

Q

?Q

++

q

?q

?

?

=

-

Q

?q

(3)電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化由3式可得到：2

d2

R

dRrdr

2+

2r

-

l(l

+1)R

=

0dr（4）dq2d2QdQ+

ctgq

+

l(l

+1)Q

=

0dq（5）常數(shù)方程4的解可寫成：R

=

b

rl

+

b

r

-(l

+1)1

2（6）令x=cosθ,則式5變成Legendre方程，即22(1-

x

)dx

dxd2Q

dQ-

2x

+

l(l

+1)Q

=

0（7）電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理方程7的解為L(zhǎng)egendre函數(shù)：Q

=

al

Pl

(

x

)

=

al

Pl

(cos

q

)（8）所以球內(nèi)與球外介質(zhì)中的電位函數(shù)分別為：(

)(

)12llllllllA

rCrP

cosqP

cosq￥-(l

+1)-(l

+1)f

=

RQ

=+

B

rf

=

RQ

=+

D

rl

=0￥l

=0（9）電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理待定系數(shù)可根據(jù)邊界條件求定:1.離球中心無(wú)限遠(yuǎn)處偶極矩的作用可以忽略不計(jì)，即:(

)(

)12lllD

rP

cosqr

=￥￥-

l

+l

=0f2

=

0C

=

0,f

=2.球面上兩邊電位相等，即:(

)(

)11llllAaf1-

l

+-

l

+r

=a

=

f2

r

=a+

B

a=

D

a電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化lr

llr

rl

lrrA

lal

-1

-(l

+2)-(l

+2)a2l

+1

Bl電介質(zhì)物理3.球面上電感應(yīng)強(qiáng)度垂直球面分量連續(xù)，即:

?f1

=

e

?f2

?r

r

?r

r

=a

r

=a-(l

+1)B

a=

-e

(l

+1)D

al(er

-1)(l

+1)

1A

=-e

l

+

e

+

lD

=

2l

+1

Be

l

+

e

+

l電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化04pe

r

2f1

a=￥電介質(zhì)物理4.球面半徑無(wú)限增大時(shí)，偶極子處于無(wú)限真空空間時(shí)有：=

m

cosq113004lr

0r3

m4pe2(er

-1)

1

m

mA

=-,

B

=,

D

=2e

+1

a

pe2e

+1

4pe由以上邊界條件可解得：20lBlm

cosqr

4pe

ra=￥a=￥l

+1l

=0P

(cosq)

=

Al

=

0￥\

f1

=電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理將所求得的待定系數(shù)Al,Bl,Cl,Dl值代入式9得：1332rrzrr3m4pe4pe

m

1

2(er

-1)

f

=-

zr

(2e

+1)a0

m

1

2(e

-1)

f

=-

m(2e

+1)r30

球內(nèi)電場(chǎng)強(qiáng)度E1為：1

1

r

rr3m4pe1

mr

m

2(e

-1)r5E

=

-Gradf

=r

-

+(2e

+1)a30

3

電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理位于球心的電偶極矩μ對(duì)電場(chǎng)強(qiáng)度的貢獻(xiàn)為：10r5

r34peE￠=1

(

m

r

r

-

m

)3

球表面感應(yīng)電荷對(duì)球內(nèi)電場(chǎng)強(qiáng)度的貢獻(xiàn)為：130

rm1

2(er

-1)E

=4pe

(2e

+1)a￠所以球內(nèi)電場(chǎng)由二部分組成，即

E

=

E￠+

E1￠電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理二：偶極矩在電場(chǎng)中的勢(shì)能和力矩E+q-qf-fμ

θ圖4在均勻外電場(chǎng)中的電偶極矩偶極矩在電場(chǎng)中的勢(shì)能為：

U

=

-qEl

cosq

=

-m

E偶極子在電場(chǎng)中的力矩為：M

=

fl

sinq

=

qEl

sinq

=

mE

sinqM

=

m

·

E

當(dāng)θ=0時(shí)，U

=-mE,M

μ=與0,E同方向,狀態(tài)最為穩(wěn)定;能最大，最不穩(wěn)。當(dāng)θ=π時(shí)，U

=mE,此M

時(shí)=0勢(shì),電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理第三節(jié)電介質(zhì)的極化強(qiáng)度與介電常數(shù)1.極化形成的三種主要情況1）電子位移極化外電場(chǎng)作用下，原子外圍電子相對(duì)原子核在微觀范圍發(fā)生相對(duì)位移圖5電子位移極化模型電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化2）離子位移極化圖6離子極化模型(a)無(wú)外電場(chǎng)，(b)在外電場(chǎng)作用下介質(zhì)分子由異性離子構(gòu)成，在電場(chǎng)作用下，正負(fù)離子在有限范圍內(nèi)相對(duì)位移從而改變介質(zhì)內(nèi)部電荷分布產(chǎn)生感應(yīng)偶極矩電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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and

