Definitions & Examples

d

A

- - - - -+ + + +a

b L

C1 C2

a

b

C3

C a b

VQC

Today…• Energy stored in the electric field

– as distinguished from electric potential energy of a charge located in an electric field.

• Definition of Capacitance

• Example Calculations• Parallel Plate Capacitor• Cylindrical Capacitor

• Combinations of Capacitors

• Capacitors in Parallel• Capacitors in Series

Text Reference: Chapter 25.2, 4 Examples: 25.2,3,5,6 and 7

Last time…• Two ways to say same thing: E(x,y,z) and V(x,y,z)

• Potential energy = potential x test charge (q)

• Conductors in E-fields “become” equipotential surfaces/volumes

– E-field always normal to surface of conductor

( , , ) ( , , )E x y z V x y z

00 0 0( , , ) ( , , ) ( , , )

r

rV x y z V x y z dl E x y z

U qV

Electric Potential Energy• In addition to discussing the energy of a "test" charge in a Coulomb

field, we can speak of the electric potential energy of the field itself!

• Reasons? – Work had to be done to assemble the charges (from infinity)

into their final positions. – This work is the potential energy of the field. – The potential energy of a system of N charges is defined to be

the algebraic sum of the potential energy for every pair of charges.

• This theme continues in the course with E in capacitors and B in inductors.

• Let’s start with a couple of example calculations.

Electric Potential Energy• Example 1: Nuclear Fission

What is the potential energy of the two nuclei?19

9

1 215

1.6 10 C(9 10 )( 56 )( 36 )e

14.6 10 m

e ekq qUd

8e(2 10 V) = 200 MeV

235Unucleus + n(92 protons)

Ba Kr(56 p) (36 p)

d = 14.6 •10-15m

Compare this to the typical energy released in a chemical reaction, ~10eV.By allowing the two fragments to fly apart, this potential energy kinetic energy heat drives a turbine generate electricity.

(What holds the protons together in the nucleus to begin with? Gluons!The “strong force” very short range, very very strong!)

(The electrons are all

“far” away)

Electric Potential Energy• Example 2: What is the potential energy of

this collection of charges?

Step 1: Bring in +2q from infinity. This costs nothing.

dqqU

04))(2(

Step 3: Bring in 2nd -q charge. It is attracted to the +2q, but repelled from the other -q charge. The total work (all 3 charges) is

-q

-q+2q

d

d

d2

Step 2: Bring in one -q charge. This is attracted! Work required is negative

d)q)(q(

d)q)(q(

d)q)(q(U

2442

42

000

214

4 0

2

dqU

Electrical Potential Energy• Example 2: What is the potential energy of

this collection of charges? -q

-q+2q

d

d

• What is the significance of the minus sign?

A negative amount of work was required to bring these charges from infinity to where they are now. (i.e., the attractive forces between the charges are larger than the repulsive ones).

214

4 0

2

dqU

1

Lecture 7, ACT 1• Consider the 3 collections of point charges shown

below. – Which collection has the smallest potential energy?

(a) (b) (c)

d

dd

-Q -Q

-Q

d

dd

-Q +Q

+Q

d

d

-Q +Q

+Q

• We have to do positive work to assemble the charges in (a) since they all have the same charge and will naturally repel each other. In (b) and (c), it’s not clear whether we have to do positive or negative work since there are 2 attractive pairs and one repulsive pair.

(a) dQU

2

0413

dQU

2

041

(b)d

QU24

1 2

0(c)

U0 (a)(b) (c)

Electric potential and potential energy• Consider three charges as our “sources”:

• We want to find the electric potential that these charges give rise to or “source” at some arbitrary point, P

• Use superposition of V(r) from before– now all three charges “source” V(r)

throughout all space– evaluate this anywhere and at P in

particular:

popopoPat r

QrQ

rQV

3

3

2

2

1

1

41

41

41

Q1

Q2Q3

P

r1p

r2pr3p

Electric potential and potential energy

• What about electric potential energy of an added fourth charge, qtest?

