Electric fields in material spaceBy: Dang Duy Ninh

1SSU - Wireless Communication RF System Lab

Weekly Report

Objectives• In chapter 4: electrostatic may be regarded as the “vacuum” field

• In this chapter: may be regarded as the theory of electric phenomena in material space To provide a basis for understanding the concept of conduction, electric current, and polarization

Some properties of dielectric materials: susceptibility, permittivity, linearity,…

Introduce the concept of boundary condition for electrics field existing in two different media

2

Contents1.Properties of materials2.Convection and conduction currents3.Conductors4.Polarization in dielectrics5.Dielectric constant and strength6.Linear, isotropic, and homogeneous

dielectrics7.Continuity equation and relaxation

time8.Boundary conditions

3

Properties of material• Classified in terms of their conductivity – High conductivity ( >> 1) is referred to as a metal

– Low conductivity (<< 1) is referred to as a dielectrics (or insulators)

– The rest is semiconductorNote: the major difference between a metal and dielectric lies in the amount of electrons available for conduction of current

4

Convection and conduction currents

• What is the current ?

• The current density J

J =

so the total current flowing through a surface is:

• Current classification:1. Conduction current is the current flowing though a conductor2. Convection current is the current flowing though an insulating

medium (doesn’t satisfy Ohm’s law)3. Displacement current (will be considered in chapter 9)

5

The current through a given area is the electric charge passing through the area per unit time

Convection and conduction currents

• Convection current

6

• A flow charge ( , )

The y-direction current density is given by

Hence, in generaly

Convection and conduction currents

• Conduction current

7

The force on an electron with charge -e is

F = -eEAccording to Newton’s law, we have or where is the average time interval between collisions

If there are n electrons per unit volume, the electron charge density is given by

Thus the conduction current density is

(1)

(2)

Conductors

8

An isolated conductor under the influence of a static electric field

A conductor has zero electric field under static conditions

The induced surface charge

Very quickly

• A perfect conductor cannot contain an electrostatic field within it

• Under static conditions,

Conductors

9

A conductor of uniform cross section under an

applied E field

Resistance As the electrons move, they encounter some damping forces called resistance• The electric field applied is

uniform and its magnitude is given by

• Since the conductor has a uniform cross section

• Substituting J=E and eqs. (1) into eqs. (2) gives

• Hence

S

(1)

(2)

or

where = is the resistivity of the material

(3)

• The basic definition of resistance R as the ratio of the potential difference V between the two ends of the conductor to the current I through the conductor still applies

Conductors

10

• Power P

P = F.u =

or(1)

For a conductor with uniform cross section, dv = dS dl, so eq. (1) becomes

or

Power P (in watts) is define as the rate of change of energy W (in joules) or force times velocity

Polarization in dielectrics

11

Polarization of a nonpolar atom or molecule

A dipole results from the displacement of the charges and the dielectric is said to be polarized The dipole moment is

If there are N dipoles in a volume of the dielectric, the total dipole moment due to electric field is

We define polarization P () as the dipole moment per unit volume of the dielectric

Conclude: the major effect of the electric field on the dielectric is the creation of dipole moments that align themselves in the direction of E (this type of dielectric is said to be nonpolar)

*Polarization in dielectrics

12

The potential dV at an exterior point O due to the dipole moment P dv’ is

where

Applying the vector identity

(1)

(2)

Substituting this into eq. (1) and integrating over the entire volume v’ of the electric, we obtain

We refer to and as bound (or polarization) surface and volume charge densities

explain

*Polarization in dielectrics• If is the free charge volume density, the total volume charge

density is given by

Hence

where

13

Conclusion: the net effect of the dielectric on the electric field E is to increase D inside it by amount P

Some case the polarization P vary directly as the applied electric field E

Where known as the electric susceptibility of the material

Dielectric constant and strength• Dielectric constant

• Dielectric strength

*Dielectric breakdown is said to have occurred when a dielectric becomes conducting

14

The dielectric constant (or relative permittivity) is the ratio of the permittivity of the dielectric to that of free space

The dielectric strength is the maximum electric field that a dielectric can tolerate or withstand without breakdown

Linear, isotropic, and homogeneous dielectrics

15

A material is said to be linear if D varies linearly with E and nonlinear otherwiseMaterials for which (or ) does not vary in the region being considered and there force same at all points are said to be homogeneous. They are said to be inhomogeneous when is dependent of the space coordinatesMaterial for which D and E are in the same direction are said to be isotropic. For anisotropic materials, D, E, and P are not parallelA electric material (in which D = E) is linear if does not change with the applied E field, homogenous if does not change from point to point, and isotropic if does not change with directionNote:

- the same idea holds for a conducting material in which J = E applies

- for such media, all formulas derived in Chapter 4 for free space can be applied by merely replacing with

Continuity equation and relaxation time

1.Continuity of current equation

16

Due to the principle of charge conservation, the time rate of decrease of charge within a given volume must be equal to the net outward current flow through the closed surface of the volume

(1)

(2)Invoking divergence theorem

but

(3)Substituting eqs. (1) and eqs. (3) into eq. (3) gives

the continuity of current equation

For steady currents, = 0 and hence = 0 showing that the total charge leaving a volume is the same as the total charge entering it

or

Continuity equation and relaxation time

2. The relaxation time

17

Relaxation time is the time it makes a charge placed in the interior of a material to drop to = 36.8 present of its initial value

(1)

(2)

Note: relaxation time is short for good conductors and long for good dielectrics

Boundary conditions

18

Boundary conditions is the condition that the field must satisfy at the interface separating the media• The boundary conditions at an interface separating:

– Dielectric and dielectric– Conductor and dielectric– Conductor and free space

• To determine the boundary conditions, we need to use Maxwell’s equation:

and

• Decomposing the electric field intensity E into orthogonal components

where and are, respectively, the tangential and normal components of E to the interface of interest

Boundary conditions1. Dielectric – dielectric boundary conditions

19

• and in media 1 and 2 can be decomposed as

• Applying Maxwell’s equation to the closed path abcda

• As -> 0, eq. (1) becomes

(1)

is said to be continuous across the boundary

• Since D = = + , eq. (2) can be written as

(2)

or

is said to be discontinuous across the interface

Boundary conditions

20

1. Dielectric – dielectric boundary conditions• Applying the Gauss’s law,

we have

• Allowing -> gives

or

• If no free charges exist at the interface, so

is continuous across the interface

Since D = eq. (1) can be written as

(1)

The normal component of E is discontinuous at the boundary

Boundary conditions2. Conductor – dielectric boundary conditions

21

• Applying Maxwell’s equation to the closed path abcda

As -> 0,

Similarly, by applying the Gauss’s law to the pillbox and letting we have

because D = = 0 inside the conductor, so

or

Boundary conditions2. Conductor – dielectric boundary conditions

22

Thus under static conditions, the following conclusions can be made about a perfect conductor:1. No electric field may exist within a conductor

2. Since E = - = 0, there can be no potential difference any two points in the conductor

3. The electric field E can be external to the conductor and normal to its surface

Boundary conditions

3. Conductor – free space boundary conditions

23

This is a special case of the conductor – dielectric condition. Free space is a special dielectric for which = 1Thus the boundary conditions are

Thank you !