capacitance and dielectrics - university of hawaiiplam/ph272_summer/l4/24_lecture_lam.pdfcapacitance...

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

PowerPoint® Lectures for

University Physics, Twelfth Edition

– Hugh D. Young and Roger A. Freedman

Lectures by James Pazun

Chapter 24

Capacitance andDielectrics

Modified by P. Lam 7_15_2008


Topics for Chapter 24

• I. Capacitors and capacitance

• II. Energy stored in a charged capacitor

• III. Capacitors with dielectrics

Intermission

• IV. Dielectric breakdown

• V. Capacitors in series an parallel circuits


I. Capacitor and capacitance

• If we pull a positive charge and

negative charge apart, it requires

energy; this energy is stored as

electrostatic potential energy in this

charge arrangement. If the charges

are allowed to move, they will move

toward each other and gain kinetic

energy (one can use this kinetic

energy to push something to do work

or to light up a light bulb)

• A capacitor is device which can

hold these separated charges.


How do we build a capacitor?• The basic components of a capacitor are two conductors

separated by a distance.

• The “parallel-plate capacitor is the easiest to analyze.

• Connect the plates to the two terminals of a battery (see figure

below). Electrons will move from the top plate to the battery

and then enter the bottom plate; leaving a net positive charge on

the top plate and net negative charge on the bottom plate. The

electrons will stop moving when the voltage difference between

the two plates (Vab) equal to the voltage of the battery ( ).

•Disconnect the battery and the capacitor holds the charge.

+

-


Capacitance

+

-

How much charge (Q) a capacitor can hold for a given voltage

difference (Vab) between the plates depends on the size and shape of

the capacitor and it is called the capacitance (C).

Definition : C =Q

Vab

For parallel - plate capacitor with area (A) and separation (d),

C = o

Ad

(if d << A)

Derivation : d << A E =o

=Q

oA; Vab = Ed =

Qd

oAC =

Q

Vab

= o

A

d


Unit of capacitance

Definition : C =Q

Vab

unit =Coulomb

voltFarad

Example : The separation of a parallel plate capacitor is 1 mm,

what is the area of the plate in order for C =1 Farad?

Answer : C = o

A

d; o = 8.85x10 12Farad /m

A =Cd

o

=(1Farad)(10 3m)

8.85x10 12Farad /m1x108m2 !!!

Common capacitances are pF =10-12Farad or μF =10-6Farad.

Note : Even if C =1μF, the area is still 100m2!!

(1)Large area can be accommodated by rolling the two plates into

a cylinder.

(2) Capacitance can also be increased by inserting an insulating

material(called dielectrics) between the plates (will be discussed later).

Electrolytic dielectrics capacitor can have a capacitance of 1 Farad and yet

of very small size.


Cylindrical capacitance - coaxial cable

Suppose we connect this capacitor to a battery;

when it is fully charged, the charge is Q.

If ra, rb << L, then we can use the infinite line cylinder

approximation.

Using Gauss's Law E =Q

2 rL o

for ra < r < rb

Vab =| Edra

b

|=Q

2 L o

ln rb

ra

CQ

Vab

= o

2 L

ln(rb /ra )

Note : Let d rb ra ,

ln(rb /ra ) = ln(1+d

ra)

d

ra if d << ra

C = o

2 L

ln(rb /ra ) o

2 raL

d= o

A

d (same formula as for parallel - plate)

A cylindrical capacitor is formed by two concentric

cylinders (eg. a coaxial cable); the inner conductor

is typically a solid cylinder, the outer conductor is a

thin cylindrical shell.


Commercial capacitors

• Commercial capacitors forhome electronics have sizesrange from a grain of rice tothat of a large cigar.

• Capacitors like thosementioned above andpictured at right aremicrofarad capacitors.

• Note: Even though theexterior shape is a cylinder,it doesn’t mean that it iscylindrical capacitor; itcould be a rolled-upparallel-plate capacitor.


Note on capacitance

Altough the definition C involves Q and Vab : C =Q

Vab

The capacitance depends on the size and geometry of the

capacitor,

Example :

C = o

A

d for parallel - plate capacitor

C = o

2 L

ln(rb /ra ) for cylindrical capacitor

If we plot the charge (Q) vs the voltage across the capacitor (Vab),

then C is the slope of the linear graph.Q

Vab

slope=C


Example

Example 24.2. The dimensions of a parallel-plate capacitor are

d=5mm and A=2 m2. A charge (Q) of 35.4μC is each plate

(a) What is the capacitance (C) ?

(b) What is the voltage (Vab) across the capacitor?

(c) What is the magnitude of the electric field between the plates?

(d) Now pull the plates apart to d=10mm, what are the values for

C, Q, Vab, and E? Do you have to apply energy to pull the plates

apart?


II. Energy stored in a charged capacitor

Q

Vab

slope=CThe energy stored in a charged capacitor is

U =1

2QVab =

1

2

Q2

C=

1

2CVab

2

Refer to the example in the previous slide, find the energy stored in

the capacitor when d=5mm and the energy stored after you pull it

apart to d=10mm.

How much energy did you have to apply to pull the plates apart?

If the capacitor is connected to a battery (10 kV) while you are

pulling the plates apart, what is the stored energy after you have

pulled the plates to d=10mm?


Capacitor with dielectrics

• The potential differencebetween the parallel plates ofa capacitor decreases when adielectric material is insertedbetween the plates becausethe dielectric weakens theelectric field (explanation innext slide).

Since CQ

V,

decreasing V while Q remains the same

increaseing C.


Dielectric partially shields the electric field

E =Eo

K; K = dielectric constant of the material C =K o

A

d


Table 24.1—Dielectric constants


Intermission


IV. Dielectric breakdown

• A very strong electrical field can exceed the strength of thedielectric to contain it. Table 24.2 at the bottom of the page listssome limits.

air ~1 3x106

•For a given capacitor separation(d), the dielectric strength sets alimit on the maximum voltage(max V=Emd) that can be sustainedby the capacitor.


V. Capacitors in series and parallel circuits

Series :

Q1 =Q2

V1 +V2 =V

Parallel :

Q1 +Q2 =Q

V1 =V2 =V


Calculations regarding capacitance


Energy stored in a charge capacitor re-visit.

The energy stored in a charged capacitor is

U =12Q2

C

Consider a parallel - plate capacitor, C = o

A

d

U =12Q2

C=

12Q2d

oA=

12

Q2

o2A2

d * A* o

E =o

=Q

oAU =

12 oE

2

*Volume

1

2 oE2

= energy density (energy/volume)

That is, one can think of energy being stored as separated

positive and negative charges OR being stored in the E - field.

The second interpretation is useful when we discuss

energy carried by electromagnetic waves.

capacitance and dielectrics - university of hawaiiplam/ph272_summer/l4/24_lecture_lam.pdfcapacitance...

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