• Those three charges source V(r)

• The potential energy of an added test charge is just qtestV (its position)

• Use V(r) from previous slide– multiply by qtest

31 2

1 2 3

1 1 14 4 4

testof q at P test

o p o p o p

QQ QU qr r r

qtest

Q1

Q2Q3r1p

r2pr3p

Potential energy, potential and potential energy

• Potential energy stored in a static charge distribution– work we do to assemble the charges

• Electrostatic potential at any point in space– sources give rise to V(r)

• Electric potential energy of a charge in the presence of a set of source charges– potential energy of the test charge equals the

potential from the sources times the test charge

Capacitance• A capacitor is a device whose purpose is to store electrical

energy which can then be released in a controlled manner during a short period of time.

• A capacitor consists of 2 spatially separated conductors which can be charged to +Q and -Q respectively.

• The capacitance is defined as the ratio of the charge on one conductor of the capacitor to the potential difference between the conductors.

VQC

• The capacitance belongs only to the capacitor, independent of the charge and voltage.

Example: Parallel Plate Capacitor

• Calculate the capacitance. We assume +, - charge densities on each plate with potential difference V:

d

A

- - - - -+ + + +

VQC

• Need Q:• Need V: from def’n:

– Use Gauss’ Law to find E

AQ

B

AAB ldEVV

Recall: Two Infinite Sheets (into screen)

• Field outside the sheets is zero

• Gaussian surface encloses zero net charge

E=0 E=0

E

+

+

++

++

+

+

++

-

-

--

-

--

-

-

-+

+

---

A

AAQ

insideAESdE

• Field inside sheets is not zero:

• Gaussian surface encloses non-zero net charge

0

E

Example:Parallel Plate Capacitor

d

A

- - - - -

+ + + +

•Calculate the capacitance:•Assume +Q, -Q on plates with potential difference V.

• As hoped for, the capacitance of this capacitor depends only on its geometry (A,d).

• Note that C ~ length; this will always be the case!

00

AQE

dAQldEVV

B

AAB

0

dA

VQC 0

Practical Application: Microphone (“condenser”)

Sound waves incident pressure oscillations oscillating plate separation d oscillating capacitance ( ) oscillating charge on plate oscillating current in wire ( ) oscillating electrical signal

1~Cd

dQIdt

d

Moveable plate Fixed plate

Current sensor

Battery

See this in action at http://micro.magnet.fsu.edu/electromag/java/microphone/ !

Example: Cylindrical Capacitor

• Calculate the capacitance:

• Assume +Q, -Q on surface of cylinders with potential difference V.

a

b L

r

Recall: Cylindrical Symmetry

• Gaussian surface is cylinder of radius r and length L

• Cylinder has charge Q

LrQE

02

• Apply Gauss' Law:

0

2

QrLESdE

+ + + +

Er

L

Er

+ + +Q

Example:Cylindrical Capacitor

• Calculate the capacitance:

• Assume +Q, -Q on surface of cylinders with potential difference V.

rLQE

02

ab

LQdr

rLQEdrldEV

b

a

a

b

a

b

ln22 00

If we assume that inner cylinder has +Q, then the potential V is positive if we take the zero of potential to be defined at r = b:

abL

VQC

ln

2 0

2

a

b L

r

+Q

- Q

Lecture 7, ACT 2• In each case below, a charge of +Q is placed on a solid

spherical conductor and a charge of -Q is placed on a concentric conducting spherical shell. – Let V1 be the potential difference between the spheres with (a1, b). – Let V2 be the potential difference between the spheres with (a2, b). – What is the relationship between V1 and V2? (Hint – think about

parallel plate capacitors.)

• What we have here are two spherical capacitors.• Intuition: for parallel plate capacitors: V = (Q/C) = (Qd)/(A0).

Therefore you might expect that V1 > V2 since (b-a1) > (b-a2).• In fact this is the case as we can show directly from the definition of V!

(a) V1 < V2 (b) V1 = V2 (c) V1 > V2

a2

b

+Q-Q

a1

b

+Q-Q

Capacitors in Parallel

• Find “equivalent” capacitance C in the sense that no measurement at a, b could distinguish the above two situations.

• Aha! The voltage across the two is the same….

Parallel Combination:

Equivalent Capacitor:

2

2

1

1

CQ

CQV

1

212 CCQQ

VCCCQ

VQQ

VQC

1

21121 )(

21 CCC

C1 C2V

a

b

Q2Q1

-Q1 -Q2

C V

a

b

Q

-Q

Capacitors in Series

• Find “equivalent” capacitance C in the sense that no measurement at a, b could distinguish the above two situations.

