Remember: Exam this Thursday, Feb 12 at the regular class time.

Please bring at least two sharpened pencils – the exams are not to be done in pen!

It is open book, open note.

Don’t forget your calculator!

Definitions

• Voltage—potential difference between two points in space (or a circuit)

• Capacitor—device to store energy as potential energy in an E field

• Capacitance—the charge on the plates of a capacitor divided by the potential difference of the plates C = q/V

• Farad—unit of capacitance, 1F = 1 C/V. This is a very large unit of capacitance, in practice we use F (10-6) or pF (10-12)

Definitions cont

• Electric circuit—a path through which charge can flow

• Battery—device maintaining a potential difference V between its terminals by means of an internal electrochemical reaction.

• Terminals—points at which charge can enter or leave a battery

Capacitors

• A capacitor consists of two conductors called plates which get equal but opposite charges on them

• The capacitance of a capacitor C = q/V is a constant of proportionality between q and V and is totally independent of q and V

• The capacitance just depends on the geometry of the capacitor, not q and V

• To charge a capacitor, it is placed in an electric circuit with a source of potential difference or a battery

CAPACITANCE AND CAPACITORS

Capacitor: two conductors separated by insulator and charged by opposite and equal charges (one of the conductors can be at infinity)

Used to store charge and electrostatic energy

Superposition / Linearity: Fields, potentials and potential differences, or voltages (V), are proportional to charge

magnitudes (Q)

(all taken positive, V-voltage between plates)

Capacitance C (1 Farad = 1 Coulomb / 1 Volt) is determined by pure geometry (and insulator properties)

1 Farad IS very BIG: Earth’s C < 1 mF

QC

V

Calculating Capacitance

1. Put a charge q on the plates

2. Find E by Gauss’s law, use a surface such that

3. Find V by (use a line such that V = Es)

4. Find C by

0encq

EAAdE

EssdEV

Vq

C

Energy stored in a capacitor is related to the E-field between the plates Electric energy can be regarded as stored in the field itself.

This further suggests that E-field is the separate entity that may exist alongside charges.

Parallel plate capacitor

d

SC

S

QdEdV

S

QE

SQ

0

000

;

area/ charge density

Generally, we find the potential differenceVab between conductors for a certain charge Q

Point charge potential difference ~ Q

This is generally true for all capacitances

Capacitance configurations

sphere individualan of ecapacitanc

- /,With

)(

)11

(2

e

e

e

b

a

e

kaCb

abk

abC

baQk

r

drQkV

Cylindrical capacitor

)ln(2

)ln(22

ab

k

lC

a

b

l

Qk

r

drkV

e

e

b

a

e

Spherical Capacitance

Definitions

• Equivalent Capacitor—a single capacitor that has the same capacitance as a combination of capacitors.

• Parallel Circuit—a circuit in which a potential difference applied across a combination of circuit elements results in the potential difference being applied across each element.

• Series Circuit—a circuit in which a potential difference applied across a combination of circuit elements is the sum of the resulting potential differences across each element.

Capacitors in Series

ac 1 cb 21 2

1 2

1 2

V ; V

Total voltage

1 1 1Equivalent

Q QV V

C C

V V V

V

C Q C C

Capacitors in Parallel

21

21

Equivalent

charge Total

CCV

QC

QQQ

Example: Voltage before and after

if

ffii

iiii

VCC

CC

C

QV

CCC

QQQQQ

VCQVCQ

21

21

21

2121

2211

:after Voltage

Equivalent

charge Total

;

:polarity opposite ofbut voltagesame the

by charged are capacitorsInitially

Energy Storage in Capacitors Electric Field Energy

Electric potential energy stored = amount of work done to charge the capacitori.e. to separate charges and place them onto the opposite plates

QV

C 2

0 0

2 2

Total work ( )2

1Stored energy

2 2 2

Q Qq Q

W V q dq dqC C

Q CVU QV

C

Charged capacitor – analog to stretched/compressed spring

Capacitor has the ability to hold both charge and energy

2 220 0( / )( )

2 2 2ES d Ed ECV

uSd Sd

To transfer charge dq between conductors, work dW=Vdq

Density of energy (energy/volume)Energy is conserved in the E-field

Applications of Capacitors: Energy Storage

Z-machine for controlled nuclear fusion Sandia National Labs

14

9

~ 10

~ 2 10

P Watt

T K

In real life we want to store more charge at lower voltage, hence large capacitances are needed

Increased area, decreased separations, “stronger” insulators

Electronic circuits – like a shock absorber in the car, capacitor smoothes power fluctuations

Response on a particular frequency – radio and TV broadcast and receiving

Undesirable properties – they limit high-frequency operation

Example: Transferring Charge and Energy Between Capacitors

Switch S is initially open

1) What is the initial charge Q0?2) What is the energy stored in C1?3) After the switch is closed what is the voltage across each capacitor? What is the charge on each? What is the total energy?

a) 0 1 0Q C V 0 01

2iU Q Vb)

c) when switch is closed, conservation of charge

1 2 0Q Q Q Capacitors become connected in parallel 1 0

1 2

C VV

C C

d) 1 21 1

2 2f iU Q V Q V U Where had the difference gone?

It was converted into the other forms of energy (EM radiation)

Definitions

• Dielectric—an insulating material placed between plates of a capacitor to increase capacitance.

• Dielectric constant—a dimensionless factor that determines how much the capacitance is increased by a dielectric. It is a property of the dielectric and varies from one material to another.

• Breakdown potential—maximum potential difference before sparking

• Dielectric strength—maximum E field before dielectric breaks down and acts as a conductor between the plates (sparks)

Most capacitors have a non-conductive material (dielectric) between the conducting plates. That is used to increase the capacitance and potential across the plates.

Dielectrics have no free charges and they do not conduct electricity

Faraday first established this behavior

Capacitors with Dielectrics

• Advantages of a dielectric include:1. Increase capacitance

2. Increase in the maximum operating voltage. Since dielectric strength for a dielectric is greater than the dielectric strength for air

3. Possible mechanical support between the plates which decreases d and increases C.

• To get the expression for anything in the presence of a dielectric you replace o with o

airdiairdi VVEE maxmaxmaxmax

00; decreases: /

k SC V Ed E E E k

d

Field inside the capacitor became smaller – why?

There are polarization (induced) charges

– Dielectrics get polarized

We know what happens to the conductor in the electric field

Field inside the conductor E=0outside field did not change

Potential difference (which is the integral of field) is, however, smaller.

( )o

V d b

0

[1 / ]

AC

d b d

Properties of Dielectrics

0EE

K

Redistribution of charge – called polarization

We assume that the induced charge is directly proportional to the E-field in the material

0

CK

C dielectric constant of a material

0VV

K

when Q is kept constant

In dielectrics, induced charges do not exactly compensate charges on the capacitance plates

00 0

; iE E

1

1i K

Induced charge density

0K Permittivity of the dielectric material

E

E-field, expressed through charge density on the conductor plates (not the density of induced charges) and permittivity of the dielectric (effect of induced charges is included here)

21

2u E Electric field density in the dielectric

Example: A capacitor with and without dielectric

Area A=2000 cm2

d=1 cm; V0 = 3kV;

After dielectric is inserted, voltage V=1kV

Find; a) original C0 ; b) Q0 ; c) C d) K e) E-field