Power

Equipment電介質(zhì)物理電介質(zhì)的極化3）轉(zhuǎn)向極化圖7轉(zhuǎn)向極化模型極性電介質(zhì)的分子，在電場(chǎng)作用下，沿電場(chǎng)方向轉(zhuǎn)向而產(chǎn)生感應(yīng)偶極矩的極化電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理由上述三種不同的極化機(jī)制，可得到相同的結(jié)果：在電場(chǎng)作用下，在介質(zhì)內(nèi)部出現(xiàn)在感應(yīng)電矩，對(duì)外呈現(xiàn)電性電介質(zhì)極化的定義：在電場(chǎng)作用下，電介質(zhì)內(nèi)部沿電場(chǎng)方向出現(xiàn)宏觀

偶極子，在電介質(zhì)表面出現(xiàn)束縛電荷（極化電荷）的現(xiàn)象電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理2.極化強(qiáng)度P(Polarization

Rector)1)

P的定義：?jiǎn)挝惑w積電介質(zhì)的電偶極矩矢量總和.DVP

=

SmiDV

fi

0

DVP

=

lim

Smi或性質(zhì)：宏觀物理量，很多粒子μi的平均值;P與E方向有關(guān)；單位為Cm/m3=C/m2;電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理2)極化強(qiáng)度P與表面束縛電荷密度σ的關(guān)系表面出現(xiàn)束縛電荷σ′電介質(zhì)極化

的兩種反應(yīng)：內(nèi)部感應(yīng)偶極矩——極化強(qiáng)度P電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化圖8極化強(qiáng)度與表面束縛電荷的關(guān)系電介質(zhì)物理以圓柱體的偶極矩為例加以說(shuō)明：各參數(shù)如右圖：Smi

=

P

DV

=

sDSLDV

=

DSL

cosqs

￠=

P

cosq

=

Pn結(jié)論：束縛電荷面密度大小等于極化強(qiáng)度在ΔS面法線方向上的分量,二者單位都是C/m2電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化s

￠=

Ps

￠=

0q

=

00q

=

900q

=1800s

￠=

-P電介質(zhì)物理例：求均勻極化的電介質(zhì)球表面上極化電荷的分布，已知電極化強(qiáng)度為P。解：s

￠=P

cosq電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理3.介質(zhì)極化的宏觀參數(shù)—介電常數(shù)電介質(zhì)的介電常數(shù)(εr)是描述電介質(zhì)極化的宏觀參數(shù).0re

Ee

=

DD、E——分別為電介質(zhì)中電感應(yīng)強(qiáng)度、宏觀電場(chǎng)強(qiáng)度介電常數(shù)的意義：用平板電容器為例進(jìn)行說(shuō)明電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理1）平板電容器上的自由電荷σ0恒定，以介質(zhì)代替真空時(shí)，電場(chǎng)下降的倍數(shù)。在真空中(如圖a):由高斯定理可知E

=

s

0e0在介質(zhì)中(如圖b)000;seE

=0dE=

s

￠

es

-s

￠E

=

E0

-

Ed=

0

e0

0

0rE

s

/

e

se===0

0

0

E

s

0

-s

￠e0s

-s

￠圖9平板電容器中的電荷與電場(chǎng)分布(a)真空(b)充以介質(zhì)s

￠=

P

s

er

=0s

0

-

Ps

￠?,

E

fl,er

?電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理2)平板間E0保持不變，介質(zhì)代替真空時(shí)，電容增加的倍數(shù)在真空中有：00000VVse

dE

==

VQ0

=

s

0

SC

=Q

=

s

0

S充介質(zhì)時(shí)有：000dEse=

s

￠,

E

=e

e0E

-

Ed

=

E0=e

ds

-s

￠

s

0

=

Vs

=

s

0

+s

￠er

=

C

/

C0

=

(s

0

+s

￠)

