Lecture 9-Lecture 9-11Capacitors

• A capacitor is a device that is capable of storing electric charges and thus electric potential energy. => charging• Its purpose is to release them later in a controlled way. => discharging• Capacitors are used in vast majority of electrical and electronic devices.

+Q -Q

Q

Q

Q

Typically made of two conductors and, when charged, each holds equal and opposite charges.

Lecture 9-Lecture 9-22 Polar and Nonpolar Dielectrics

• Eext = 0: the orientations of the permanent electric dipoles are

distributed randomly, so the net dipole moment of the material is zero.

• Eext 0 : the molecular dipoles try to align themselves with the field

against random thermal motion, resulting in a net dipole moment.

Polar dielectrics: dielectric material whose molecules have permanent electric dipole moments, such as water.

Nonpolar dielectrics: molecules with no permanent electric dipole moments.

• Eext 0 : charges separate, induced dipole moment emerges.

Lecture 9-Lecture 9-33Dielectrics between Capacitor Plates

+ Q - Qfree charges

neutral-q +q

• Electric field E between plates can be calculated from Q – q.

0

( ) /,

Q q AE V Ed

0

0

( )

1

1

/

Q QC

V Q q d A

AqdQ

Lecture 9-Lecture 9-44Inserting Dielectrics (= Insulator)

• Inserting a dielectric between the plates of a capacitor increases capacitance

• Dielectric constant of a dielectric is the ratio of the capacitance when filled with it to that without it:

Q=CV holds more charges at fixed V

0

C

C > 1 always

(dimensionless) Material air (1 atm) 1.00059

paper 3.7

pyrex 5.6

water (20C) 80.4

strontium titanate

310 Breakdown potential determined by dielectric strength (Emax)

•Under the same Q,

•V = V0/k

•E = E0/k

Lecture 9-Lecture 9-55Capacitors in Parallel

Equivalent Capacitor:

V is common 1 2 3

1 2 3

q q qV

C C C

qC

V where 1 2 3q q q q

23

11 2

3eq

q q qC C

VC C

Lecture 9-Lecture 9-66Capacitors in Series

Equivalent Capacitor:

qC

V where

1 2 3V V V V

1 2 3

1 2 3

1 1 1 1

eq

V V V

qC C C C

1 1 2 2 3 3q C V C V C V q is common

Lecture 9-Lecture 9-77Conductor inserted between plates

• A parallel-plate capacitor with conductor inserted in the middle

E=0 outside capacitors

+q

-q

0

qE

A

E=0 in conductor

0

qE

A between platesbetween plates

0 0 0

1 1 1

a b

a b a b

C C C A A A

0 AC

a b

• Two capacitors of area A in series

-Q-Q

a

b

+q

-q

+q

-q -q

+q

a

b

Lecture 9-Lecture 9-88Warm-up quiz

All the capacitors have the same capacitance C, What is the total capacitance?

A). 4/3 CB). 3/4 CC). 3/2 CD). 1/2 CE). 2/3

Lecture 9-Lecture 9-99Non-uniform Parallel-plate Capacitor 1

1 0 2 01 2

0 1 2

/ 2 / 2

2

A AC C C

d dA

d

Equivalent to 2 capacitors in parallel

0 1 2

2

( )

Q d QV

C A

Potential drop V in each is the same.

11 1

1 2

,Q C V Q

2

2 21 2

Q C V Q

Note 1 2Q Q if 1 2

even though 1 2 1 2,V V E E

Lecture 9-Lecture 9-1010Non-uniform Parallel-plate Capacitor 2

1 2 0 0

0

1 1 1 / 4 3 / 4

1 3 1

4 4

d d

C C C A A

d

A

Equivalent to 2 capacitors in series

0

1 3

4 4

Q d QV

C A

(Free) charge in each Q is the same.

01 1 0

0 1

( / ) 1/ ,

( / ) 4

Q A dV Q C V

A d C

Note 1 2/ 1/ 3V V if 1

02 2 0

0 2

( / ) 3/ ,

( / ) 4

Q A dV Q C V

A d C

+Q

-Q

-Q+Q

Lecture 9-Lecture 9-1111Inserting Dielectric Material with Battery Disconnected

1. Charge a parallel plate capacitor filled with air (or vacuum) to potential difference V0.

Deposits charge 0 0Q C V

3. Insert a dielectric of dielectric constant

2. Disconnect the battery

Q remains fixed

00

QC C

V

0Q VV

C

So, V and E decreases from V0 , E0 to

0EVE

d and

Q

-Q

Q

-Q

20

2

UQU

C and

20 0

200

1122

Euu E

Lecture 9-Lecture 9-1212Inserting Dielectric Material with Battery Connected

1. Charge a parallel plate capacitor filled with vacuum (air) to potential difference V.

Deposits charge 0 0Q C V

3. Insert a dielectric of dielectric constant

2. Keep the battery connected

V remains fixed

00

QC C

V

0Q CV Q

So, Q increases from Q0 and E remains fixed

0

VE E

d and

0Q

Q

0Q

Q

20

1

2U CV U 2

0 0

1

2u u E and

0 0

Lecture 9-Lecture 9-1313Force between the plates of a capacitor

What is the force between these charged plates?

2

0

QQE

AIs it?

Not quite!

Is it attractive?+Q

-Q

0 0

QE

A

Battery disconnected

Yes!

Force F acting on a charge q in the electric field E created by other source charges is given by:

F qE����������������������������

E created by the charge on the upper plate at the location of the lower plate is 02

2

0 02 2

QF Q

A

Lecture 9-Lecture 9-1414Energy stored in a capacitor revisited

+Q

-Q

0 0

QE

A

Battery disconnected

separation d

separation 0

+Q

-Q

External work required to separate the plates from zero to d m apart is

2

0

2

2

2

QW F d d

A

Q

C

This is also the potential energy stored in capacitor.What if a battery

remains connected?

Lecture 9-Lecture 9-1515Force and energy with battery connected, a mystery?

+Q

-Q

0 0

QE

A

Battery connected

V

Force between plates?

attractive2

0 02 2

QF Q

A

Work to change separation from 0 to d?

separation 0

+Q

-QQ changes

2

2

0

1

2 2

d d d d

d d

CV dW F dl dl CV

A d

separation d

separation d+d

separation d However,

2 21 1

2 2

dU CV CV

d

Lecture 9-Lecture 9-1616 Lecture quiz A.

Five identical capacitors have the same capacitance C. Then capacitor (e) is filled with a material with dielectric constant k. What is the total capacitance?

A). 2CB). 1/(1+k) CC). (1+k)CD). 5 CE). [4/(4+k)] C

a b

e

c d

Lecture 9-Lecture 9-1717 Lecture quiz B.

Five identical capacitors have the same capacitance C. Then capacitor (e) is filled with a material with dielectric constant k. What is the total capacitance?

A). 2CB). [k/(1+k)] CC). (1+k)CD). [1/(1+k)] C CE). (4+k) C

a b

e

c d

Lecture 9-Lecture 9-1818 Lecture quiz C.

Four identical capacitors have the same capacitance C. Then capacitor (a) is filled with a material with dielectric constant k. What is the total capacitance?

A). (3/2 + k) CB). (3+k) CC). 5/2 CD). (2/3 +k) CE). (3+k) C

a

b

c d