Electric Fields in Material Space

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EngineeringElectromagnetics

Electric Fields in

Material Space


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Electric Fields inMaterial Space

Objectives:

1.

To determine the properties of materials and theelectromagnetic characteristics of each

2. To differentiate convection and conduction currents

3. To understand the concept of dielectric constant

and strength for different materials4. To determine the different boundary conditions of

materials

5. To understand the concept of resistance

6. To understand the concept of capacitance


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Introduction

In the last chapter, we considered electrostatic fieldsin free space or a space that has no materials in it.Thus what we have developed so far under

electrostatics may be regarded as the "vacuum" field

theory. We shall develop in this chapter what may be

regarded as the theory of electric phenomena inmaterial space.

Objectives:

1. To determine the properties of materials and theelectromagnetic characteristics of each


3. To understand the concept of dielectric constant andstrength for different materials

4. To determine the different boundary conditions of materials





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Just as electric fields can exist in free space, theycan exist in material media. Materials are broadlyclassified in terms of their electrical properties asconductors and nonconductors. Nonconductingmaterials are usually referred to as insulators ordielectrics.

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Properties of Materials

In a broad sense, materials may be classified interms of their conductivity !, in mhos per meteror Siemens per meter (S/m), as conductors andnonconductors, or technically as metals andinsulators (or dielectrics). The conductivity of amaterial usually depends on temperature and

frequency. A material with high conductivity (! >> 1)is referred to as a metal whereas one with lowconductivity (!


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Approximate Conductivity of Some CommonMaterials at 20°C


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The conductivity of metals generally increases with

decrease in temperature. At temperatures nearabsolute zero (T = 0°K), some conductors exhibit

infinite conductivity and are called

superconductors. Lead and aluminum are typical

examples of such metals. The conductivity of lead at

4°K is of the order of 1020

mhos/m.

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Convection and Conduction Currents

Electric voltage (or potential difference) and currentare two fundamental quantities in electricalengineering. Before examining how electric field

behaves in a conductor or dielectric, it is appropriate

to consider electric current. Electric current is

generally caused by the motion of electric charges.

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The current (in amperes) through a given area is the

electric charge passing through the area per unittime.

That is,

Thus in a current of one ampere, charge is beingtransferred at a rate of one coulomb per second.

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We now introduce the concept of current density J. If

current "I flows through a surface "S, the currentdensity is

assuming that the current density is perpendicular to

the surface.

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4.

To determine the different boundary conditions of materials





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If the current density is not normal to the surface,

Thus, the total current flowing through a surface S is

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4.






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Depending on how I is produced, there are different

kinds of current densities: convection current density,conduction current density, and displacement current

density. What we need to keep in mind is that the

equation applies to any kind of current density.

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4.






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Convection current, as distinct from conduction

current, does not involve conductors andconsequently does not satisfy Ohm's law. It occurs

when current flows through an insulating medium

such as liquid, rarefied gas, or a vacuum. A beam of

electrons in a vacuum tube, for example, is a

convection current.

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4.






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Consider a filament of the figure. If there is a flow of

charge, of density #v, at velocity , thecurrent through the filament is

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The y-directed current density Jy is given by

Hence, in general

The current I is the convection current and J is the

convection current density in amperes/square meter(A/m2).

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Conduction current requires a conductor. A

conductor is characterized by a large amount of freeelectrons that provide conduction current due an

impressed electric field. When an electric field E is

applied, the force on an electron with charge -e is

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Since the electron is not in free space, it will not be

accelerated under the influence of the electric field.Rather, it suffers constant collision with the atomic

lattice and drifts from one atom to another. If the

electron with mass m is moving in an electric field E

with an average drift velocity u, according to

Newton's law, the average change in momentum ofthe free electron must match the applied force.

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Thus

where ! is the average time interval betweencollisions. This indicates that the drift velocity of the

electron is directly proportional to the applied field.

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

Thus the conduction current density is

where " = ne2!/m is the conductivity of theconductor. This is known as the point form ofOhm’s law.

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4.






