• Conductors are commonly used as places to store charge• You can’t just “create” some positive charge somewhere, you have to

have corresponding negative charge somewhere elseDefinition of a capacitor:• Two conductors, one of which stores charge +Q, and the other of

which stores charge –Q.

Capacitors

Vab

+Q

–Q

Can we relate the charge Q that develops to the voltage difference V?• Gauss’s Law tells us the electric field

between the conductors• Integration tells us the potential difference

2ˆek Q

rE r

2

ae

b

k QV dr

r

a

e

b

k Q

r 1 1

eV k Qa b

Ch 26

• The relationship between voltage difference and charge is always linear• This allows us to define capacitance• Capacitance has units of Coulomb/Volt

• Also known as a Farad, abbreviated F• A Farad is a very large amount of capacitance

• Let’s work it out for concentric conducting spheres:

Capacitance1 1eV k Q

a b

Q C V

QC

V

e

ab

k b a

What’s the capacitance of the Earth, if we put the “other part” of the charge at infinity?

6370 kma

b

e

abC

k b a

e

ab

k b

6

9 2 2

6.37 10 m

8.988 10 N m / C

4

2

7.09 10

N m / C

CF

V

709 F

• A “more typical” geometry is two large, closely spaced, parallel conducting plates• Area A, separation d.

Parallel Plate Capacitors

Let’s find the capacitance:• Charge will all accumulate on the inner surface

• Let + and – be the charges on each surface• As we already showed using Gauss’s law, this means

there will be an electric field given by• If you integrate the electric field over the distance d,

you get the potential difference

A

dQ A

0

ˆE n

V dE s 0

dx

0

d

0

Qd

A

QC

V

0 A

Cd

To get a large capacitance, make the area large and the spacing small

Circuit symbol for a capacitor:

Warmup 08

Ex A 20.0 F spherical capacitor is composed of two metallic speheres, one having twice the radius of the other. If the region between the spheres is a vacuum, determine the volume of this region.

Solve on board

Capacitors in Parallel

2 V 3 F 6 F

When we close the switch, how much charge flows from the battery?A) 36 C B) 4 C C) 18 CD) 8 C E) 10 C

• The voltage difference across each capacitor will be 2 V.

1 1 3 F 2 V 6 CQ C V

tot 1 2Q Q Q 18 C

2 2 6 F 2 V 12 CQ C V

• When capacitors are connected like this at both ends, we say they are connected in parallel

• The combined capacitors act like a single capacitor with capacitance:

tot 1 2C C C

Q C V

Capacitors in Series

V C1

• Charge flows to the left side of the first capacitor• Since capacitors have balanced charge, the right

side must have the equal and opposite charge• The only place it can get this charge is the left side of the right

capacitor• The charges on the second capacitor are also balanced

C2

When we close the switch, which capacitor gets more charge Q on it?A) The one with the bigger capacitanceB) The one with the smaller capacitanceC) They get the same amount of chargeD) Insufficient information

+Q -Q +Q -Q

1 1Q C V 2 2C V

V1

• The voltage difference on each capacitor is not the same, since only one end is connected. But the charge is the same.

• The two voltages differences can be added together

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

1 1Q

C C

V2

QC

V

1

1 2

1 1C

C C

Warmup 08

Series and Parallel• When two circuit elements are connected at one end, and nothing else

is connected there, they are said to be in series

1 2

1 1 1

C C C

• When two circuit elements are connected at both ends, they are said to be in parallel

1 2C C C

C1 C2

C1 C2

• These formulas work for more than two circuit elements as well.

C1

C2 C3 C4

C5 1 2 3 4 5

1 1 1 1 1 1

C C C C C C

Ct1- Two capacitors C1 and C2 are connected in a series connection. Suppose that their capacitances are in the ratio C2/C1 = 2/1. When a potential difference, V, is applied across the capacitors, what is the ratio of the charges Q2 and Q1 on the capacitors? Q2/Q1 = A. 2 B. 1 C. 1/2 D. none of the above E. Need more information

CT2 - For the capacitors above the ratio of the voltage drops across each one is V2/V1 = A. 2 B. 1 C. 1/2 D. none of the above E. Need more information CT3 - If the capacitors were connected in parallel, how would you answer the above two questions?

V

C1 C2

Sample Problems60 F

60 F

60 F

What is the capacitance of the structure shown at left?A) 180 F B) 120 F C) 30 FD) 20F E) none of the above

• They are connected end-to-end, so they are in series

1 2 3

1 1 1 1

C C C C 1 1 1

60 60 60

3 1

60 20

• Now add a wire across the middle. The two ends of the middle capacitor are at the same potential; therefore, there is no charge on this capacitor

• You might as well ignore it

1 3

1 1 1

C C C

1 1 1

60 60 30

Complicated Capacitor Circuits

4

The capacitance of the capacitors in pF below is marked.What is the effective capacitance of all the capacitors shown?

