Capacitors and DielectricsCapacitors

•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 theother of which stores charge –Q.

V

ab+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 2

ˆek Q

rE r

2

ae

b

k QV dr

r

a

e

b

k Q

r

1 1eV k Q

a b

•The relationship between voltage difference and charge is normally 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 1

eV k Qa 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

Parallel Plate Capacitors•A “more typical” geometry is two large, closely spaced, parallel conducting plates

•Area A, separation d.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, thismeans there will be an electric field given by:•If you integrate the electric field over the distance d, you get the potential difference

A

d

Q 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:

Capacitors in Parallel

V C1 C2

Q C V

•When capacitors are joined at both ends like this, they are said to be in parallel

•They have the same voltage across them•They can be treated like a single capacitor:

1 1

2 2

Q C V

Q C V

1 2Q Q Q 1 2C C V

1 2C C C

•When capacitors are joined at one end, with nothing else, they are said to be in series

•They have the same voltage across them•They can be treated like a single capacitor:

V C1

C2

1 1

2 2

Q C V

Q C V

1 2

Q Q

C C 1 2V V V Q

C

1 2

1 1 1

C C C

Capacitors in Series

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

Complicated Capacitor Circuits

4The capacitance of the capacitors in pF

at right are 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

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

q

U dU

0

Qq dq

C

2

02

Qq

C

2

2

Q

C

2

2

QU

C Q C V

22

2 2

C VQU

C

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

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 dielectric

What 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 voltagewhere the capacitor fails

0 AC

d

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

Dipoles•We’ve done a lot with charges in electric fields•However, in nature, neutral combinations are much more common than charged objects

•This doesn’t mean there are no electric effects!•A dipole is any collection of 2+ charges that have no total charge, but the charge is lopsided on one side or the other

•Many molecules are dipoles•The dipole moment for a pair of charges, is just a vector equal to the charge q times the separation vector r•For more complicated objects, it is harder

C+ O-

H+O-

H+

+q -q

r qp r

i ii

qp r

dVp r

Dipoles in Uniform Electric Fields•Electric fields are often uniform, or nearly uniform, as seen by a molecule

•After all, molecules are pretty small!•A dipole in a uniform electric field feels no total force•However, there is a torque, or twisting force, on a dipole

0i i ii i i

q q F F E E+q

-q r F

F

i ii

τ r F

i ii

q r E

i ii

q r E

τ p E

•There is also an energy associated with a dipole in an electric field

sinτ p E

i ii

U V q i ii

q E r

i ii

qp r

i ii

q E rU E p

cosU E p

Dipoles as Dielectrics

•In the absence of an electric field, dipoles will orient randomly in different directions due to random thermal motion•When you turn the electric field on, therandom motions still continue, but afraction of the molecules will reorient tomatch the electric field•Now there is an excess of positive chargeon the right and negative charge on the left•This creates a weaker counter-balancingelectric field that partly cancels theimposed field