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Conductors

A conductor has abundance of charge that is free tomove. Consider an isolated conductor, such asshown in Figure (a). When an external electric fieldEe is applied, the positive free charges are pushedalong the same direction as the applied field, whilethe negative free charges move in the oppositedirection. This charge migration takes place veryquickly. The free charges do two things. First, theyaccumulate on the surface of the conductor and forman induced surface charge. Second, the inducedcharges set up an internal induced field Ei, whichcancels the externally applied field Ee. The result isillustrated in Figure (b).

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A perfect conductor cannot contain an electrostatic

field within it. A conductor is called an equipotential body, implying

that the potential is the same everywhere in theconductor. This is based on the fact that

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Another way of looking at this is to consider Ohm's

law,

To maintain a finite current density J, in a perfectconductor (" $ %), requires that the electric fieldinside the conductor must vanish. In other words, E

$ 0 because " $ % in a perfect conductor. If somecharges are introduced in the interior of such aconductor, the charges will move to the surface and

redistribute themselves quickly in such a manner that

the field inside the conductor vanishes.

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4.






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According to Gauss's law, if E = 0, the charge

density #v must be zero. We conclude again that aperfect conductor cannot contain an electrostaticfield within it. Under static conditions,

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We now consider a conductor whose ends are

maintained at a potential difference V, as shown inthe Figure. Note that in this case, E & 0 inside theconductor.

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There is no static equilibrium in the Figure since the

conductor is not isolated but wired to a source ofelectromotive force, which compels the free charges

to move and prevents the eventual establishment of

electrostatic equilibrium. Thus in the case of the

Figure, an electric field must exist inside the

conductor to sustain the flow of current. As theelectrons move, they encounter some damping

forces called resistance.

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4.






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Based on Ohm's law, we will derive the resistance of

the conducting material. Suppose the conductor hasa uniform cross section S and is of length '. Thedirection of the electric field E produced is the same

as the direction of the flow of positive charges or

current I. This direction is opposite to the direction of

the flow of electrons. The electric field applied isuniform and its magnitude is given by

Objectives:




4.

To determine the different boundary conditions of materials5. To understand the concept of resistance




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Since the conductor has a uniform cross section,

By substitution,

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Hence

where #c = 1/" is the resistivity of the material. Thisequation is useful in determining the resistance ofany conductor of uniform cross section.

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For a conductor having a nonuniform cross section,the resistance is

Note that the negative sign before V = - ( E • dl isdropped in the equation because ( E • dl < 0 if I > 0.

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Power P (in watts) is defined as the rate of change ofenergy W (in joules) or force times velocity. Hence,

which is known as Joule's law.

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The power density wP (in watts/m3) is given by the

integrand; that is,

For a conductor with uniform cross section, dv =

dSdl,

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Polarization in Dielectrics

To understand the macroscopic effect of an electricfield on a dielectric, consider an atom of the dielectricas consisting of a negative charge -Q (electron

cloud) and a positive charge +Q (nucleus) as shown

in Figure (a).

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A similar picture can be adopted for a dielectricmolecule; we can treat the nuclei in molecules as

point charges and the electronic structure as a single

cloud of negative charge. Since we have equal

amounts of positive and negative charge, the whole

atom or molecule is electrically neutral. When an

electric field E is applied, the positive charge isdisplaced from its equilibrium position in the direction

of E by the force F+ = QE while the negative charge

is displaced in the opposite direction by the force F-

= QE.

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A dipole results from the displacement of thecharges and the dielectric is said to be polarized. In

the polarized state, the electron cloud is distorted by

the applied electric field E. This distorted charge

distribution is equivalent, by the principle of

superposition, to the original distribution plus a dipole

whose moment is

where d is the distance vector from -Q to +Q of thedipole as shown in Figure (b).

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

As a measure of intensity of the polarization, we

define polarization P (in coulombs/meter square) as

the dipole moment per unit volume of the dielectric;that is,

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Thus we conclude that the major effect of the electricfield E on a dielectric is the creation of dipole

moments that align themselves in the direction of E.

This type of dielectric is said to be nonpolar.

Examples of such dielectrics are hydrogen, oxygen,

nitrogen, and the rare gases. Nonpolar dielectric

molecules do not possess dipoles until theapplication of the electric field as we have noticed.

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Other types of molecules such as water, sulfurdioxide, and hydrochloric acid have built-in

permanent dipoles that are randomly oriented as

shown in the figure and are said to be polar.

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When an electric field E is applied to a polarmolecule, the permanent dipole experiences a

torque tending to align its dipole moment parallel

with E as shown in the figure.

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Let us now calculate the field due to a polarizeddielectric. Consider the dielectric material shown in

the figure as consisting of dipoles with dipole

moment P per unit volume.

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The potential dV at an exterior point O due to thedipole moment P dv' is

where R2 = (x – x’)2 + (y – y’)2 + (z – z’)2 and R isthe distance between the volume element dv' at (x’,

y', z') and the field point O (x, y, z).

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The gradient of 1/R with respect to the primedcoordinates is

Thus,

Applying the vector identity

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By substitution and integrating over the entirevolume v' of the dielectric, we obtain

Applying divergence theorem to the first term leads

finally to

where a'n is the outward unit normal to surface dS' ofthe dielectric.

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It shows that the two terms denote the potential dueto surface and volume charge distributions with

densities (upon dropping the primes)

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It reveals that where polarization occurs, anequivalent volume charge density #pv is formed

throughout the dielectric while an equivalent surfacecharge density #ps is formed over the surface of thedielectric. We refer to #ps and #pv as bound (orpolarization) surface and volume charge densities,respectively, as distinct from free surface andvolume charge densities #s and #v. Bound charges

are those that are not free to move within thedielectric material; they are caused by thedisplacement that occurs on a molecular scaleduring polarization. Free charges are those that arecapable of moving over macroscopic distance aselectrons in a conductor; they are the stuff we

control.

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Electric Fields in

Material Space


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The total positive bound charge on surface Sbounding the dielectric is

while the charge that remains inside S is

Thus the total charge of the dielectric materialremains zero, that is,

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Electric Fields in

Material Space


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We now consider the case in which the dielectricregion contains free charge. If #

v

is the free charge

volume density, the total volume charge density #t isgiven by

Hence,

where

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Electric Fields in

Material Space


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We conclude that the net effect of the dielectric onthe electric field E is to increase D inside it by

amount P. In other words, due to the application of E

to the dielectric material, the flux density is greater

than it would be in free space. It should be noted that

the definition of D for free space because P = 0 in

free space.

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Electric Fields in

Material Space


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We would expect that the polarization P would varydirectly as the applied electric field E. For some

dielectrics, this is usually the case and we have

where # e, known as the electric susceptibility of thematerial, is more or less a measure of how

susceptible (or sensitive) a given dielectric is to

electric fields.

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Electric Fields in

Material Space


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Dielectric Constant and Strength

Sinceand

then

where

and

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Electric Fields in

Material Space


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) is called the permittivity of the dielectric, )o is thepermittivity of free space, approximately 10-9/36$ F/m

or 8.854x10-12 F/m, and )r is called the dielectricconstant or relative permittivity.

The dielectric constant (or relative permittivity) )r isthe ratio of the permittivity of the dielectric to that of

free space.The dielectric strength is the maximum electric fieldthat a dielectric can tolerate or withstand without

breakdown.

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Electric Fields in

Material Space


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Electric Fields in

Material Space


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The theory of dielectrics we have discussed so farassumes ideal dielectrics. Practically speaking, no

dielectric is ideal. When the electric field in adielectric is sufficiently large, it begins to pullelectrons completely out of the molecules, and thedielectric becomes conducting. Dielectric breakdownis said to have occurred when a dielectric becomesconducting. Dielectric breakdown occurs in all kinds

of dielectric materials (gases, liquids, or solids) anddepends on the nature of the material, temperature,humidity, and the amount of time that the field isapplied. The minimum value of the electric field atwhich dielectric breakdown occurs is called thedielectric strength of the dielectric material.