• For complex combinations of capacitors, you can replace small structures by equivalent capacitors, eventually simplifying everything

10 V

3

2 1 5

• Capacitors 1 and 5 are connected at both ends- therefore they are parallel

1 5 6C 6

• Capacitors 3 and 6 are connected at just one end – therefore they are series

1 1 1

3 6C

1

2 2C

2

• All three capacitors are now connected at both ends – they are all in parallel 2 4 2 8C

Exs – Serway 26 - 13 and 14. Two capacitors C1 = 5.00 F and C2 = 12.0 F are connected in parallel, and the resulting combination is connected to a 9.00 V battery. (a) What is the value of the equivalent capacitance of the combination, (b) what are the potential differences across each capacitor?, and (c) what are the charges on each capacitor? Repeat if they are in series.

Solve on board

Warmup09

Energy in a capacitor• Suppose you have a capacitor with charge q already on it,

and you try to add a small additional charge dq to it, where dq is small. How much energy would this take?

• The side with +q has a higher potential• Moving the charge there takes energy• The small change in energy is:

q C V

+q –q

dq

dU dq V dq q

C

• Now, imagine we start with zero charge and build it up gradually to q = Q• It makes sense to say an uncharged capacitor has U = 0

0

q Q

qU dU

0

Q q dq

C

2

02

Qq

C

2

2

Q

C

2

2

QU

C Q C V

22

2 2

C VQU

C

Energy in a capacitor (2)

20 V 1 F 2 F

20 V1 F 2 F

For each of the two circuits, which capacitor gets more energy in it?

A) The 1 F capacitor in each circuitB) The 2 F capacitor in each circuitC) The 1 F capacitor in the top circuit,

the 2 F capacitor in the bottom circuitD) The 2 F capacitor in the top circuit,

the 1 F capacitor in the bottom circuitE) They are equal in each circuit

22

2 2

C VQU

C

• Capacitors in parallel have the same voltage difference V

• The larger capacitor has more energy

• Capacitors in series have the same charge Q

• The smaller capacitor has more energy

CT4- Consider a simple parallel-plate capacitor whose plates are given equal and opposite charges and are separated by a distance d. Suppose the plates are pulled apart until they are separated by a distance D > d. The electrostatic energy stored in the capacitor is A. greater than B. the same as C. . smaller than before the plates were pulled apart.

Energy density in a capacitorSuppose you have a parallel plate capacitor with area A, separation d, and charged to voltage V.

(1) What’s the energy divided by the volume between the plates?(2) Write this in terms of the electric field magnitude

A

d

212U C V 20

2

AV

d

Uu

V

U

Ad

• Energy density is energy over volume

2

022

A V

Ad

2

0

1

2

V

d

V

d

E

2102u E

• We can associate the energy with the electric field itself• This formula can be shown to be completely generalizable

• It has nothing in particular to do with capacitors

Ex- A uniform electric field E = 3000 V/m exists within a certain region of space. What volume of space contains an energy equal to 1.00 x 10-7 J? Express your answer in m3 and liters.

Solve on Board

Warmup09

Dielectrics in Capacitors• What should I put between the metal

plates of a capacitor?• Goal – make the capacitance large

+ +

+ +

+ +

+ +

+ +

+ +

• The closer you put the plates together, the bigger the capacitance• It’s hard to put things close together –

unless you put something between them• When they get charged, they are also very

attracted to each other• Placing an insulating material – a dielectric –

allows you to place them very close together• The charges in the dielectric will also shift• This partly cancels the electric field• Small field means smaller potential

difference• C = Q/V, so C gets bigger too

0 AC

d

0 AC

d

1

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

––––––––––––

Choosing a dielectricWhat makes a good dielectric?• Have a high dielectric constant

• The combination 0 is also called , the permittivity• Must be a good insulator

• Otherwise charge will slowly bleed away• Have a high dielectric strength

• The maximum electric field at which the insulator suddenly (catastrophically) becomes a conductor

• There is a corresponding breakdown voltage where the capacitor fails

0 AC

d

CT -5-Consider a capacitor made of two parallel metallic plates separated by a distance d. The top plate has a surface charge density +s, the bottom plate –s. A slab of metal of thickness l < d is inserted between the plates, not connected to either one. Upon insertion of the metal slab, the potential difference between the plates A. increases. B. decreases. C. remains the same. CT - 6 - A dielectric is inserted between the plates of a capacitor. The system is then charged and the dielectric is removed. The electrostatic energy stored in the capacitor is A. greater than B. . the same as C. . smaller than it would have been if the dielectric were left in place. CT - 7- A parallel-plate capacitor is attached to a battery that maintains a constant potential difference V between the plates. While the battery is still connected, a glass slab is in-serted so as to just fill the space between the plates. The stored energy A. increases. B. . decreases. C. . remains the same.

Consider isolated capacitor

Consider isolated capacitor after charging

What are capacitors good for?• They store energy

• The energy stored is not extremely large, and it tends to leak away over time

• Gasoline or fuel cells are better for this purpose• They can release their energy very quickly

• Camera flashes, defibrillators, research uses• They resist changes in voltage

• Power supplies for electronic devices, etc.• They can be used for timing, frequency filtering, etc.

• In conjunction with other parts