di electrics

Dielectrics

OutlineI. Polarizability: From atom to bulk; linear dielectricsII. Electric potential of a polarized object, bound volume and surface charge densities III. Electric field inside a dielectric and electric displacement, boundary conditions in electrostaticsIV. Applications

Dielectrics

Learning ObjectivesI. To learn about polarizability.II. To learn about potential due to a polarized object, bound surface and volume charge densities.III. To learn about electric field inside a dielectric and the boundary conditions in electrostatics

Dielectrics

Learning OutcomesI. To be able to calculate the potential due to a polarized object given the polarization or the bound volume and surface charge densities.II. To be able to calculate the electric field inside a dielectric.III. To be able to apply the electrostatics boundary conditions in the presence of dielectrics.

Dielectrics

DielectricsThe kinds of material based on their conducting properties are:

Material Resistivity (Ω-m)

Silver 1.6x10-8

Copper 1.7x10-8

Gold 2.4x10-8

Iron 1.0x10-7

Sea water 0.2

Polyethylene 2.0x1011

Glass ~1012

Fused quartz 7.5x1017

Solids such as wood, plastic, glass, rubber etc. are insulators.

~Eext= 0

~p = 0

Dielectrics

A Neutral Atom

DielectricsAtomic Polarizability: A Neutral Atom in an External Electric Field

~Eext= 0

~p = 0

H He Li Be C Ne Na Ar K Cs

0.667 0.205 24.3 5.60 1.76 0.396 24.1 1.64 43.4 59.6

Atomic polarizabilities

~p = ®~Eext

d

~Eext= E0x

® =jpjjEextj

) Coulomb¡meters

Newtons=Coulomb=

Coulombs2

Newtons=m

®

4¼²0) Coulombs2

Newtons=m

1

Coulombs2=Newton¡m2 = m3

Atomic Polarizability: A Neutral Atom in an External Electric Field

Dielectrics

DielectricsMolecular Polarizability: A Molecule in an External Electric Field

CO O

CO2

z

H2O

H+ H+

Non-Polar Molecules

Polar Molecules

Dielectrics

~P = ²0Âe~E

² ´ ²0(1 + Âe)

²r ´ 1 + Âe =²

²0

The relative permittivity or the dielectric constant is given by

Linear Dielectrics

where is the electric susceptibility. We also define the permittivity of a dielectric material as

Âe

For an ideal, linear, homogeneous and isotropic dielectric the polarization is related to the electric field through~P ~E

Vacuum 1

Helium 1.000065

Hydrogen 1.00025

Air (dry) 1.0054

Water vapour 1.00587

Diamond 5.7

Salt 5.9

Water 80.1

Dielectric constants (1 atm; 200C)

Linear Dielectrics

Dielectrics: Potential of a Polarized Object

~r

x

y

z

Consider a polarized material with polarization What is the potential at a point due to the polarized material? .

~P :~r

We know that the potential at due to a single dipole is

Using , we get

~r

~r ¡~r 0

~r0

~Pd¿0

x

y

z


V (~r) =1

4¼²0

r ¢ ~pr2

~r

Applying it to the dipole moment in volume at , we get,

Identifying , we can rewrite the potential

as

The first term can be rewritten as a surface integral using the divergence theorem,


Since

Hence, the potential due to a polarized material reduces to

On the RHS, the first term looks like the potential due to a surface charge density

while the second term is the potential due to a volume charge density

Thus, the potential due to a polarized object is


Visualizing Bound Surface Charge Density in a Dielectric

Electric Field Inside a Dielectric

x

Consider a polarized material with polarization What is the potential at a point inside the polarized dielectric?

~P :~r

~r

y

z

~r

y

z

Electric Field Inside Dielectrics


Since the average field over the sphere due to charges outside the sphere is equal to the field they produce at the center of the sphere, it is correctly given by

The dipoles inside produce an average field given by

Close to point , the dipole expression doesn't work. Therefore, we surround the point by a sphere of radius R, then the field at consists of the average field over the sphere due to all the charges outside, and the average field due to charges inside,

~r

~r


As R is small, the polarization is uniform within the sphere. We have seen that the electric field is uniform inside the sphere and it is given by

The total dipole moment can be written in terms of polarization,

Hence, the field is correctly given by the full integral

So that,

We need to revisit Gauss's law in case dielectrics are present. Let us denote by subscripts f and b the free charges and the bound charges, respectively. The total charge density inside the dielectric can be written as

Applying Gauss's law,

We get, in the presence of dielectrics,

Dielectrics: Gauss's Law

Defining electric displacement as, ~D

Since

Dielectrics: Gauss's Law

Integrating over volume V bounded by surface S,

we get,

Now applying the divergence theorem, we have the integral form of the Gauss's law

Integrating over volume V bounded by surface S,

~r£ ~Pnote that is not always zero.

Dielectrics: Boundary Conditions

The above boundary conditions for the electric field across the charged surface can be written as

How do the following boundary conditions on the electric field change in the presence of dielectrics?

We get,

That is, the tangential components of are continuous while the perpendicular component is discontinuous across a charged surface.

~E

Using electric displacement in the presence of a dielectric with polarization ,

Dielectrics: Boundary Conditions

we get,

~D

In the presence of dielectrics, the above two boundary conditions are more useful than the previous ones.

~P

Applications

I. Microwave oven

Dielectrics

Dr. Percy Spencer invented microwave oven in 1946

WolframCDFPlayer MicrowaveOven.cdf

A schematic diagram of a microwave oven

Summary

Dielectrics

where the bound surface charge density

and the bound volume charge density

II. The electric displacement is defined as

III. In the presence of dielectrics, the two boundary conditions of electrostatics can be written as

I. The potential due to a polarized object with polarization is ~P

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