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Lecture 24: Capacitance and Energy Storage, Study notes of Physics

The concept of capacitance, the relationship between capacitance and geometry, and the calculation of capacitance for parallel plate, cylindrical, and spherical capacitors. It also discusses capacitors in parallel and series, and the energy stored in the electric field. Problem-solving examples and exercises.

Typology: Study notes

Pre 2010

Uploaded on 08/31/2009

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Lecture 24 - Capacitance
Review
Spherical Capacitor
Capacitors in Parallel
Capacitors in Series
Storing Energy in the Electric Field
Other Expressions for the Energy
Energy Density
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Lecture 24 - Capacitance

  • Review • Spherical Capacitor • Capacitors in Parallel • Capacitors in Series • Storing Energy in the Electric Field • Other Expressions for the Energy • Energy Density

Summary

A Capacitor is an object with two spatially separated conductingsurfaces.

The definition of the capacitance of such an object is:

Q V

C

The capacitance depends on the geometry :

d

A

+ + + + - - - - -

Parallel Plates

A d

C

a b

L

r

+

Q

-

Q

Cylindrical

)

/

ln(

a

b L

C

a

b

+

Q

-

Q

Spherical

a

b

ab

C

Problem 25-

The plates of a spherical capacitor have radii 38.0 mm and 40.0 mm. (a)

Calculate the capacitance.

(b)

What must be the plate area of a

parallel-plate capacitor with the same plate separation and capacitance?

Capacitors in Parallel

Equivalent capacitor

: single capacitor

that has the same capacitance as acombination

Capacitors in parallel

when a potential

difference that is applied across theircombination results in the same potentialdifference across each capacitor

Figure 25-

(

)

=

=

=

=

    • = + + =

=

=

=

n j

j

eq eq

C

C

C

C

C

V

q

C

V C C C q q q q

V C q V C q V C q

1

3

2

1

3 2 1 3 2 1

3 3 2 2 1 1

Capacitors in Series

Capacitors in series

when a potential

difference that is applied across theircombination is the sum of the resultingpotential differences across each capacitor

Equivalent capacitor

: single capacitor

that has the same capacitance as acombination

Figure 25-

=

=

=

=

⎞ ⎟⎟ ⎠

⎛ ⎜⎜ ⎝

    • = + + =

=

=

=

n j

j

eq eq

C

C

C

C

C

q V

C

C C C q V V V V

C

q

V

C

q

V

C

q

V

1

3

2

1

3 2 1 3 2 1

3 3 2 2 1 1

1

1

1

1

1

1

1

1

1

Checkpoint 25-

A battery of potential V stores charge

q

on a combination of two

identical capacitors. What are the potential difference and the chargeon either capacitor if the capacitors are

(a)

in parallel and

(b)

in

series?

(a) V

1

= V

2

= V

q

1

= q

= q/

( q

1

  • q

= q )

(b) V

1

= V

2

= V/

( V

1

  • V

2

=V )

q

1

= q

= q

Example

(a)

V

= (2/3)

V

0

(b)

V

=

V

0

(c)

V

= (3/2)

V

0

What is the relationship between

V

0

and

V

in the

systems shown below?

d

(Area

A

)

V

0

+

Q

-

Q

conductor

(Area

A

)

V

+

Q

-

Q

d

/ d

/

3B

The arrangement on the right is equivalent to capacitors (each withseparation =

d

/3) in SERIES!!

C

C

eq

1 2

=

conductor

d

/

(Area

A

)

V

+

Q

-

Q

d

/

d

/

+

Q

-

Q

d

/

(

)

0

0

0

3 2

(^32)

3 /

1 2

C

d A

A d

C

eq

=

=

=

ε

ε

(

)

0

0

2 3

2 /

3

V

C

Q

C

Q

V

eq

=

=

=

14

Sample Problem 25-

(a) Find the equivalent capacitance for the combination of capacitancesshown in Fig. 26-9a, across which the potential difference V is applied.Assume

C

1

= 12.

μ

F, C

2

= 5.

μ

F,

and

C

= 4.

μ

F.

F F F C C C

μ

μ

μ

  1. 7 3. 5 0

.

12

2

1

12

=

=

=

1

3

12

123

28 . 0

5 . 4

1

3 .

17

1

1

1

1

− = + = + =

F F F C C C

μ

μ

μ

F

F

C

57 .

3

)

28 .

0

/(

1

123

=

=

(b) The potential difference that is applied to the input terminal in Fig.26-9 is V = 12.5 V. What is the charge on C

1

? C V F V C q

μ

μ

6 .

44

)

5 .

12

)(

57 .

3 (

123

123

=

=

=

V

C F

q C

V

q

q

58 .

2

3 .

17

6 .

44

12 12

12

123

12

=

=

=

=

μ μ

C

V

F

V

C

V

C

q

μ

μ

0 .

31

)

58 .

2

)(

0 .

12 (

12

1

1

1

1

=

=

=

=

=

Energy Stored in an Electric Field

  • Energy stored equals the work done by an

external agent to charge the capacitor

C
q
U
C
q
dq
q
C
dW
W
dq
q C
dq
V
dW

q

2

2

0

Where is the Energy Stored?

Claim: energy is stored in the electric field itself.Think of the energy needed to charge the capacitoras being the energy needed to create the field.

The electric field is given by:

A

Q

E

0

0

ε

σ ε

=

=

2

0

U
E
Ad

ε

The energy density

u

in the field is given

by:

2

0

E
W Ad
volume
W
u

ε

3

m

J

Units:

This is the energydensity,

u

, of the

electric field….

To calculate the energy density in the field, first consider theconstant field generated by a parallel plate capacitor, where

2

2

1

1

Q

Q

U

=

=

0

2

2 (

/

)

C

A

d

ε

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

+

Q

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

Q

Energy Density

  • Energy stored in the volume between the

plates

2

0

2

0

0

2

1 2

1 2

2

E

u

d V

u

d

A

C

Ad

CV

U Ad

VOL

U

u

ε

ε

ε

=

⎞ ⎟ ⎠

⎛ ⎜ ⎝

=

=

=

=

=

Energy Density

Example

(and another exercise for the student!)

Consider

E

-field between surfaces of cylindrical

capacitor:

Calculate the energy in the field of the capacitor byintegrating the above energy density over the volume ofthe space between cylinders.

is general and is not restricted to the special case of theconstant field in a parallel plate capacitor.

Claim: the expression for the energy density of theelectrostatic field

2

0

1 2

E

u

ε

=

2

1 2

CV

W

=

2

2

0

0

1

1

2

.

2

2

U

E dV

E

rdrdl

etc

ε

ε

π

=

=

=

∫ ∫

Compare this value with what you expect from thegeneral expression: