PHL110

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Multipole expansion

SUMMARY

- Potential of an arbitrary localized charge distribution.

V = monopole term+dipole term+quadrapole term+octopole term+……

• For a charged body, dominant term is monopole term.

• When total charge zero, dominant term is dipole term.

Dipole moment,

For a dipole,+ -p

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Electric field of a dipole

For a dipole at origin, potential,

Using, V E

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Dielectrics

Ch. 4

Electric Fields in Matter

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Insulators and Dielectrics

• All dielectrics are insulators but all insulators are not dielectric.

Dielectric: materials where the polarization is important, such as the

insulating material in a capacitor.

Insulator: when a material is being used to prevent a current flow.

•Since charges tend not to move easily in nonmetallic solids it's

possible to have "islands" of charge in glass, ceramics, and plastics.

The latin word for island is insula.

•Inserting a layer of nonmetallic solid between/across the plates of a

capacitor increases its capacitance. The greek prefix di or dia means

"across". The material placed across the plates of a capacitor like a

little nonconducting bridge is a dielectric.

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• A dielectric medium possess no (or very few) free

electrons to provide currents in an electric field.

• Although there is no macroscopic migration of

charge when a dielectric is placed in an electric

field, microscopic displacements (on the order of

the size of atoms or molecules) of charge occur

resulting in the appearance of induced electric

dipoles.

What happens to dielectric in an electric field?

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• In the absence of an

applied electric field,

the positively charged

nucleus is surrounded

by a spherical electron

cloud with equal and

opposite charge.• Outside the atom, the

electric field is zero.

electron

cloud nucleus

In the absence of an applied electric field:

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• In the presence of an

applied electric field,

the electron cloud is

distorted such that it is

displaced in a direction

(w.r.t. the nucleus)

opposite to that of the

applied electric field.

E app

In the presence of an applied electric field:

+-p

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• A dielectric is said to be polarized when induced

electric dipoles are present.

• The presence of induced electric dipoles within the

dielectric causes the electric field both inside and

outside the material to be modified.

• There are two principal methods by which a dielectric

can be polarized: stretching and rotation.

• Stretching an atom or molecule results in an induced

dipole moment added to every atom or molecule.

• Rotation occurs only in polar molecules — those with a

permanent dipole moment like the water molecule.

(Animation: http://physics.info/dielectrics/)

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Induced dipoles

Nucleus pushed in the direction of the field and electroncloud in the opposite direction

Equilibrium is soon established between the two opposite forces

Electrostatic force holding

the atom together External field that is

distorting the atom

The atom now has a tiny dipole moment

The constant of proportionality is called atomic polarizability

It’s value depends on the detailed structure of the atom.

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• The net effect is that

each atom becomes a

small charge dipole

which affects the

total electric field

both inside andoutside the material.

E p

dipole

moment

(C-m)polarizability

(F-m2)

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Molecular Polarizability• Polarizability is a measure of the ability of

a material to become polarized in the

presence of an applied electric field.

• Polarization occurs in both polar and

nonpolar materials. (displace, stretching,

rotation etc.)

1 p

2 p

1 2 p p p

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The external field pushing the nucleus to the right exactly balances

the internal field pulling it to the left.

Assume that electron cloud retains spherical shape (only moves).

Field at a distance from a uniformly charged sphere is [Pr. 2.12]

Example 4.1: Polarizability of an atom

At equilibrium

We have, and we have,

Atomic polarizability,

= volume of the atom

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General p and E relation

Molecules polarize more readily in some directions than others.

When the field is at some angle to the molecular axis

For asymmetric molecules the situation is more complicated

For molecules, this induced dipole moment is easy to create in

certain directions and not in others.

CO2 molecule

E E p

N mC

N mC

/.

/.

105.4

1022

2

40

40

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Alignment of polar molecules

p

E

E

•

is generally induced by .

• Some molecules have built-in permanent dipole

moment which is much larger, e.g. H2O.

mC p .101.6 30

p

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About the centre of the dipole

Dipole in a uniform Electric Field

There is torque,

Net force is zero.

VIDEO: Dipole

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Non-Uniform field

There will be a net force on the dipole in addition to the torque

dz z

E dy

y

E dx

x

E z y x

z E y E x E E d E x y xˆˆˆ

z E y E x E z

z y

y x

x z d yd xd z y x z y xˆˆˆˆˆˆˆˆˆ

Using,

POLARIZATION (P)In any dielectric material when we apply a field, lots of dipoles

point along the direction of field

The material is said to be polarized.

Polarization, P dipole moment per unit volume

N dipoles per unit volume.

Thus polarization is a property which is true as an average

over volumes much larger than the atomic size. In practicewe will use it as a continuous field just like the electric or

magnetic field and assume it is defined everywhere.

V

p Np i

i

V

d P p

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Polarization

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PHL110