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Capacitance and Capacitance and DielectricsDielectrics

CapacitanceCapacitors in combination #Series#Parallel

Energy stored in the electric field of capacitors and energy density

DielectricsDielectric Strength

Lesson 4

Field Above ConductorField above surface of

charged conductor

Does not depend on thickness of conductor

E Q

A 0 0

charge =

Area AE

conductor in electrostatic equilibrium

A 0

E dA EdA

A

closedcylinder

E dA

A

E

A

A 0

0

Charged Plates

+ -

d

E

WFdQEd

UQEd U UVEdV V

+Q -Q

Potential drops Ed in

going from + to -V- is Ed lower than V+

PD between Plates

How does one make such a separation of charge? Must move positive chargeWork is done on positive charge in producing separation

Q -Q+QF

Work Done in Moving Charge

What forms when we have separation of charge?An Electric Field

+Q -QE

Electric Field

Capacitorb

The work done on separating charges to fixed positionsis stored as potential energy in this electric field, which can thus DO work

This arrangement is called a CAPACITORCAPACITOR

How do we move charge?With an electric fieldalong a conduction pathconduction path

Moving Charge

Picture

The charge separation is

maintained by removing the conduction pathonce a charge separation has been

producedAn electric component that does

this is called A Capacitor

Charge Separation

Capacitor Symbol

+ -

Battery Symbol

Charging CapacitorCan charge a capacitor by

connecting it to a battery

+

+ -

-

CapacitancePlates are conductorsEquipotential surfacesLet V = P.D. (potential difference)

between platesQ (charge on plates) ~ V (why?)Thus Q = CVC is a constant called

CAPACITANCECAPACITANCE

SI Units

FaradsVolts

Coulombs

V

C

V

QC

Calculation of Capacitance

assume charge Q on platescalculate E between plates using Gauss’ LawFrom E calculate V Then use C = Q/V

Capacitors

Electric Field above Plates

0

00

plates to is

EAQ

A

QE

Calculating Capacitance in General

going from positive to negative plate

V = Vf Vi E ds

i

f

0

E ds 0 choose path from + plate to - plateV = - V ( PD across plates )

Thus V = Eds+

-

( choose path | | to electric field )

C EA 0

Eds+

-

In order that

for Parallel Plates Capacitor

-

+

CQ

V EA0

EdsEA0

EdA0

d

C Q

V 2 0 L

lnb

a

•a = radius of inner cylinder•b = radius of outer cylinder•L = length of cylinder

for Cylindrical Capacitor

Combination of Capacitors Parallel

Combinations of Capacitors in Combinations of Capacitors in equilibriumequilibrium

Parallelsame electric potential felt by

each elementSerieselectric potential felt by the

combination is the sum of the potentials across each element

Picture

Calculation of Effective Capacitance

V Q1

C1

Q2

C2

Total charge Q Q1 Q2

VC1 VC2 VCeq

Ceq C1 C2

In general

Ceq Ci

i

Combination of Capacitors Series

PictureNet charge zero

Net charge zero

Why are the charges on the plates of equal magnitude ?

Calculation of Effective Capacitance I

If net charge inside these Gaussian surfaces is not zero

Field lines pass through the surfaces

and cause charge to flowThen we do have not equilibrium

Calculation of Effective Capacitance II

Vtotal V1 V2 Q

C1

Q

C2

Q1

C1

1

C2

Q

1

Ceq

In general

1

Ceq

1

Cii

Question I

Is this parallel or series?

=

Question II

Is this parallel or series?

+

+ -

-

Work Done in Charging Capacitor

Work done in charging capacitor

I +

+ -

-q

CalculationV q q

Cif dq of charge is then transfered the work done is

dW V q dq

Thus total work done on charging is

W V q dq0

Q

1

Cqdq

0

Q

Q2

2C1

2CV 2

Energy DensityThis work is stored as P.E.

EnergyDensity U

Volume

for parallel plate capacitor U

Ad

CV 2

2Ad1

2 0

V

d

2

1

20 E2

DielectricsDielectrics

Picture

Picture

Picture

Polarization

Induced Electric Field

Polarization

Charge Q stays the same, Total electric Field is less,

thus P.D. Veffective across plates is less

C Q

V C

Q

Veffective

C C

Dielectric Constant 1.00

Dielectric Constant

C0 A

d

C C 0

A

d

C A

dwhere, PERMITTIVITY of the dielectric 0

Permitivity

Permitivity in Dielectrics

For conductors ( not dielectrics )

For regions containing dielectrics all electrostatic equations containing

0 are replaced by e .g . Gauss ' Law

E dAQ

surface

The Dielectric Strength Dielectric Strength of a non conducting material is the value of the Electric Field that causes it to be a conductor. When dielectric strength of air is surpassed we get lightning

Dielectric Strength