4. electric fields in matter
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4. Electric Fields in Matter. 4.1 Polarization. Insulators: All charge is attached to the atoms or molecules. ( v - Volume). Point charge in a homogenously charged sphere. Molecules. Polar molecules have a permanent dipole. Take a small volume v that contains, say, N =1000 atoms. - PowerPoint PPT PresentationTRANSCRIPT
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4. Electric Fields in Matter
4.1 Polarization
Insulators: All charge is attached to the atoms or molecules.
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Ep
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Point charge in a homogenously charged sphere
vo 3, Ep (v - Volume)
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Molecules
perpperpEEp ||||
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Polar molecules have a permanent dipole.
EpN EdF )(
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Ep
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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.
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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
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Example 4.2Electric field of a uniformly polarized sphere.
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Pp
rp
3
2
34
ˆ4
1
R
Rrr
Vo
out
dipole at the center of the sphere
RrzPVin 03
constant field
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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.
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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
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Example 4.4
Long straight wire with uniform line charge is surroundedby a rubber insulation. Find the electric displacement.
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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
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Dielectric constants
Table
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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
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Charge embedded in a homogenous dielectric material.
rE ˆ4
12r
q
Bound charges partially screen q.
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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
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A cut section of a multiplayer capacitorwith a ceramic dielectric.
Ceramic capacitors
mmkVEc
r
/7.57
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Foil wound capacitor.Frequently used dielectrics:PaperMicaPolysterene mmkVE
mmkVEmmkVE
cr
cr
cr
/246.2/1001104.5
/167.3
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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.
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Displacement at a boundary without free charge.
belowbelowaboveabovebelowabove DDDD // ||||||
abovebelow
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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
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Example 4.7A sphere of homogeneousdielectric material is placedinto an otherwise uniformelectric field. Find the fieldinside the sphere.
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Example 4.8
Find the electrical fieldinside and outside the dielectric and the force on the charge.
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4.5 Energy in a dielectric system
Capacitor vacuumrdielectric CCCVW 2
21
dW ED21For linear dielectrics
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Force on a dielectric