Electric Fields in Matter Chapter 4: (Griffiths)

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Electric Fields in Matter Chapter 4 (Griffiths)

• Polarization

• Field of polarized object

• Electric displacement

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Conductors
Matter
Insulators/Dielectrics
All charges are attached to specific
atoms/molecules and can only have a restricted
motion WITHIN the atom/molecule.
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When a neutral atom is placed in an external
electric field (E)

• The positively charged core (nucleus) is pushed
  along E

• The centre of the negatively charged cloud is
  pushed in the opposite direction of E

• If E is large enough

? the atom gets pulled apart completely
gt the atom gets IONIZED
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• For less extreme fields

? an equilibrium is established
. the attraction between the nucleus and the
electrons AND . the repulsion between them
caused by E
gt the atom gets POLARIZED
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Induced Dipole Moment
(pointing along E)
Atomic Polarizability
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To calculate ? (in a simplified model)
The model an atom consists of a point charge
(q) surrounded by a uniformly charged spherical
cloud of charge (-q).
a
q
-q
At equilibrium,
( produced by the negative charge cloud)
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At distance d from centre,
(where v is the volume of the atom)
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Molecules always have a preferred direction of
polarization
Example CO2
? (when E is along axis ) gt ? (when E is ? to
axis)
when E is at some angle to the axis
( and p may not be directed along E )
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For completely asymmetrical molecules
In General
elements of polarizability tensor
(values depend on the orientation of the chosen
axis)
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Pr. 4.2 Griffiths
The electron cloud for a hydrogen atom in the
ground state has a charge density
where a is the Bohr radius.
Find the atomic polarizability of such an atom
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Sol. Pr. 4.2
At first, to find the field at radius r, using
Gauss law
The field of the electron cloud is
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The proton will be shifted from r 0 to the
point d where EeE (the external field).
Expand
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Compare with the simplified model result
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Pr. 4.4 Griffiths
A point charge q is situated a large distance r
from a neutral atom of polarizability ?.
r
q
Find the force of attraction between them.
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Sol. Pr. 4.4
Induced dipole moment of atom
Field of this dipole at location of q
Force on q due to this field
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Alignment of Polar Molecules
Polar molecules molecules having permanent
dipole moment

• when put in a uniform external field

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Alignment of Polar Molecules

• when put in a non-uniform external field

q
F
d
-q
F-
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q
F
E
d
-q
F-
E-
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assuming the dipole to be very short
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For perfect dipole of infinitesimal length,
the torque about the centre
the torque about any other point
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Pr. 4.9 Griffiths
A dipole p is a distance r from a point charge
q, and oriented so that p makes an angle ? with
the vector r from q to p.
(i) What is the force on p?
(ii) What is the force on q?
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Sol. Pr. 4.9
(i)
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(force on the dipole)
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(ii)
(force on the point charge)
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Recall Divergence theorem
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Defining
Surface Bound Charge
Volume Bound Charge
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Potential produced by
a surface charge density ??b
a volume charge density ??b
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Field/Potential of a polarized object

Field/Potential produced by a surface bound
charge ?b

Field/Potential produced by a volume bound charge
?b
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Physical Interpretation of Bound Charges
are not only mathematical entities devised for
calculation
but represent
perfectly genuine accumulations of charge !
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Surface Bound Charge
d
P
A dielectric tube
Dipole moment of the small piece

-q q
A
Surface charge density
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If the cut is not ? to P
?
P
A
A
In general
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Volume Bound Charge
A non-uniform polarization
accumulation of bound charge within the volume
diverging P
pile-up of negative charge
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Net accumulated charge with a volume
Opposite to the amount of charge pushed out of
the volume through the surface

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(using divergence theorem)
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Potential of a uniformly polarized sphere
Potential of a polarized sphere at a field point
( r )
P is uniform
P is constant in each volume element
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Electric field of a uniformly charged sphere
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For r lt R
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For r gt R
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Field of a uniformly polarized sphere
Choose z-axis P
P is uniform
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For r lt R
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Inside the sphere the field is uniform
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For r gt R
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Total dipole moment of the sphere
potential due to a dipole at the origin
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Uniformly polarized sphere A physical analysis
Without polarization
Two spheres of opposite charge, superimposed and
canceling each other
With polarization
The centers get separated, with the positive
sphere moving slightly upward and the negative
sphere slightly downward
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At the top a cap of LEFTOVER positive charge and
at the bottom a cap of negative charge
Bound Surface Charge ?b
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Recall Pr. 2.18 Griffiths
Two spheres , each of radius R, overlap
partially.
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Electric field in the region of overlap between
the two spheres
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For an outside point
the charges are concentrated at the respective
centers
A DIPOLE
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Pr. 4.10 Griffiths
A sphere of radius R carries a polarization
where k is a constant and r is the vector from
the center.
(i) Calculate the bound charges ?b and ?b.
(ii) Find the field inside and outside the
sphere.
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Sol. Pr. 4.10
(i)
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(ii) For r lt R,
For r gt R,
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Pr. 4.14 Griffiths
When a neutral material is polarized, charge
moves a bit, but the total remains zero.
Prove that the total bound charge vanishes.
Sol.
From divergence theorem
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Gauss Law in the presence of dielectrics
Within the dielectric the total charge density
free charge
bound charge
caused by polarization
NOT a result of polarization
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Gauss Law
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Electric Displacement ( D )
Gauss Law
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D E
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Pr. 4.15 Griffiths
A thick spherical shell is made of dielectric
material with a frozen-in polarization
where k is a constant and r is the distance from
the center. There is no free charge.
a
b
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Find E in three regions by two methods

• Locate all the bound charges
• and use Gauss law.

ii) Find D and then get E from it.
a
b
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Sol. i)
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For r lt a
For r gt b
For a lt r lt b
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Sol. ii)
Everywhere
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For a lt r lt b
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?? External electric field
Cause of Polarization
If E is not TOO strong
Electric susceptibility of the medium
LINEAR DIELECTRICS
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In linear dielectrics
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Permittivity of the material
A dimensionless quantity
Relative permittivity or Dielectric constant of
the material
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If the susceptibility of the medium doesnt vary
with position
a homogeneous medium
In such a homogeneous linear dielectric
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When the medium is filled with a homogeneous
linear dielectric, the field is reduced by a
factor of 1/ke.