/s

0

=

(s

0

+

P)

/s

0電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理以上分析說(shuō)明，εr與E無(wú)關(guān)（E不大時(shí)），有以下特性：εr是相對(duì)介電常數(shù)，與真空的比值；介質(zhì)絕對(duì)介電常數(shù)ε=ε0εr

εr是無(wú)量綱，且εr≥1在工程上，希望電容器的εr大，電纜的εr??；電機(jī)定子線圈，出糟口和套管，εr小可以提高沿面放電電壓.電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理例1：平行板電容器兩極板分別帶異號(hào)等值電荷，面密度為9.0x10-6C/m2,兩極板間介質(zhì)εr=4,求：1）極板上自由電荷產(chǎn)生的場(chǎng)強(qiáng)；2)電介質(zhì)內(nèi)的場(chǎng)強(qiáng)；3）介質(zhì)表面極化電荷面密度；4）極化電荷產(chǎn)生的場(chǎng)強(qiáng)；解：1）600-12-6

2s

0

9.0

·10

C

/

mE

== =1.02

·10

V

/

me

8.85·10

F

/

m64E

1.02

·106V

/

m2）E

=

0

=er=

0.25·10

V

/

m電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)物理電介質(zhì)的極化6

6

6

Ed

=

E0

-

E

=1.02

·10

-

0.25·10

=

0.75·10

V

/

m6010

V

/

ms

￠6.75·10-6

C

/

m2Ed

=

e==

0.75·8.85·10-12

F

/

m4）0rs

0

e

=s

-s

￠所以40

rres

(e

-1)

9.0

·10-6

·(4

-1)s

￠===

6.75·10-6

C

/

m2660-12=

(9.0

·10-

0.25·10

)·8.85·10s

￠=

s

0

-s

=

E0e0

-

Ee=

6.75·10-6

C

/

m23）電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理例2：平板電容器的電容為100PF，極板面積為1×10-2m2,極板間介質(zhì)εr=5.4，兩極板間電位差恒定為50V，求：1）介質(zhì)中的場(chǎng)強(qiáng) 2）極板上自由電荷面密度3）介質(zhì)面上極化電荷面密度解：1）r

0

r

0-12-2

2\

d

10·1·10

m

/100PF

C

=

ee

S

/

d

,

=

ee

S

/

C

=

5.4

·8.85·40

0

0

0

r

0

0

r-12·5.4

·1.05

·10=

5

·10-7

C

/

m2s

=

e

E

,\

s

=

e

e

E

=

e

e

E

=

8.85

·10=4.78

·10-3

m\

E

=V

/

d

=

50

/

4.78·10-3

=1.05·104V

/

m2）s

=

s

0

+s

￠=

ers

00r

0e3）s

￠=

(e

-1)s

=

er

-1s

=

4.07

·10-7

C

/

m2電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理4.電感應(yīng)強(qiáng)度D,電場(chǎng)強(qiáng)度E與極化強(qiáng)度P三者之間的關(guān)系：由Gauss’s

Law定理可知，?s

Eds

=

(q+q￠)

/

e0其中q為自由電荷，q′為極化電荷\蜒s

Eds

=(q

-s

Pds)/e0蜒s

e0

Eds

=q

-s

PdsQ

q￠=-蜒s

s

￠ds

=-s

Pds?s

(e0

E

+

P)ds

=

qe0

E

+

P

=

D?s

Dds

=

q電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理性質(zhì)：電感應(yīng)強(qiáng)度D與自由電荷分布有關(guān)且為矢量，其方向起始于正自由電荷，終于負(fù)自由電荷；D在數(shù)值上等于該點(diǎn)自由電荷面密度;各極板上自由電荷為q，極板附近D=q/s極化強(qiáng)度P只與極化電荷有關(guān)，P是矢量，起始于于負(fù)極化電荷，終于正極化電荷E與實(shí)際存在的所有電荷（包括自由電荷和極化電荷）有關(guān)，E是描述電場(chǎng)的基本矢量，D和P均可用E來(lái)表示電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理通過(guò)以上分析可以知道電感應(yīng)強(qiáng)度D,電場(chǎng)強(qiáng)度E與極化強(qiáng)度P三者之間的關(guān)系如下：D

