1

Chapters 24 and 26.4-26.5

2

Capacitor

++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - -

+q

-q

Any two conductors separated by either an insulator or vacuum for a capacitor

The “charge of a capacitor” is the absolute value of the charge on one of conductors.

This constant is called the “capacitance” and is geometry dependent. It is the “capacity” for holding charge at a constant voltage

Potential difference=V

constantV

q

3

Units

1 Farad=1 F= 1 C/VSymbol:

Indicates positive potential

4

Interesting Fact

When a capacitor has reached full charge, q, then it is often useful to think of the capacitor as a battery which supplies EMF to the circuit.

5

Simple Circuit

S

H L

Initially, H & L =0

+q

i -i

-q

After S is closed, H=+qL=-q

6

Recalling Displacement Current

i

+q -q

iid

This plate induces a negative charge hereWhich means the positive charge carriers are moving here and thus a positive current moving to the right

Maxwell thought of the capacitor as a flow device, like a resistor so a “displacement current” would flow between the plates of the capacitor like this

7

If conductors had area, A

Then current density would beJd=id/A

8

Calculating Capacitance

Calculate the E-field in terms of charge and geometrical conditions

Calculate the voltage by integrating the E-field.

You now have V=q*something and since q=CV then 1/something=capacitance

9

Parallel plates of area A and distance, D, apart

CD

A

AqD

C

q

A

qDV

A

qEandEDV

EDsdEsdEV

A

qE

qEA

qAdE

f

i

0

0

0

0

00

0

1

++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - -

Distance=D

Area, A

10

Coaxial Cable—Inner conductor of radius a and thin outer conductor radius b

+q

-q

abL

C

C

ab

L

ab

L

a

b

L

qV

r

dr

L

qrdrEsdEV

rL

qE

qrLE

qAdE

b

a

f

i

ln

2

ln

2

ln2

11

ln2

2ˆ

22

0

0

0

0

0

00

0

11

Spherical Conductor—Inner conductor radius A and thin outer conductor of radius B

CAB

AB

ABAB

AB

ABq

AB

qV

r

drqrdrEsdEV

r

qE

qrE

qAdE

B

A

f

i

0

0

00

20

02

0

2

0

4

41

1

4

11

4

4ˆ

44

12

Isolated Sphere of radius A

AC

A

qV

BLetAB

qV

0

0

0

4

4

11

4

13

Capacitors in Parallel

E C1

C2 C3

CeqE

E

i1 i2 i3i

i

i=i1+i2+i3 implies q=q1+q2+q3

321

321

CCCC

VCVCVCVC

eq

eq

q

q1 q2 q3

14

Capacitors in Series

EC1

i

i q

q

qC2

C3

By the loop rule,E=V1+V2+V3

E Ceq

321

321

321

1111

CCCC

C

q

C

q

C

q

C

q

C

q

C

q

C

qE

C

qE

eq

eq

eq

15

Energy Stored in Capacitors

2

2

2

1

2

1

charge

Work

CVWork

Or

qC

dqC

qWork

C

qVAnd

VdqWork

d

dV

Technically, this is the potential to do work or potential energy, U

U=1/2 CV2 or U=1/2 q2/C

Recall Spring’s Potential EnergyU=1/2 kx2

16

Energy Density, u

u=energy/volume Assume parallel

plates at right Vol=AD U=1/2 CV2

++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - -

Distance=D

Area, A

Volume wherein energy resides

202

2

0

20

0

2

1

2

1

2

1

ED

Vu

VD

A

ADu

D

AC

17

Dielectrics

++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - -

Distance=D

Area, A

++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - -

++++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - - - - -

Insulator

Voltage at which the insulating material allows current flow (“break down”) is called the breakdown voltage1 cm of dry air has a breakdown voltage of 30 kV (wet air less)

18

The capacitance is said to increase because we can put more voltage (or charge) on the capacitor before breakdown.

