The Method of Images

President University Erwin Sitompul EEM 8/1

Dr.-Ing. Erwin SitompulPresident University

Lecture 8

Engineering Electromagnetics

http://zitompul.wordpress.com


The Method of ImagesChapter 5 Current and Conductors

One important characteristic of the dipole field developed in Chapter 4 is the infinite plane at zero potential that exists midway between the two charges.

Such a plane may be represented by a thin infinite conducting plane.

The conductor is an equipotential surface at a potential V = 0. The electric field intensity, as for a plane, is normal to the surface.



Thus, we can replace the dipole configuration (left) with the single charge and conducting plane (right), without affecting the fields in the upper half of the figure.

Now, we begin with a single charge above a conducting plane. ► The same fields above the plane can be maintained by

removing the plane and locating a negative charge at a symmetrical location below the plane.

This charge is called the image of the original charge, and it is the negative of that value.



The same procedure can be done again and again.Any charge configuration above an infinite ground plane may

be replaced by an arrangement composed of the given charge configuration, its image, and no conducting plane.



ExampleFind the surface charge density at P(2,5,0) on the conducting plane z = 0 if there is a line charge of 30 nC/m located at x = 0, z = 3, as shown below.

• We remove the plane and install an image line charge

• The field at P may now be obtained by superposition of the known fields of the line charges



2 3x z R a ax = 0, z = 3

x = 0, z = –3

P(2,5,0)

2 3x z R a a

02L

RR

E a9

0

2 330 10

2 13 13x z

a a

02L

RR

E a9

0

2 330 10

2 13 13x z

a a

E E E9

0

180 10

2 (13) z

a

249 V mz a

0D E 22.20 nC mz a

• Normal to the plane

S ND 22.20nC m at P


SemiconductorsChapter 5 Current and Conductors

In an intrinsic semiconductor material, such as pure germanium or silicon, two types of current carriers are present: electrons and holes.

The electrons are those from the top of the filled valence band which have received sufficient energy to cross the small forbidden band into conduction band.

The forbidden-band energy gap in typical semiconductors is of the order of 1 eV.

The vacancies left by the electrons represent unfilled energy states in the valence band. They may also move from atom to atom in the crystal.

The vacancy is called a hole, and the properties of semiconductor are described by treating the hole as a positive charge of e, a mobility μh, and an effective mass comparable to that of the electron.



The conductivity of a semiconductor is described as:e e h h

As temperature increases, the mobilities decrease, but the charge densities increase very rapidly.

As a result, the conductivity of silicon increases by a factor of 100 as the temperature increases from about 275 K to 330 K.



The conductivity of the intrinsic semiconductor increases with temperature, while that of a metallic conductor decreases with temperature.

The intrinsic semiconductors also satisfy the point form of Ohm's law: the conductivity is reasonably constant with current density and with the direction of the current density.

J E


The Nature of Dielectric MaterialsA dielectric material in an electric field can be viewed as a free-

space arrangement of microscopic electric dipoles, a pair of positive and negative charges whose centers do not quite coincide.

These charges are not free charges, not contributing to the conduction process. They are called bound charges, can only shift positions slightly in response to external fields.

All dielectric materials have the ability to store electric energy. This storage takes place by means of a shift (displacement) in the relative positions of the bound charges against the normal molecular and atomic forces.

Chapter 6 Dielectrics and Capacitance


The Nature of Dielectric MaterialsThe mechanism of this charge displacement differs in various

dielectric materials.Polar molecules have a permanent displacement existing

between the centers of “gravity” of the positive and negative charges, each pair of charges acts as a dipole.

Dipoles are normally oriented randomly, and the action of the external field is to align these molecules in the same direction.


Nonpolar molecules does not have dipole arrangement until after a field is applied.

The negative and positive charges shift in opposite directions against their mutual attraction and produce a dipole which is aligned with the electric field.


The Nature of Dielectric MaterialsEither type of dipole may be described by its dipole moment p:


Qp d

total1

n v

ii

p p

If there are n dipoles per unit volume, then there are nΔv dipoles in a volume Δv. The total dipole moment is:

We now define the polarization P as the dipole moment per unit volume:

01

1lim

n v

iv

iv

P p

The immediate goal is to show that the bound-volume charge density acts like the free-volume charge density in producing an external field ► We shall obtain a result similar to Gauss’s law.

n nQ p d


The Nature of Dielectric MaterialsTake a dielectric containing nonpolar molecules. No molecules

has p, and P = 0 throughout the material.Somewhere in the interior of the dielectric we select an

incremental surface element ΔS, and apply an electric field E.The electric field produces a moment p = Qd in each molecule,

such that p and d make an angle θ with ΔS.


Due to E, any positive charges initially lying below the surface ΔS and within ½dcosθ must have crossed ΔS going upward.

Any negative charges initially lying above the surface ΔS and within ½dcosθ must have crossed ΔS going downward.



The Nature of Dielectric MaterialsFor n molecules/m3, the net total charge (positive and negative)

which crosses the elemental surface in upward direction is:

cosbQ nQd S

bQ P S

b SQ d P S

bQ nQ d S

The notation Qb means the bound charge. In terms of the polarization, we have:

If we interpret ΔS as an element of a closed surface, then the direction of ΔS is outward.

The net increase in the bound charge within the closed surface is:


The Nature of Dielectric MaterialsSeeing some similarity to Gauss’s law, we may now generalize

the definition of electric flux density so that it applies to media other than free space.

We write Gauss’s law in terms of ε0E and QT, the total enclosed charge (bound charge plus free charge):


0T SQ d E ST bQ Q Q

T bQ Q Q 0( )S

d E P SCombining the last three equations:

We may now define D in more general terms:

0 D E P• There is an added term to D when a

material is polarized


The Nature of Dielectric MaterialsFor equations with volume charge densities, we now have:


b bvQ dv

vvQ dvT TvQ dv

With the help of the divergence theorem, we may transform the equations into equivalent divergence relationships:

b P

0 T Ev D


The Nature of Dielectric MaterialsChapter 6 Dielectrics and Capacitance

To utilize the new concepts, it is necessary to know the relationship between E and P.

This relationship will be a function of the type of material. We will limit the discussion to isotropic materials for which E and P are linearly related.

In an isotropic material, the vectors E and P are always parallel, regardless of the orientation of the field.

The linear relationship between P and E can be described as:0e P E

0 0e D E E

1r e

0 r D E E

0 r

χe : electric susceptibility, a measure of how easily a dielectric polarizes in response to an electric

field

0( 1)e E

We now define:

εr : relative permittivity


The Nature of Dielectric Materials In summary, we now have a relationship between D and E

which depends on the dielectric material present:


D E

0 r

v D

Sd Q D S


The Nature of Dielectric MaterialsChapter 6 Dielectrics and Capacitance

ExampleWe locate a slab of Teflon in the region 0 ≤ x ≤ a, and assume free space where x < 0 and x > a. Outside the Teflon there is a uniform field Eout = E0ax V/m. Find the values for D, E, and P everywhere.

in 0 in2.1D E

in 0 in1.1P E

,teflon 2.1r 2.1 1 1.1e

out 0 xEE a

out 0 0 xED a

out 0P • No dielectric materials outside 0 ≤ x ≤ a

• No relations yet established over the boundary• This will be discussed in the next section


Homework 7D5.6. D5.7.D6.1. (7th Edition) or D5.8 (6th Edition)

Deadline: 17.03.11, at 07:30 am.

Chapter 5 Current and Conductors

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The Method of Images