• The charge on C1 must be the same as the charge on C2 since applying a potential difference across ab cannot produce a net charge on the inner plates of C1 and C2 – assume there is no net charge on node between C1 and C2

C a b

+Q -Q

C1

C2a b

+Q

-Q

CQVab

2121 C

QCQVVVab

RHS:

LHS:

21

111CCC

-Q

+Q

Examples:Combinations of Capacitors

C1 C2C3

C

a

b

a b

• How do we start??

• Recognize C3 is in series with the parallel combination on C1 and C2. i.e.,

213

111CCCC

321

213 )(CCC

CCCC

3

Lecture 7, ACT 3• What is the equivalent capacitance, Ceq, of the

combination shown?

(a) Ceq = (3/2)C (b) Ceq = (2/3)C (c) Ceq = 3C

o

o

C CCCeq

3A

• What is the relationship between V0 and V in the systems shown below?

conductor

(Area A)

V

+Q

-Q

d/3

d/3

(a) V = (2/3)V0 (b) V = V0 (c) V = (3/2)V0

d

(Area A)

V0

+Q

-Q

3B

Lecture 7, ACT 3• What is the equivalent capacitance, Ceq, of the

combination shown?

(a) Ceq = (3/2)C (b) Ceq = (2/3)C (c) Ceq = 3C

o

o

C CCCeq

3A

CCC C C1

CCC111

1

21CC CCCCeq 2

32

Lecture 7, ACT 3

• The electric field in the conductor = 0.• The electric field everywhere else is: E = Q/(A0)• To find the potential difference, integrate the electric field:

EdldEV

0

30

3dEdEldEV

EdV32

(a) V = (2/3)V0 (b) V = V0 (c) V = (3/2)V0

• What is the relationship between V0 and V in the systems shown below?

d

(Area A)

V0

+Q

-Qconductor

(Area A)

V

+Q

-Q

d/3

d/3

3B

Lecture 7, ACT 3, Another Way

(a) V = (2/3)V0 (b) V = V0 (c) V = (3/2)V0

• What is the relationship between V0 and V in the systems shown below?

d

(Area A)

V0

+Q

-Qconductor

(Area A)

V

+Q

-Q

d/3

d/3

3B

• The arrangement on the right is equivalent to capacitors (each with separation = d/3) in SERIES!!

CCeq 21

conductord/3

(Area A)

V

+Q

-Qd/3

d/3+Q

-Qd/3

000

23

23

3/21 C

dA

dACeq

00 3

22/3

VC

QCQVeq

Summary•A Capacitor is an object with two spatially separated conducting surfaces.•The definition of the capacitance of such an object is:

VQC

• The capacitance depends on the geometry :

d

A

- - - - -+ + + +

Parallel Plates

dAC

a

b L

r

+Q

-Q

Cylindrical

)/ln( abLC

a

b

+Q-Q

Spherical

ababC

Next time:

• Energy stored in a capacitor

• Dielectrics

Reading assignment: Ch. 25.3-6Examples: 25.4,5,6,7,8,9 and 10

Appendix: Another example• Suppose we have 4

concentric cylinders of radii a,b,c,d and charges +Q, -Q, +Q, -Q

• Question: What is the capacitance between a and d?

• Note: E-field between b and c is zero! WHY??• A cylinder of radius r1: b < r1< c

encloses zero charge!

-Q+Q

-Q+Q

ab

c

d

Note: This is just the result for 2 cylindrical capacitors in series!

d

c

b

aad dr

rLQdr

rLQV

00 20

2

b

a

d

c

ad

rdr

rdr

LVQC 02

cd

ab

LClnln

2 0

Conductors versus InsulatorsCharges move to Charges cannot cancel out electric move at allin the conductor

E=0 equipotential Charge distributionsurface on insulator

unaffected by external fields

All charge on surface

What does grounding do?1. Acts as an “infinite” source or sink of charge.

2. The charges arrange themselves in such a way as to minimize the global energy (e.g., E0 at infinity, V0 at infinity).

3. Typically we assign V = 0 to ground.