The “dielectric strength” of vacuum is 1 Dry air is 1.00059

So we can replace, our old capacitance, Cair, by a capacitance based on the dielectric strength, , which is Cnew=*Cair

An example is the white dielectric material in coaxial cable, typically polyethylene (=2.25) or polyurethane (=3.4)

Dielectric strength is dependent on the frequency of the electric field

19

Induced Charge and Polarization in Dielectrics

++++++++++++++++++++++

- - - - - - - - - - - - - - - - - - - - - - -

-- - - - - - - - - - - - - - - - - -

+++++++++++++++++++

E0Ei

ETotal=E0-Ei

11

00

0

00

0

i

i

E

EE

Note that the charges have separated or polarized

20

Permittivity of the Dielectric

0

For real materials, we define a “D-field” where D=0E

For these same materials, there can be a magnetization based on the magnetic susceptibility, , : H= B

adD

tisdH

qadD

D

Dfree

free

enclosed

enclosed

21

Capacitor Rule

For a move through a capacitor in the direction of current, the change in potential is –q/C If the move opposes the current then the

change in potential is +q/C.

i

move

VaVb

Va-Vb= -q/CVa-Vb= +q/C

22

RC Circuits

Initially, S is open so at t=0, i=0 in the resistor, and the charge on the capacitor is 0.

Recall that i=dq/dt

B

A

V

S

R

C

23

Switch to A

Start at S (loop clockwise) and use the loop rule

B

A

V

S

R

C

C

q

dt

dqRV

C

qiRV

VC

qiR

0

24

An Asatz—A guess of the solution

)1()(

0

00

0,0

:

0

RC

t

RCp

RC

t

RC

t

RC

tpRC

t

p

eCVtqCVB

BCV

BeAq

qtat

CVAorC

AV

C

BeAe

C

BV

C

q

dt

dqRV

eRC

B

dt

dqBeAqansatzMy

25

Ramifications of Charge

At t=0, q(0)=CV-CV=0 At t=∞, q=CV (indicating fully charged) What is the current between t=0 and the time

when the capacitor is fully charged?

RC

t

RC

t

RC

t

eR

Ve

RC

CVi

eCVdt

dtq

dt

di

1)(

26

Ramifications of Current At t=0, i(0)=V/R (indicates full current) At t=∞, i=0 which indicates that the current has

stopped flowing. Another interpretation is that the capacitor has an

EMF =V and thus

B

A

V

S

R

~V

Circuit after a very long time

27

Voltage across the resistor and capacitor

B

A

V

S

R

C

RC

t

RC

t

R VeeR

ViRV

Potential across resistor, VR

RC

t

C

RC

t

C

eVV

C

eCV

C

qV

1

1

Potential across capacitor, VC

At t=0, VC=0 and VR=V

At t=∞, VC=V and VR=0

28

RC—Not just a cola

RC is called the “time constant” of the circuit

RC has units of time (seconds) and represents the time it takes for the charge in the capacitor to reach 63% of its maximum value

When RC=t, then the exponent is -1 or e-1

=RC

29

Switch to B

The capacitor is fully charged to V or q=CV at t=0

RC

t

RC

t

eRC

AtiandAetq

kRC

tqdt

RCq

dqC

q

dt

dqR

C

qiR

)()(

ln1

0B

A

V

S

R

C

CV

CVA

AeqtatCVqIf

0)0(,0

30

Ramifications

At t=0, q=CV and i=-V/RAt t=∞, q=0 and i=0 (fully discharging)Where does the charge go?

The charge is lost through the resistor

31

Three Connection Conventions For Schematic Drawings

A

B

C

Connection Between Wires

No Connection

32

Ground Connectors

Equivalently

33

Household Wiring

“hot” or black

“return”/ “neutral” or white

“ground” or green

Normally, the “return” should be at 0 V w.r.t. ground

Single PhaseRated 20 A (NW-14)Max V 120 VAC

In THEORY, but sometimes no!

34

The Death of Little Johnny

Washerhot

neutral

Little Johnny

A short develops between the hot lead and the washer case

RG

120VRLittle Johnny

RG

If RG=∞, then Johnny is safe

Uhoh! It leaks!

If RG=0, then Johnny is dead!

X X

35

Saving Little Johnny

Washerhot

neutral

Little Johnny

A short develops between the hot lead and the washer case

RG

120VRLittle Johnny

RG

Uhoh! It leaks!

No Path to Johnny!