=

e0

E

+

PD

=

e0er

EP

=

D

-e0

E

=

e0

(er

-1)E電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理例3：平行板電容器的兩極板上帶等值異號(hào)電荷,極板間距離為5mm,其中介質(zhì)εr=3,介質(zhì)中場(chǎng)強(qiáng)E=106V/m,求:1.介質(zhì)中的電位移矢量0

r-12

6

2D

=

e

e

E

=

8.85·10

·3·10

=

2.66

·10-5

C

/

m平板上自由電荷密度s

=

D

=

2.66

·10-5

C

/

m2介質(zhì)中的極化強(qiáng)度620

r10

C

/

m-12-6P

=

e

(e

-1)E

=

8.85·10·(3

-1)

·10=1.77

·電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理介質(zhì)面上極化電荷面密度s

￠=

P

=1.77

·10-6

C

/

m2平板上自由電荷及介質(zhì)面上極化電荷分別產(chǎn)生的場(chǎng)強(qiáng)000dE

=

s

/

e=

2.66

·10-5

/

8.85·10-12

=

3·106V

/

mE

=

E

-

E

=

3·106

-106

=

2

·106V

/

m電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理例4：平板電容器d=5mm,極板帶異號(hào)電荷其面密度為20μC/m2,其中雙層介質(zhì)

d1=2mm,εr1=3;d2=3mm,εr2=3;求1)雙層介質(zhì)中的DD

=

D

=

s

=

20mC

/

m21

22)雙層介質(zhì)中的場(chǎng)強(qiáng)65151

0

r110

V

/

m10

V

/

m-12-6-12E

=

D

/

ee=

20

·10-

/(8.85·10·3)

=

7.5·E2

=

D

2

/

ee

1r0

=

20

·10/(8.85·10·

4)

=

5.6

·電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理3.兩介質(zhì)面上極化電荷面密度521

11522

2

20

r10

C

/

m10

C

/

m-12-5-12-5￠s

=

P=

e

(e

-1)E

=

8.85·10·(3

-1)

·7.5·10=1.33·￠s

=

P=

e0

(er-1)E

=

8.85·10

·(4

-1)

·5.6

·10

=1.5·交界面上的極化電荷面密度為：1

2-6

2￠

￠s

-s

=

-1.7

·10

C

/

m交界面上的自由電荷面密度為：D1

-

D2

=

0電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理第四節(jié)電介質(zhì)的宏觀與微觀參數(shù)和關(guān)系極化強(qiáng)度P可表示為：DV

DVi

iP

=

Sμi

=

n

μ

=

Nμ其中：N為單位體積內(nèi)電介質(zhì)組成的基本粒子數(shù)；μi是感應(yīng)偶極矩；μi為分子的平均感應(yīng)介極矩電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理當(dāng)外加電場(chǎng)E低于原子或分子內(nèi)電場(chǎng)時(shí)，μi

與作用在分子的有效電場(chǎng)強(qiáng)度Ei成正比，m

=

a

Eiα—極化率，單位是Fm2所以極化強(qiáng)度P又可表示為：P

=

Nμi

=

Na

E

iP

=

e0

(er

-1

E

=

Na

Ei(Clausion

方程)0re

-1

=iNa

Ee

E電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化Clausion

方程電介質(zhì)物理意義:宏觀參數(shù)εr與其分子微觀參數(shù)N,α的關(guān)系.作用：預(yù)測(cè)極化性能；設(shè)計(jì)新介質(zhì)，提高或降低或控制εr注：由Clausions方程，要求得εr，必須要研究各種粒子的α—極化率和作用在粒子上的內(nèi)電場(chǎng)E。電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

Key

Laboratory

of

Electrical

Insulation

and

Power

Equipment電介質(zhì)的極化電介質(zhì)物理第五節(jié)電介質(zhì)的極化機(jī)理與極化率1.極化機(jī)理根據(jù)電介質(zhì)的極化微觀機(jī)理，極化可分為：①電子位移極化③轉(zhuǎn)向極化②原子（或離子）位移極化④熱離子極化⑤夾層（界面）極化說(shuō)明：在實(shí)際介質(zhì)中，往往是多種機(jī)化并存！電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化Ei=0Eixμe極化前

極化后圖10電子位移極化原理示意圖2.電子位移極化及電子極化率αe1)電子位移極化以單原子為例，如右圖所示:當(dāng)Ei=0,電子云中心與原子核重合，μ=0；當(dāng)Ei≠0,電子云中心與原子核距離為x，m

=me

=Zex

=ae

Ei特點(diǎn)：①μe不是原子固有，在Ei作用下感應(yīng)；②所有介質(zhì)在電場(chǎng)作用下均會(huì)產(chǎn)生電子極化；③極化建立時(shí)間很短，約為10-14~10-15s;電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化；圖11電子位移極化計(jì)算模型電介質(zhì)物理2)電子極化率αe用球狀原子模型來(lái)(如右圖)計(jì)算電子位移極化率。假設(shè)：①Ei作用下，電子云球狀不變，球半徑為a,電子云在其中均勻分布②元素序數(shù)為Z，原子核的電荷量為Ze半徑為x球內(nèi)電子云對(duì)原子核的庫(kù)侖引力為：001Z

2e2

xx24p

x34pe4pe

a33

(Ze)Ze

4pa33F

=

=電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理電子云受到有效電場(chǎng)Ei的電場(chǎng)力為：Fe

=

ZeEi電子云平衡時(shí)有：F

=Fe即:0iZ

2e2

x4pe

a3=

ZeEZe4pe

a3x

=

0

Eie

0

im

=

Zex

=

4pe

a3

E意義：①已知a,就可計(jì)算出αe；②αe與溫度基本無(wú)關(guān).3ae=

4pe0

a

Ei電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理不足：①模型簡(jiǎn)單，電子云認(rèn)為均勻分布；②

從量子力學(xué)角度，原子核與電子的聯(lián)系小,αe較大其它計(jì)算αe模型①球狀原子模型單原子模型：②圓周軌道模型③介質(zhì)球模型①雙原子分子的簡(jiǎn)化模型非球狀分子模型：②回轉(zhuǎn)橢球分子模型電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理3.離子極化及離子極化率αa1)離子極化定義：在E的作用下，引起正負(fù)離子相對(duì)位移而產(chǎn)生感應(yīng)電矩特點(diǎn)：建立時(shí)間約為10-12~10-13s,當(dāng)交變電場(chǎng)的頻率f<紅外光時(shí)，離子極化來(lái)得及建立。電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化Ei圖12離子位移極化計(jì)算模型Ei=0

正負(fù)離子相距為a；負(fù)離子位移為Δx-正離子位移為Δx+Δx=Δx-+Δx+Ei≠02）離子極化率αa的計(jì)算利用孤立的正負(fù)離子對(duì)來(lái)計(jì)算如右圖:電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理假設(shè)D

x

=

a,兩異性離子的彈性恢復(fù)力：f

=

-

kD

x離子受到的電場(chǎng)力大小為：f

=

qEi正負(fù)離子處于平衡狀態(tài)時(shí)：qEi

=

kDx

D

x

=

qEi

/

kq2m

=

qDx

=

Ei

=

aa

Eikq2Ei

kmaa

==求解k是關(guān)鍵電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理異性離子的相互作用勢(shì)能為：nq2b4pe0

x

4pe0

xu

(x)=-

+b為常數(shù)，n=7~11沒(méi)有外電場(chǎng)時(shí)，離子處于平衡位置，x＝a，?u

(x‰

＝0?x

c＝aan-1q2b

=n0

x

4pe0

xnq2

an-1q2u

(x)=

-

4pe

+則有：電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Electrical

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Equipment電介質(zhì)的極化電介質(zhì)物理當(dāng)場(chǎng)強(qiáng)不大，D

x

=

a時(shí),

μ(x)展開(kāi)成Taylor形式,2x=a?u(x) 1

?2u(x)u

(x)=

u

(a)+x=a

(x

-

a)

+

2

(x

-

a)+......?x

?x2離子間的彈性能為：u

(x)-

u

(a

)=

1

k

(x

-

a

)2

=

1

kDx2211222

21

?2u(x)x=a

(x

-

a)u

(x)=

u

(a)+

2?x2\

u

(x)-u

(a)=

k

(x

-

a)2

=

kDx2?2u

(xk

=‰c

=a?x2電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化解得：04pe

a3(n

-1

q2k

=離子中心距離a可以認(rèn)為是正、負(fù)離子的半徑之和優(yōu)點(diǎn)：圖像清晰缺點(diǎn)：比較粗略特點(diǎn)：①αa與(r++r-)3成正比，在數(shù)量級(jí)上與αe相同，約為10-40Fm2②αa與溫度T關(guān)系不大、r+,r-—分別為正、負(fù)離子的半徑a4pe

