Capacitor filling

Filling a capacitor with dielectric 1

Filling a capacitor with a dielectric:

I. Complete filling

Filling a capacitor with a dielectric:

I. Complete filling

© Frits F.M. de Mul


Filling a capacitor with dielectricFilling a capacitor with dielectric

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 filling the capacitor with dielectric material (with r

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


AnalysisAnalysis

A

d

+Q

-Q

Important : Does the capacitor remain connected to the battery that loaded it?

Important : Does the capacitor remain connected to the battery that loaded it?

V

Case A : yes

Case B : no

Case A : yes

Case B : no

Consequences :

in A : voltage remains constant in B : charges remain constant.

Consequences :

in A : voltage remains constant in B : charges remain constant.


Approach (1) Approach (1)

A

d

+Q

-Q

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

planar symmetry everywhere : Gauss’ Law is applicable


Approach (2)Approach (2)

A

d

+Q

-Q

Relevant variables :

Q , , E, D, V and C

Relevant variables :

Q , , E, D, V and C

Material constants :

r , A and d .

Material constants :

r , A and d .

Task :

Establish the relations between the variables.

Task :

Establish the relations between the variables.


Approach (3)Approach (3)

A

d

+Q

-Q

Relations between variables :Relations between variables :

Qf

Gauss:

D dA Q dVf fVA

Gauss:

D dA Q dVf fVA

D

EMaterial constants:

D =rE

Material constants:

D =rE

Potential:

Vpath

E dl

Potential:

Vpath

E dl

V

Capacitance:

C Q Vf /

Capacitance:

C Q Vf /


Calculations (1)Calculations (1)

Gauss:

D dA Q dVf fVA

Gauss:

D dA Q dVf fVA

DTake Gauss pill box, top/bottom area = dA , enclosed charge = f..dA

Take Gauss pill box, top/bottom area = dA , enclosed charge = f..dA

dA

dADdA

. dAD dADdA

. dAD => D = f=> D = f



E

path

V dlE :Potential path

V dlE :Potential

Take integration path from bottom to top, length d

Take integration path from bottom to top, length d

dEtop

bottom

. dlE dEtop

bottom

. dlE => V = E.d=> V = E.d

d



Qf

D

E

V

D = f = Qf /AD = f = Qf /A

D = r ED = r EV = E.dV = E.d

C = Qf /VC = Qf /V

Case A : capacitor connected to battery

Case A : capacitor connected to battery

V remains constant !!V remains constant !!

QffD E V C b= begin

= end

1. V = const

startstart

2. E = const

3. D : r x as large

4. f : r x as large

5. Qf : r x as large

6. C : r x as large 7. b = (r-1) f, “old”

Where to start?????Where to start?????



Qf

D

E

V

D = f = Qf /AD = f = Qf /A

D = r ED = r EV = E.dV = E.d

C = Qf /VC = Qf /V

Case B : capacitor NOT connected to battery

Case B : capacitor NOT connected to battery

Qf remains constant !!Qf remains constant !!

QffD E V C b

1. Qf = const

startstart

2. f = const

3. D = const

4. E : r x as small

5. V : r x as small

6. C : r x as large 7. b = (1- r)f, “old”

Where to start?????Where to start?????


ConclusionsConclusions

if connected to battery • V = const• E = const

• D,f , Qf : r x as large

• C : r x as large

if connected to battery • V = const• E = const

• D,f , Qf : r x as large


if NOT connected

• Qf = const

• f , D = const

• E ,V: r x as small


if NOT connected

• Qf = const

• f , D = const

• E ,V: r x as small


the endthe end

Technology

Capacitor filling