Lecture 5 Capacitance Chp. 26

•Cartoon - Capacitance definition and examples.

•Opening Demo - Discharge a capacitor •Warm-up problem •Physlet •Topics •Demos • Circular parallel plate capacitior •Cylindrical capacitor •Concentric spherical capacitor •Leyden jar capacitor •Dielectric Slab sliding into demo •Show how to calibrate electroscope

•

Capacitance

Definition of capacitance A capacitor is a useful device in electrical circuits that allows us to store charge and electrical energy in a controllable way. The simplest to understand consists of two parallel conducting plates of area A separated by a narrow air gap d. If charge +Q is placed on one plate, and -Q on the other, the potential difference between them is V, and then the capacitance is defined as C=Q/V. The SI unit is C/V, which is called the Farad, named after the famous and creative scientist Michael Faraday from the early 1800’s.

• Applications – Radio tuner circuit uses variable capacitor – Blocks DC voltages in ac circuits – Act as switches in computer circuits – Triggers the flash bulb in a camera – Converts AC to DC in a filter circuit

Parallel Plate Capacitor



Electric Field of Parallel Plate Capacitor

Area A +++++++++++ S E ------------------- + q - q Gauss Law

E

   0  

q A V C



ES

 

q V

 

qS

 0

A q qS

 0

A

  0 

A S



E



q

 0

A



q

  0

EA Coulomb/Volt = Farad

Show Demo Model, calculate its capacitance , and show how to charge it up with a battery.

Circular parallel plate capacitor r s r r = 10 cm A =  r 2 =  (.1) 2 A = .03 m 2 S = 1 mm = .001 m

C

  0

A S C

 ( 10  11 ) .

03 .

001

Coulomb Volt



Farad C

 3  10  10

F C

 300

pF

p = pico = 10 -12

Demo Continued Demonstrate

1.

As S increases, voltage increases.

2. As S increases, capacitance decreases.

3. As S increases, E 0 and q are constant.

Dielectrics

• A dielectric is any material that is not a conductor, but polarizes well. Even though they don’t conduct they are electrically active.

– Examples. Stressed plastic or piezo-electric crystal will produce a spark.

– When you put a dielectric in a uniform electric field (like in between the plates of a capacitor), a dipole moment is induced on the molecules throughout the volume. This produces a volume polarization that is just the sum of the effects of all the dipole moments. If we put it in between the plates of a capacitor, the surface charge densities due to the dipoles act to reduce the electric field in the capacitor.

Dielectrics

• The amount that the field is reduced defines the dielectric constant  from the formula E = E 0 /  , where E is the new field and E 0 is the old field without he dielectric.

• Since the electric field is reduced and hence the voltage difference is reduced (since E = Vd), the capacitance is increased.

–

C = Q / V = Q / (V 0 /



) =



C 0

– 

is typically between 2 – 6 with water equal to 80

–

Show demo dielectric slab sliding in between plates. Watch how capacitance and voltage change. Also show aluminum slab.

Permanent dipoles

_ ++ _

Induced dipoles

E 0 = the applied field E’ = the field due to induced dipoles E = E 0 E’

S

C



q V V E

0  

E

 0

S

 0  

q A E

0   0

q A E



E

0 

V



E

0

S



V



V

 0

V



qS

 0

A C

  0

A S C



q V C

 

q V

0

C

 

C

0

Find the capacitance of a ordinary piece of coaxial cable (TV cable) For a long wire we found that

E r

 2

k



r

where r is radial to the wire.

• r

V a



V b E

.

ds

  

a



E

.

ds

  2

k



a



b



b Eds

cos180  

Eds



dr r

  2

k

 ln

Edr r

a b outer insulator radius b metal braid with - q • signal wire radius a with + q Insulator (dielectric  )    ds = - dr because path of integration is radially inward

V a



V b

  2

k

 ln

a b

or

V

 2

k

 ln

b a

V a is higher than V b  

Q L k

 1 4  0   air 

V C C

   2

Q



Q V

0 

L

ln

b a Q

2  0

L



Q

ln

b a

2  0

L

ln

b a

a = 0.5 mm

C L

 2  0

b

ln

a

b = 2.0 mm   2

C L

 6  10  11 ln 4  6  10  11 1 .

38

C L C L

 43

PF

 86

PF m m

 0 (air)  = 2  

Model of coaxial cable for calculation of capacitance

Outer metal braid Signal wire

Spherical capacitor or sphere

Recall our favorite example for E and V is spherical symmetry Q R The potential of a charged sphere is V = (kQ)/R with V = 0 at r =  .

The capacitance is

C



Q V



Q kQ R



R k

 4  0

R

Where is the other plate (conducting shell)?

It’s at infinity where it belongs, since that’s where the electric lines of flux terminate.

k = 10 10 and R in meters we have 

C



R

10 10

C



R

(  10  10

R

(

m

)  10  12

R

(

cm

)

cm

)

PF

Earth: C = (6x10 8  F cm)PF = 600 Marble: 1 PF Basketball: 15 PF You: 30 PF

Demo

: Leyden jar capacitor

Demo

: Show how you measured capacitance of electroscope

Capacitance of two concentric spherical shells

- q Integration path a +q b  E

V a



V b E

.

ds

  

b a



E

.

ds Eds

cos180  

a



Edr b

 

Eds



Edr

 ds = - dr

V a



V b

 

b a



Edr

 

b a



kq

/

r

2

dr V



kq

1

r b a



kq

( 1

a

 1

b

) 

kq

(

b



ab a

)  

kq b a



dr r

2

C



q

/

V



k

(

b ab



a

)  4  0

ab b



a



•

Electric Potential Energy of Capacitor

As we begin charging a capacitor, there is initially no potential difference between the plates. As we remove charge from one plate and put it on the other, there is almost no energy cost. As it charges up, this changes.