a3∴離子極化率為：a=

0

n

-1(

)3a0

+-4pe

r

+

ra

=n

-1電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理例：對(duì)于三維的NaCl離子晶體（參看圖13），經(jīng)過(guò)類似的運(yùn)算，可得NaCl晶體的離子極化率圖13NaCl離子晶體a4pe

a3a=

0

0.58(n

-1)習(xí)題:一雙原子分子，原子半徑為R，單原子極化率為αe（電子極化率），兩原子中心距離為L(zhǎng)，求電場(chǎng)方向與兩原子中心線平等及垂直時(shí)，雙原子的極化率。電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化圖14轉(zhuǎn)向極化a)無(wú)外電場(chǎng)；b)有外電場(chǎng)電介質(zhì)物理4.轉(zhuǎn)向極化及轉(zhuǎn)向極化率αa1)轉(zhuǎn)向極化E=0

由于熱運(yùn)動(dòng)，宏觀偶極矩μ=0,即P=0;E≠0

μ0轉(zhuǎn)向與E的方向，整體出現(xiàn)宏觀極矩P≠0.Ei

=

-m0

Ei

cosq宏觀極矩大?。簎

=-m0電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)物理電介質(zhì)的極化μ0受到：①熱運(yùn)動(dòng)，無(wú)序化的作用②電場(chǎng)有序化的作用特點(diǎn)：①轉(zhuǎn)向極化建立需要較長(zhǎng)時(shí)間，約為10-6~10-2s;②針對(duì)有固有偶極矩μ0的極性分子而言電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理2)轉(zhuǎn)向極化率αa的計(jì)算偶極子分布函數(shù)滿足:f

(u=

Ae-u

kT

=

Aeu0Ei

cosq

/

kT設(shè)電介質(zhì)單位體積的分子數(shù)為n0在夾角q

fi

(q

+dq00

im

E

cosq

kTdn

=

n

Ae之間的偶極分子數(shù)為：d

Wr

2dW

=

2p

sinqrdq

=

2p

sinqdq00m

E

cosq

kTpn

Asin

qedqdn

=

2圖15偶極分子的定向立體角電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化00

000

i電介質(zhì)物理球內(nèi)偶極分子沿電場(chǎng)方向的總偶極矩為：pm

E

cosq

kT2

n

Am

cosqsinqedqM

=m

cosqdn

=

pW

000

i球內(nèi)偶極分子的總數(shù)為：pm

E

cosq

kTpn

Asinqedqn

=dn

=

2W

因此，偶極分子沿電場(chǎng)方向的平均偶極矩為:0000

i0

iMnpm

E

cosq

kTpm

E

cosq

/

kTm

cosqesinqedqm

=

=電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理令m0

Ei

kT

=

a，cosq

=

y,

則有dy

=

-sinqdq20ayeay

dy1

eaa

(

)ma

-a-a+1-1+1eay

dy-a-1

1

1e

-

em0

(e

+

e

)-(

)

a

am

==-

e(

)000aa

-a1

ea

+

e-a1

=

m-=

mctha

-=

m

L

ae

-

e

aLangevin函數(shù)電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理當(dāng)m0

Ei

<kT

或a

<1

時(shí)，3

45

945aa32a5L

(a)=

-+ -i如分子之間聯(lián)系很弱，認(rèn)為E

=

E2003kT15k

2T

2m2

m2m

=E

1-E

+圖16平均偶極矩與電場(chǎng)的關(guān)系電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化電介質(zhì)物理當(dāng)E

<107

V

m、在室溫，a

= 1

時(shí)，Langevin函數(shù)可以寫成:3

3kTL

(a

)=

a

=

m0

E

E3kTm2于是

m

=

0

EddE

3kTm2m轉(zhuǎn)向極化率a

為:a==

0

，說(shuō)明：①一般情況下滿足a

= 1

,如E

<107

V

m

,

m

=

10-

30

C

?m,

k

1.38?

10-

23

J

/

K

,T

300K

,

a

=

0.0024

=

10②沒(méi)有考慮分子間作用力，即E

=Ei電力設(shè)備電氣絕緣國(guó)家重點(diǎn)實(shí)驗(yàn)室State

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Equipment電介質(zhì)的極化圖17弱離子聯(lián)系的勢(shì)能曲線實(shí)線—未加電