Partly filling a capacitor with dielectric

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Filling a capacitor with a dielectricII: Partial filling

Filling a capacitor with a dielectricII: Partial filling

© Frits F.M. de Mul

Partly filling a capacitor with dielectric

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Partial filling a capacitor (1)Partial filling a capacitor (1)

Available: Flat capacitor : = surface area A , = distance of plates d , = no fill (r .

Available: Flat capacitor : = surface area A , = distance of plates d , = no fill (r .

A

d Assume: initially, plates are charged with +Q , -Q.

Assume: initially, plates are charged with +Q , -Q.

+Q

-Q

Question: What will happen with Q , E, D, V and C upon PARTIAL filling the capacitor with dielectric material (with r

Question: What will happen with Q , E, D, V and C upon PARTIAL filling the capacitor with dielectric material (with r

Partly filling a capacitor with dielectric

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Partial filling a capacitor (2)Partial filling a capacitor (2)

A. SeriesA. Series B. ParallelB. Parallel

I. FreeI. Free

II. Connected to battery

II. Connected to battery

Options:Options:

Partly filling a capacitor with dielectric

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Partial filling a capacitor (3)Partial filling a capacitor (3)

AA BB

II

IIII

Suppose:• when empty: Q0 , V0 .....etc.• fillingfilling for 1/3 of volumevolume• with r = 5

Suppose:• when empty: Q0 , V0 .....etc.• fillingfilling for 1/3 of volumevolume• with r = 5

Assumption :

no “edge effects” : d << plate dimensions

Assumption :

no “edge effects” : d << plate dimensions

Consequences :

no E-field leakage

Consequences :

no E-field leakage

• planar symmetry everywhere

• Gauss’ Law is applicable

• straight field lines (E and D)

• planar symmetry everywhere

• Gauss’ Law is applicable

• straight field lines (E and D)

Partly filling a capacitor with dielectric

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RelationsRelationsA

d

+Q

-Q

Qf

Gauss:

D dA Q dVf fVA

Gauss:

D dA Q dVf fVA

D

E Material constants:

D =rE

Material constants:

D =rE

Potential:

Vpath

E dl

Potential:

Vpath

E dl

V

Capacitance:

C Q Vf /

Capacitance:

C Q Vf /

AADQ ff .. AADQ ff ..

dEV . dEV .

Partly filling a capacitor with dielectric

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Options: main featuresOptions: main featuresOptions: main featuresOptions: main features

A.I and B.I: total charge unchanged

A.I and B.I: total charge unchanged

AA BB

II

IIIIA.II and B.II: total potential unchanged

A.II and B.II: total potential unchanged

A.I and A.II: potentials in series

A.I and A.II: potentials in series

B.I and B.II: potentials parallel

B.I and B.II: potentials parallel

Suppose:• when empty: Q0 , V0 ......• when filled: Q’, V’ , ......• fillingfilling for 1/3 of volumevolume

(depth or surface area)• with r = 5

Suppose:• when empty: Q0 , V0 ......• when filled: Q’, V’ , ......• fillingfilling for 1/3 of volumevolume

(depth or surface area)• with r = 5

Partly filling a capacitor with dielectric

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A.I. Horizontal fill, not A.I. Horizontal fill, not connectedconnected

Qf

D

E

V

db , dt : bottom and top layerFill: db = d0 /3 with r = 5

dt = 2d0 /3 with r = 1

db , dt : bottom and top layerFill: db = d0 /3 with r = 5

dt = 2d0 /3 with r = 1db

dt

Qf,t’

-Qf,b’

Qf’ D’ E’ d’

51

31

V’

151

32

32

1511

V’ C’

1115

o = oldt = topb = bottom} total

11

Start

Partly filling a capacitor with dielectric

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A.II. Horizontal fill, connectedA.II. Horizontal fill, connectedQf

D

E

VFill: db = d0 /3 with r = 5

dt = 2d0 /3 with r = 1

Fill: db = d0 /3 with r = 5dt = 2d0 /3 with r = 1

Qf’ D’ E’ d’51

31

V’

151

1510

32 3

2

V’

C’

1115

db

dt

Qf,t

-Qf,b

V0

V’

111

1110

11

E’113

1115

32

1110 :

1115

1115

D’ Qf’

o = oldt = topb = bottom} total

Consider ratios:

11

1115

<01115

Partly filling a capacitor with dielectric

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B.I. Vertical fill, not connectedB.I. Vertical fill, not connectedQf

D

E

VFill: Ar = A0 /3 with r = 5

Al = 2A0 /3 with r = 1

Fill: Ar = A0 /3 with r = 5Al = 2A0 /3 with r = 1

Qf’D’E’

73

51

715 :

73

E’ C’

37

o = oldl = leftr = right } total

Qf,l’

-Qf,r’

Qf,r’

-Qf,l’

Al Ar

Consider ratios:

V’

11

32

31

A’

32

Qf’

757

235

Qf’

715

31

75 :

73

32

72 :

D’

5

73

73

V’

Partly filling a capacitor with dielectric

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B.II. Vertical fill, connectedB.II. Vertical fill, connectedQf

D

E

VFill: Ar = A0 /3 with r = 5

Al = 2A0 /3 with r = 1

Fill: Ar = A0 /3 with r = 5Al = 2A0 /3 with r = 1

Qf’D’E’

5

C’

37

o = oldl = leftr = right } total

Consider ratios:

V’

11 3

2

31

A’

32

Qf’

37

35

Qf,l’

-Qf,r’

Qf,r’

-Qf,l’

Al Ar

V0

Partly filling a capacitor with dielectric

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Options: overviewOptions: overviewOptions: overviewOptions: overview

BBB.I and B.II: potentials parallel

B.I and B.II: potentials parallel

C : 15/11 xC : 15/11 xQf : unchangedV : 11/15 x

Qf : unchangedV : 11/15 x

V : unchangedQf : 15/11 x

V : unchangedQf : 15/11 x

C : 7/3 xC : 7/3 xQf : unchangedV : 3/7 x

Qf : unchangedV : 3/7 x

V : unchangedQf : 7/3 x

V : unchangedQf : 7/3 x

AAA.I and A.II: potentials in series

A.I and A.II: potentials in series

II IIII

Partly filling a capacitor with dielectric

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With combination rulesWith combination rulesWith combination rulesWith combination rules

BB B.I and B.II: parallelB.I and B.II: parallel

AA A.I and A.II: seriesA.I and A.II: series

II IIII d

AC r0 d

AC r0 Filling with r=5

in 1/3 of volume

Filling with r=5in 1/3 of volume

A

d

A

d

A

d

CCC bt .15

11

.

3/2

5.

3/11

'

1

000 A

d

A

d

A

d

CCC bt .15

11

.

3/2

5.

3/11

'

1

000

01115' CC 01115' CC

d

A

d

A

d

ACCC rl

.

3

73/.5.3/2.' 000

d

A

d

A

d

ACCC rl

.

3

73/.5.3/2.' 000

037' CC 037' CC

Partly filling a capacitor with dielectric

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Finally...Finally...Finally...Finally...

BB B.I and B.II: parallelB.I and B.II: parallel

AAA.I and A.II: seriesA.I and A.II: series

II IIII

bt CCC

11

'

1

bt CCC

11

'

1

rl CCC ' rl CCC ' the end

fQ dAD fQ dAD

D =rE

D =rE

dlEV dlEV

Qf

DE

V

VQC f / VQC f / AADQ ff .. AADQ ff ..

dEV . dEV .