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Electric Fields in

Material Space


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Linear, Isotropic, and Homogeneous Dielectrics

A material is said to be linear if D varies linearly withE and nonlinear otherwise. Materials for which ) (or!) does not vary in the region being considered andis therefore the same at all points (i.e., independent

of x, y, z) are said to be homogeneous. They are

said to be inhomogeneous (or nonhomogeneous)when ) is dependent of the space coordinates. Theatmosphere is a typical example of an

inhomogeneous medium; its permittivity varies with

altitude.

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Electric Fields in

Material Space


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Materials for which D and E are in the same directionare said to be isotropic. That is, isotropic dielectrics

are those which have the same properties in alldirections. For anisotropic (or nonisotropic)materials, D, E, and P are not parallel; ) or # e hasnine components that are collectively referred to as atensor. For example,

for anisotropic materials. Crystalline materials and

magnetized plasma are anisotropic.

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Electric Fields in

Material Space


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A dielectric material (in which D = )E applies) islinear if ) does not change with the applied E field,homogeneous if ) does not change from point topoint, and isotropic if ) does not change withdirection.

The same idea holds for a conducting material in

which J = !E applies. The material is linear if ! doesnot vary with E, homogeneous if ! is the same at allpoints, and isotropic if ! does not vary with direction.

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Electric Fields in

Material Space

Obj ti


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For most of the time, we will be concerned only withlinear, isotropic, and homogeneous media. For such

media, all formulas derived last Chapter for free

space can be applied by merely replacing )o with)o)r . Thus Coulomb's law, for example, becomes

and work expended becomes

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Electric Fields in

Material Space

Obj ti


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Continuity Equation and Relaxation Time

Due to the principle of charge conservation, the timerate of decrease of charge within a given volumemust be equal to the net outward current flow

through the closed surface of the volume. Thus

current Iout coming out of the closed surface is

where Qin is the total charge enclosed by the closed

surface.

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Electric Fields in

Material Space

Obj ti


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Invoking divergence theorem

But

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Electric Fields in

Material Space

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Substituting the equation gives

which is called the continuity of current equation.

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Electric Fields in

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It must be kept in mind that the continuity equation isderived from the principle of conservation of charge

and essentially states that there can be no

accumulation of charge at any point. For steady

currents, and hence showing

that the total charge leaving a volume is the same asthe total charge entering it. Kirchhoff's current law

follows from this.

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Electric Fields in

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Having discussed the continuity equation and theproperties ! and ) of materials, it is appropriate toconsider the effect of introducing charge at some

interior point of a given material (conductor or

dielectric). We make use of the equation

in conjunction with Ohm's lawand Gauss’s law

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Electric Fields in

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Substituting the equation yields

This is a homogeneous linear ordinary differential

equation.

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Electric Fields in

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By separating variables in the equation, we get

and integrating both sides gives

where In#vo is a constant of integration.

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Electric Fields in

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Thus

where

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In the equation, #vo is the initial charge density (i.e.,#v at t = 0). The equation shows that as a result ofintroducing charge at some interior point of the

material there is a decay of volume charge density

#v. Associated with the decay is charge movementfrom the interior point at which it was introduced tothe surface of the material. The time constant T

r (in

seconds) is known as the relaxation time or

rearrangement time.

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Electric Fields in

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Relaxation time is the lime it takes a charge placedin the interior of a material to drop to e-1 = 36.8

percent of its initial value.

It is short for good conductors and long for gooddielectrics. For example, for copper ! = 5.8 X 107 mhos/m, )r = 1, and

showing a rapid decay of charge placed inside

copper.

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Electric Fields in

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This implies that for good conductors, the relaxationtime is so short that most of the charge will vanish

from any interior point and appear at the surface (assurface charge). On the other hand, for fused quartz,

for instance, ! = 10-17 mhos/m, )r = 5.0,

showing a very large relaxation time. Thus for good

dielectrics, one may consider the introduced charge

to remain wherever placed.

j




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Electric Fields in

Material Space

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Boundary Conditions

So far, we have considered the existence of theelectric field in a homogeneous medium. If the fieldexists in a region consisting of two different media,

the conditions that the field must satisfy at the

interface separating the media are called boundary

conditions. These conditions are helpful indetermining the field on one side of the boundary ifthe field on the other side is known.