+q -q At some point during the charging, we have a charge q on the positive plate.

+ The potential difference between the plates is V = q/C. As we transfer an amount dq of positive charge from the negative plate to the positive one, its potential energy increases by an amount dU.

dU



Vdq



q C dq

.

The total potential energy increase is

U



Q

0 

q C dq



q

2 2

C



Q

2 2

C

Also

U

 1 2

QV

 1 2

CV

2  1 2

Q

2

C

using C = Q/ V



Graphical interpretation of integration



U dU



Vdq

 0 

Q Vdq

where V = q/C V V = q/c q/c dq Q q

U

 1

C i N

  1

q i



q i

= Area under the triangle Area under the triangle is the value of the integral

Q

0 

q C dq

Area of the triangle is also = 1/2 bh Area = 1 2 (

b

)(

h

)  1 2 (

Q

)(

Q C

)  1 2

Q

2

C Q

0 

q C dq



q

2 2 0 Q 

Q

2 2

C

Where is the energy stored in a capacitor?

• Find energy density for parallel plate capacitor. When we charge a capacitor we are creating an electric field. We can think of the work done as the energy needed to create that electric field. For the parallel plate capacitor the field is constant throughout, so we can evaluate it in terms of electric field E easily. Use U = (1/2)QV

E

   0    

Q



A

and V = ES Solve for Q =  AE, V = ES and substitute in We are now including dielectric effects: 

U

 1 2

QV U AS

 1 2 

E

2  1 2 ( 

AE

)(

ES

)  1 2 

E

2 (

AS

)   volume occupied by E   1 2 

E

2 Electrostatic energy density general result for all geometries.

To get total energy you need to integrate over volume.

How much energy is stored in the Earth’s atmospheric electric field?

(Order of magnitude estimate) atmosphere h Earth R R = 6x10 6 m 20 km

E U

 100

V m

 1 2  0

E

2  10 2 

Volume Volume

 4 

R

2

h Volume

 4  ( 6  10 6 ) 2 ( 2  10 4 )  8 .

6  10 18

m

3

U U

  1 ( 10  11 )( 10 2 2 4 .

3  10 11

J

)( 8 .

6  10 18 ) This energy is renewed daily by the sun. Is this a lot?

The total solar influx is 200 Watts/m 2

U sun

 200  3 .

14 ( 6  10 6 )  2  10 16

J s

 2  10 21

J day

World consumes about 10 18 J/day. This is 1/2000 of the solar flux.

U U sun

 2  10  10 Only an infinitesimal fraction gets converted to electricity.

Parallel Combination of Capacitors

Typical electric circuits have several capacitors in them. How do they combine for simple arrangements? Let us consider two in parallel.

C 1 Q 1 C 2 Q 2 C 1 V + V + C = Q/V We wish to find one equivalent capacitor to replace C 1 Let’s call it C.

and C 2 . The important thing to note is that the voltage across each is the same and equivalent to V. Also note what is the total charge stored by the capacitors? Q.

Q



Q

1 

Q

2 

C

1

V



C

2

V

 (

C

1 

C

2 )

V Q



C

1 

C

2 

C



C

1 

C

2

V

Series Combination of Capacitors

V + Q + Q + C 1 V 1 C 2 V 2 V + Q + C V

C



Q V V



Q C

What is the equivalent capacitor C?

Voltage across each capacitor does not have to be the same.

The charges on each plate have to be equal and opposite in sign by charge conservation.

The total voltage across each pair is:

V



V

1 

V

2 

Q C

1 

Q C

2  1

Q

(

C

1  1

C

2 )  1

Q

(

C

) So 1

C

1  1

C

1  1

C

2 ; Therefore,

C



C

1

C

2

C

1 

C

2

Sample problem

V + C 1 C 2 C 3 C 1 = 10  F C 2 = 5.0  F C 3 = 4.0  F a) Find the equivalent capacitance of the entire combination.

C 1 and C 2 are in series.

C

1

C

12 12  1

C

1   10 10   5 5 1

C

2   50 15

C

12  3 .



C

1

C

2

C

1 

C

2 3 

F

C 12 and C 3 are in parallel.

C eq



C

12 

C

3  3 .

3  4 .

0  7 .

3 

F

Sample problem (continued)

V + C 1 C 2 C 3 C 1 = 10  F C 2 = 5.0  F C 3 = 4.0  F b) If V = 100 volts, what is the charge Q 3 on C 3 ?

C = Q/V

Q

3 

C

3

V

 4 .

0  10  6  100

Q

3  4 .

0  10  4

Coulombs

c) What is the total energy stored in the circuit?

U

 1 2

C eq V

2  1 2  1 .

3  10  6  10 4  3 .

6  10  2

J U

 3 .

6  10  2

J

Warm up set 5

1. HRW6 26.TB.03. [119752] A capacitor C "has a charge Q". The actual charges on its plates are: Q/2, Q/2 Q, -Q Q/2, -Q/2 Q, 0 Q, Q 2. HRW6 26.TB.13. [119762] Pulling the plates of an isolated charged capacitor apart: increases the potential difference increases the capacitance does not affect the capacitance decreases the potential difference does not affect the potential difference 3. HRW6 26.TB.25. [119774] Let Q denote charge, V denote potential difference and U denote stored energy. Of these quantities, capacitors in series must have the same: Q only Q and U only U only V and U only V only

What is the electric field in a sphere of uniform distribution of positive charge. (nucleus of protons)

R  

Q

4 3 

R

3 r E    

EdA



E

4 

r

2 

q enc

 0  4 

r

3  0 3

E

 

r

3  0 

Q

4  0

R

3

r

