Chapter 24

Capacitance, Dielectrics,Electric Energy Storage

Physics for Scientists & Engineers, 3rd EditionDouglas C. Giancoli

© Prentice Hall

P07 -

Capacitors and Capacitance

P07 -

Capacitors: Store Electric EnergyCapacitor: two isolated conductors with equal and opposite charges Q and potential difference ∆V between them.

QCV

=∆

Units: Coulombs/Volt or Farads

20P07 -

Parallel Plate Capacitor

0=E

0=E

Q Aσ+ =

Q Aσ− = −

d?E =

21P07 -

Parallel Plate CapacitorWhen you put opposite charges on plates, charges move to the inner surfaces of the plates to get as close as possible to charges of the opposite sign

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/35-capacitor/35-capacitor320.html

22P07 -

Calculating E (Gauss’s Law)

0S

inqdε

⋅ =∫∫ E A00 εε

σAQE ==( )

0

GaussGauss

AE A σε

=

Note: We only “consider” a single sheet! Doesn’t the other sheet matter?

23P07 -

Alternate Calculation Method

+ + + + + + + + + + + + + +Top Sheet:02

E σε

= −

02E σ

ε=

- - - - - - - - - - - - - -Bottom Sheet: 02E σ

ε=

02E σ

ε= −

0 0 0 02 2QE

Aσ σ σε ε ε ε

= + = =

24P07 -

Parallel Plate Capacitor

top

bottom

V d∆ = − ⋅∫ E S0

Q dAε

=Ed=dA

VQC 0ε=∆

=

C depends only on geometric factors A and d

Figure 24-5 (b)

Figure 24-6

26P07 -

Spherical CapacitorTwo concentric spherical shells of radii a and b

What is E?

Gauss’s Law E ≠ 0 only for a < r < b, where it looks like a point charge:

rE ˆ4 2

0rQπε

=

27P07 -

Spherical Capacitor

For an isolated spherical conductor of radius a:

20

ˆ ˆ4

b

a

Q drrπε

= − ⋅∫r r

( )1104−− −

=∆

=baV

QC πε

outside

inside

V d∆ = − ⋅∫ E S0

1 14Q

b aπε⎛ ⎞= −⎜ ⎟⎝ ⎠

Is this positive or negative? Why?

aC 04πε=

28P07 -

Capacitance of EarthFor an isolated spherical conductor of radius a:

aC 04πε=

mF1085.8 120

−×=ε m104.6 6×=a

mF7.0F107 4 =×= −C

A Farad is REALLY BIG! We usually use pF (10-12) or nF (10-9)

29P07 -

1 Farad Capacitor

How much charge?

( )( )1F 12 V12C

Q C V= ∆==

33P07 -

Energy To Charge Capacitor

1. Capacitor starts uncharged.2. Carry +dq from bottom to top.

Now top has charge q = +dq, bottom -dq3. Repeat4. Finish when top has charge q = +Q, bottom -Q

+q

-q

34P07 -

Work Done Charging CapacitorAt some point top plate has +q, bottom has –q

Potential difference is ∆V = q / CWork done lifting another dq is dW = dq ∆V

+q

-q

35P07 -

Work Done Charging CapacitorSo work done to move dq is:

dW dq V= ∆1qdq q dq

C C= =

Total energy to charge to q = Q:

0

1 Q

W dW q dqC

= =∫ ∫ +q

-q212

QC

=

36P07 -

Energy Stored in CapacitorQCV

=∆

Since

22

21

21

2VCVQ

CQU ∆=∆==

Where is the energy stored???

37P07 -

Energy Stored in Capacitor

Energy stored in the E field!

ando AC V Eddε

= =Parallel-plate capacitor:

212

U CV= ( )2

21 ( )2 2

o oA EEd Addε ε

= = × ( )Eu volume= ×

2

field energy density2

oE

Eu E ε= =

38P07 -

1 Farad Capacitor - EnergyHow much energy?

( )( )

2

2

121 1F 12 V272 J

U C V= ∆

=

=

Compare to capacitor charged to 3kV:

( )( )

( )( )

22

24 3

1 1 100µF 3kV2 21 1 10 F 3 10 V 450 J2

U C V

−

= ∆ =

= × × =

P8-

Capacitors: Store Electric Energy

QCV

=∆

To calculate:1) Put on arbitrary ±Q 2) Calculate E3) Calculate ∆V

Parallel Plate Capacitor:

0 ACdε

=

18P8-

Capacitors in Parallel

19P8-

Capacitors in Parallel

Same potential!

1 21 2,Q QC C

V V= =∆ ∆

20P8-

Equivalent Capacitance

?

( )1 2 1 2

1 2

Q Q Q C V C VC C V

= + = ∆ + ∆

= + ∆

1 2eqQC C CV

= = +∆

21P8-

Capacitors in Series

Different Voltages NowWhat about Q?

22P8-

Capacitors in Series

23P8-

Equivalent Capacitance

1 21 2

Q QV , VC C

∆ = ∆ =

1 2V V V∆ = ∆ + ∆(voltage adds in series)

1 2eq

Q Q QVC C C

∆ = = +

1 2

1 1 1

eqC C C= +

25P8-

Dielectrics

27P8-

DielectricsA dielectric is a non-conductor or insulatorExamples: rubber, glass, waxed paper

When placed in a charged capacitor, the dielectric reduces the potential difference between the two plates

HOW???

28P8-

Molecular View of Dielectrics

Polar Dielectrics : Dielectrics with permanent electric dipole moments Example: Water

29P8-

Molecular View of Dielectrics

Non-Polar DielectricsDielectrics with induced electric dipole momentsExample: CH4

30P8-

Dielectric in Capacitor

Potential difference decreases because dielectric polarization decreases Electric Field!

31P8-

Gauss’s Law for Dielectrics

Upon inserting dielectric, a charge density σ’ is induced at its surface

0S

insideqd EAε

⋅ = =∫∫ E A0

'εσσ −

=E( )0

' Aσ σε−

=

What is σ’?

32P8-

Dielectric Constant κDielectric weakens original field by a factor κ

0

0 0

' EE σ σ σε κ κε−

= ≡ = 1' 1σ σκ

⎛ ⎞= −⎜ ⎟⎝ ⎠

⇒

Gauss’s Law with dielectrics:

0S

freeinsideqdκε

⋅ =∫∫ E A

Dielectric constantsVacuum 1.0Paper 3.7Pyrex Glass 5.6Water 80

33P8-

Dielectric in a CapacitorQ0= constant after battery is disconnected

0VVκ

=Upon inserting a dielectric:

0 00

0 0/Q QQC C

V V Vκ κ

κ= = = =

34P8-

Dielectric in a CapacitorV0 = constant when battery remains connected

Upon inserting a dielectric: 0Q Qκ=

00

0

QQC CV V

κ κ= = =

36P8-

Group: Partially Filled Capacitor

What is the capacitance of this capacitor?