4. Electric Fields in Matter

4.1 Polarization

Insulators: All charge is attached to the atoms or molecules.

Ep

Point charge in a homogenously charged sphere

vo 3, Ep (v - Volume)

Molecules

perpperpEEp ||||

Polar molecules have a permanent dipole.

EpN EdF )(

Ep

4.2 The Field of a Polarized Object

Polarization P (dipole moment per unit volume)tells how strongly the atoms/moleculesare polarized and/or aligned with the electric field.

V rr ')(ˆ

41)( 2

0

dV rPrPotential generated by the microscopic dipoles:

vNpP Take a small volume v that contains, say,

N=1000 atoms.

Bound charges

')'(4

1')'(4

1)(00

ddaV bb VS rr

rrr

volume charge density Pb

surface charge density surface. theon normal theis ˆ where

,ˆnnPb

Example 4.2Electric field of a uniformly polarized sphere.

Pp

rp

3

2

34

ˆ4

1

R

Rrr

Vo

out

dipole at the center of the sphere

RrzPVin 03

constant field

The field inside a dielectric

Deriving the expressions for the bound charges we consideredpure dipoles.

The real dielectric contains physical dipoles. The electricfield is much more complicated near the molecular dipoles.

The macroscopic field is the average over a small volumethat contains many molecules.

The average field of the pure and molecular dipoles is the same.

4.3 The Electric DisplacementTotal charge bf

Free charge (at our disposal) f

Bound charge (induced, comes along) b

Electric displacement (auxiliary field) PED 0

enclosedff Qd ,aDD 0 D Butin general

Boundaryconditions

||||||||||belowabovebelowabovefbelowabove PPDDDD

Example 4.4

Long straight wire with uniform line charge is surroundedby a rubber insulation. Find the electric displacement.

4.4 Linear Dielectrics

Most macroscopic fields are weak as compared to the atomicand molecular fields. The polarization is weak.

linear dielectric EP eo ED

electric susceptibility e

permittivity )1( eo

permittivity of free space o

relative permittivity,dielectric constant o

r

Dielectric constants

Table

On may calculate D in the same way as E in the vacuum ifthe different boundary conditions for E and D do not playrole. In this case, one simply replaces This is the case if:

b) When the symmetry of the problem makes 0|| Da) When the space is filled with a homogenous dielectric.

o

Charge embedded in a homogenous dielectric material.

rE ˆ4

12r

q

Bound charges partially screen q.

Parallel plate capacitor filled with a dielectric.

vacuumrCdAC

D

D=0

dielectricfQ

fQ

Dielectrics are used to:a) Increase the capacityb) Keep the plates apartc) Increase the dielectric strength (field strength without a spark)

Air: dielectric constant dielectric strength mmkVEc

r

/300059.1

A cut section of a multiplayer capacitorwith a ceramic dielectric.

Ceramic capacitors

mmkVEc

r

/7.57

Foil wound capacitor.Frequently used dielectrics:PaperMicaPolysterene mmkVE

mmkVEmmkVE

cr

cr

cr

/246.2/1001104.5

/167.3

Example 4.5

Metal sphere of radius a carries a charge Q. It is surroundedby a linear dielectric material. Find the potential at the center.

Displacement at a boundary without free charge.

belowbelowaboveabovebelowabove DDDD // ||||||

abovebelow

4.4 Boundary Problems with Linear Dielectrics

Within a homogenous linear dielectric, Laplace’s equation holds.

02 V

Boundary conditions on the surface between twodielectrics:

fbelow

belowabove

above

belowabove

nV

nVVV

Example 4.7A sphere of homogeneousdielectric material is placedinto an otherwise uniformelectric field. Find the fieldinside the sphere.

Example 4.8

Find the electrical fieldinside and outside the dielectric and the force on the charge.

4.5 Energy in a dielectric system

Capacitor vacuumrdielectric CCCVW 2

21

dW ED21For linear dielectrics

Force on a dielectric