application of Capacitors

• Tuning circuits of radios

• Electronic timing circuits

• Electronic flashguns

• Smoothing fluctuation in the output of power supplies in radio and TVs.

• Experiment in high energy particle accelerators, and especially those in fusion power research

EXAMPLE 26.1: A parallel-plate capacitor with a plate separation of 1 mm has a capacitance of 1 F. What is the area of each plate?

unit. lare very a is farad the

10km!10kmely approximat is This

m101.13

/1085.8

)10)(1(

28

12

3

0

mF

mFCdA

EXAMPLE 26.2: A parallel-plate capacitor has plates with dimensions 3 cm x 4 cm, separated by 2 mm. The plates are connected across a 60-V battery. Find: (a) the capacitance; (b) the magnitude of the charge on each plate.

C103.19

F)(60V)110(5.3

CV(b)Q

5.31pF 102

)102.1)(/1085.8(

)(

1012

10-

12-

3

23120

24

m

mmF

d

AC

a

mA

EXAMPLE 26.3: What is the capacitance of an isolated sphere of radius R?

F710 be wouldecapacitanc its

thatpredictsit , 6370km radius of sphere

conducting a isearth that theassume we

4C

)1/(4k sin

/

/

0

0

If

R

ce

VQC

RkQV

EXAMPLE 26.4: A spherical capacitor consists of two con centric conducting spheres, as shown in Fig. 26.6. The inner sphere, of radius R1? has charge +Q. The charge on the outer shell of radius R2 is -Q. Find its capacitance.

capacitor plate parallel a of that toand sphere isolatedan

of ecapacitanc the torelated becan expression

)R-k(R

RR

/

)11

(

r

kQ--

/E

12

21

12

12

2

r

2

1

2

1

This

VQC

RRkQ

drEVV

rkQdrEsdE

sdEV

R

R

R

R

r

r

EXAMPLE 26.5: A cylindrical capacitor consists of a central conductor of radius a surrounded by a cylindrical shell of radius b, as shown in Fig. 26.7. A coaxial cable used for transmission of TV signals has this geometry. Usually, the outer sheath is grounded and shields the signal in the inner wire from electrical disturbances. A nylon, or Teflon, sleeve separates the inner wire from the sheath. Find the capacitance of a length L assuming that air is between the plates.

(26.6) )/ln(

2C

LQ

needed) is magnigtude(only .ln22-V-V

dr ErdE

conductorinner on the charge by theonly determined is field that theNote

bra 2

0

b

aab

r

ab

L

a

bk

r

drkdrE

r

kE

b

ar

r

EXAMPLE 26.6: For the circuit in Fig. 26.10a, find: (a) the equivalent capacitance; (b) the charge and potential difference for each capacitor.

FC

CCCCC

a

eq

eq

22

1

12

1

31

1

6

1

1111

)(

4321

CVFVCQ

CVFVCQ

VVVVVV

VF

C

C

QVV

F

C

C

QV

VVCQQQ

CVFVCQ

QQQQb

eq

72)24)(3(

24)24)(1(

24)816(48

812

96,16

6

96

,96

96)48)(2(

)(

333

222

32

4

44

1

11

3241

4321

EXAMPLE 26.7-1: Two capacitors, C1= 5 μF and C2=3μ F, are initially in parallel with a 12-V battery, as in Fig. 26.11a. They are disconnected and then reconnected as shown in Fig26.11b. Note carefully the numbering on the plates. Find the charges, potential differences, and energies stored (a) in the initial state, and (b) in the final state.

JUUU

JVCVQU

JVCVQU

CQCQ

VC

Q

C

Q

a

576

216)12)(36(2

1

2

1

360)12)(60(2

1

2

1

36,60

12

)(

21

222

111

21

2

2

1

1

EXAMPLE 26.7-2: Two capacitors, C1= 5 μF and C2=3μ F, are initially in parallel with a 12-V battery, as in Fig. 26.11a. They are disconnected and then reconnected as shown in Fig26.11b. Note carefully the numbering on the plates. Find the charges, potential differences, and energies stored (a) in the initial state, and (b) in the final state.

tube.- Uain

waterlike ,capacitors theof plates ebetween th oscillates

charges The ously.instantane valuesfinalr reach theinot do

capacitors on the charges theresistance no is thereIf (2)

wirsconnecting in the loss

thermalbe will theresystem realany In )1(:

36

5.132

1,5.22

2

1

9 ,15Q

53Q ,

24

)(

'

'2

'2

'2

'1

'1

'1

'2

'1

'2

'1

'2

'1

2

'2

1

'1

'2

'121

note

JU

JVQUJVQU

VV

CQC

QC

Q

C

Q

CQQQQ

b

EXAMPLE 26.8: The "breakdown" field strength, at which dry air loses its insulating ability and allows a discharge to pass through it, is about 3 × 106 V/m. What is the energy density at this field strength?

densityenergy maximum the(2)

strength field breakdown (1)the

/40

)/103)(/1085.8(2

1

2

1

3

262212

2

0

mJ

mVmNC

EuE

EXAMPLE 26.9: Use Eq. 26.10 to derive the potential energy of a metal sphere of radius R with charge Q.

field. electric itsin stored being as sphere charged theofenergy the22

2

)4()(2

1

2

1

r4d

R)(r

22

2

2

2

22

20

2

0

2

2

R

kQdrr

kQUE

drr

kQ

drrr

kQ

dVEdVudU

drVr

kQE

R

EE

EXAMPLE 26.10: A dielectric slab of thickness ; and dielec tric constant « is inserted into a parallel plate capacitor with plates of area A, separated by distance d, as shown in Fig. 26.19. Assume that the battery is disconnected before the slab is inserted. What is the capacitance?

dACas

td

AVAVQC

ttd

ttdEVV

ttEtdEV

A

QE

totaltotal

total

D

/,1

)11

( //

)( )(V

V ,)(

EE ;

0

0

00

0D0

0

D00

0D

00

0

Exercises of chapter 26

• Questions:

• Exercises:4,6,7,12,13,41,42,43

• Problems:1,2,3,4,5,6,11