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Electric Fields in

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Obviously, the conditions will be dictated by thetypes of material the media are made of. We shall

consider the boundary conditions at an interfaceseparating

– dielectric ()r1) and dielectric ()r2)

– conductor and dielectric

–

conductor and free space

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Electric Fields in

Material Space

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To determine the boundary conditions, we need touse Maxwell's equations:

and

Also we need to decompose the electric field

intensity E into two orthogonal components:

where Et and En are, respectively, the tangential and

normal components of E to the interface of interest.

A similar decomposition can be done for the electric

flux density D.







Electric Fields in

Material Space

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A. Dielectric-Dielectric Boundary Conditions

Consider the E field existing in a region consistingof two different dielectrics characterized by )1 =)o)r1 and )2 = )o)r2 as shown in the figure.




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Electric Fields in

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Electric Fields in

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E1 and E2 in media 1 and 2, respectively, can bedecomposed as

We apply the Maxwell’s equation to the closed path

abcda of the figure (a) assuming that the path is very

small with respect to the variation of E. We obtain




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Electric Fields in

Material Space

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As "h $ 0, the equation becomes

Thus the tangential components of E are the same

on the two sides of the boundary In other words, Et

undergoes no change on the boundary and it is said

to be continuous across the boundary.







Electric Fields in

Material Space

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Since D = )E = Dt + Dn, the equation can be writtenas

that is, Dt undergoes some change across theinterface. Hence Dt is said to be discontinuousacross the interface.







Electric Fields in

Material Space

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Similarly, we apply the equation to the pillbox(Gaussian surface) of the figure (b). Allowing "h $ 0gives

where #s is the free charge density placeddeliberately at the boundary.







Electric Fields in

Material Space

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The equation is based on the assumption that D isdirected from region 2 to region 1 and it must be

applied accordingly. If no free charges exist at theinterface (i.e., charges are not deliberately placed

there), #s = 0 and the equation becomes







Electric Fields in

Material Space

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T d t i th ti f t i l d th


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Thus the normal component of D is continuousacross the interface; that is, Dn undergoes no

change at the boundary. Since D = )E, the equationcan be written as

showing that the normal component of E is

discontinuous at the boundary. The equations arecollectively referred to as boundary conditions; theymust be satisfied by an electric field at the boundary

separating two different dielectrics.







Electric Fields in

Material Space

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T d t i th ti f t i l d th


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The boundary conditions are usually applied infinding the electric field on one side of the boundary

given the field on the other side. Besides this, wecan use the boundary conditions to determine the"refraction" of the electric field across the interface.Consider D1 or E1 and D2 or E2 making angles *1 and *2 with the normal to the interface as illustrated

in the figure.







Electric Fields in

Material Space

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1 T d t i th ti f t i l d th


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Electric Fields in

Material Space

Objectives:

1 To determine the properties of materials and the


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Using

we have







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Material Space

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Similarly, by applying the equations

we get







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Material Space

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Dividing the two equations gives

Since )1 = )o )r1 and )2 = )o )r2, the equation becomes







Electric Fields in

Material Space

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This is the law of refraction of the electric field at aboundary free of charge (since #s = 0 is assumed atthe interface). Thus, in general, an interface betweentwo dielectrics produces bending of the flux lines as

a result of unequal polarization charges that

accumulate on the sides of the interface.







Electric Fields in

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B. Conductor-Dielectric Boundary Conditions

As shown in the figure, the conductor is assumed tobe perfect (i.e., ! $ % or #c $ 0). Although such aconductor is not practically realizable, we may

regard conductors such as copper and silver as

though they were perfect conductors.







Electric Fields in

Material Space

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To determine the boundary conditions for aconductor-dielectric interface, we follow the same

procedure used for dielectric-dielectric interfaceexcept that we incorporate the fact that E = 0 inside

the conductor. Applying the Maxwell’s equation to

the closed path abcda of Figure(a) gives







Electric Fields in

Material Space

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1. To determine the properties of materials and the


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As "h $ 0,

Similarly, by applying the Maxwell’s equation to thepillbox of Figure(b) and letting "h $ 0, we get

because D = )E = 0 inside the conductor.







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Material Space

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The equation may be written as

p pelectromagnetic characteristics of each






Electric Fields in

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Thus under static conditions, the followingconclusions can be made about a perfect conductor:

1. No electric field may exist within a conductor; that

is,

2. Since , there can be no potentialdifference between any two points in the conductor;

that is, a conductor is an equipotential body.







Electric Fields in

Material Space

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3. The electric field E can be external to theconductor and normal to its surface; that is

An important application of the fact that E = 0 inside

a conductor is in electrostatic screening or shielding.

If conductor A kept at zero potential surroundsconductor B as shown in the figure, B is said to be

electrically screened by A from other electric

systems, such as conductor C, outside A. Similarly,

conductor C outside A is screened by A from B.







Electric Fields in

Material Space

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Thus conductor A acts like a screen or shield and theelectrical conditions inside and outside the screenare completely independent of each other.

electromagnetic characteristics of each






Electric Fields in

Material Space

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C. Conductor-Free Space Boundary Conditions

This is a special case of the conductor-dielectric

conditions and is illustrated in Figure.







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Material Space

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The boundary conditions at the interface between aconductor and free space can be obtained from the

equation

by replacing )r by 1 (because free space may be

regarded as a special dielectric for which )r = 1).







Electric Fields in

Material Space

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We expect the electric field E to be external to theconductor and normal to its surface. Thus the

boundary conditions are

It should be noted again that the equation impliesthat E field must approach a conducting surface

normally.







Electric Fields in

Material Space

Objectives:

1. To determine the properties of materials and thel t ti h t i ti f h


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Resistance“ opposition to flow of electrons”

Although resistance is a negative term, it is sometimesdesirable because it produces heat. As electrons bump intoeach other, friction releases heat energy. If resistance isincreased, more electrons bump into each other and more heatis produced. Electric heating elements such as toasters, ovens,flat irons, and electric heaters employ this principle. A high

resistance wire made of a substance such as Nichrome, analloy of iron, nickel and chromium, is used in heating elements.

As electric current flow through Nichrome, the high resistanceproduces heat. Coiled, high resistance tungsten wires are usedas filaments in electric light bulbs.







Electric Fields in

Material Space

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The unit of measurement of resistance is the ohm (+) namedafter the German physicist, Georg Simon Ohm. A resistance of

one ohm permits one ampere to flow through a circuit at an emfof one volt. A device called an ohmmeter measures resistance.

In modern electronic equipment, it sometimes becomes

necessary to increase resistance of a circuit. In such cases,

devices known as resistors are used.

At normal temperatures, all substances provide at least some

resistance to the flow of electrons. Because metals offer littleresistance to electric flow, they make good conductors. Since

glass provide a great deal of resistance, it is an insulator.







Electric Fields in

Material Space

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Factors affecting magnitude of resistance:

1. Resistivity (#) is a constant property of material dependent on

its molecular structure. The higher the resistivity of amaterial, the higher is its resistance. Thus, R , # (directlyproportional).







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Material Space

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2. Cross-sectional area (A) or diameter affects resistance since

bigger cross section offers greater number of free electrons

while smaller cross section has less. Thus, seemingly biggercross section allows flow of electrons easier than smaller one.







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3. The longer the wire leads the electron to pass through more

opposition due to the presence of several other particles on

the conductor. Thus, the longer the wire, the higher is theresistance.







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4. Given a conductor ’s resistivity, cross-sectional area, and

length, still the resistance varies dependent on the

temperature of the environment where the conductor isplaced. At a higher temperature, the kinetic molecular energy

is higher which leads the molecules to rapidly move at

random direction, which blocks the passage of electrons on it.

Therefore, the resistance is higher at a higher temperature.







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Capacitance

Generally speaking, to have a capacitor we must

have two (or more) conductors carrying equal butopposite charges. This implies that all the flux lines

leaving one conductor must necessarily terminate at

the surface of the other conductor. The conductors

are sometimes referred to as the plates of thecapacitor. The plates may be separated by freespace or a dielectric.







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Consider the two-conductor capacitor of the Figure.




4. To determine the different boundary conditions of mater

Electric Fields in Material Space

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