Electric Fields in Material Space

The charges considered up to this point have been assumed to bestationary and located in free space (vacuum) or air. If we place chargewithin a gas, solid or liquid material, the charge associated with thematerial atoms will be affected. Also, under the influence of the appliedelectric field, charges not bound by other forces (free charges) may be setin motion (electric current).

Current (I ) - net flow of positive charge in a given direction (vector)measured in units of Amperes (Ampere = Coulomb/second).

Note that, mathematically, the negative charge moving in the oppositedirection constitutes a positive component of the overall current flowingin the ax direction.

Conductor - current carrying medium.

Insulator - non-conducting medium.

Material Classification Based on Conductivity

The conductivity (�) of a given material is a measure of the ability ofmaterial to conduct current. Conductivity is measured in units of S/m or�/m. The inverse of conductivity is resistivity (�c = 1/�). For elements,the structure of the element atom dictates the conductivity of the element.Specifically, the element conductivity is related to the strength of the bondsbetween the outer (valence) electrons and the atom nucleus.

Positive nucleus charge = Total negative electron chargeCentroid of the nucleus charge - atom centerCentroid of the overall electron charge - atom center

If under the influence of an electric field, the bond between thevalence electron and the atom nucleus is broken, the electron becomes afree electron or conduction electron. Materials are classified asconductors, insulators, or semiconductors based on the strength of thesebonds between the valence electrons and the atom nucleus. The strongerthe bond between the valence electrons and the nucleus in a particularmaterial, the fewer free electrons are available for conduction.

The atom iselectrically neutral.

(�V = 0, V = 0, E = 0)�

The values of conductivity designated at the boundaries betweenmaterial types are defined differently by many authors. Since conductivityis, in general, a function of temperature, comparisons of conductivity aremade at a constant temperature (reference temperature, usually To = 20oC).The dependence of resistivity on temperature may be expressed as

where �co is the material resistivity at the reference temperature To and �is the temperature coefficient for the material. Certain conductors andoxides exhibit superconductivity at temperatures near absolute zero (0K =�273oC) where the resistivity of the material drops abruptly to zero.

Examples (Conductivity in S/m at T = 20oC)

Insulators Semiconductors Conductors

Porcelain (10�12) Silicon (4.4×10�4) Silver (6.1×107) Glass (10�12) Germanium (2.2) Copper (5.8×107) Mica (10�15) Gold (4.1×107) Wax (10�17) Aluminum (3.5×107)

Carbon (3×104)

Ideal Models

Perfect Insulator (� = 0) Perfect conductor (� = �)

Current Types

Currents that flow in conductors are only one of three different typesof currents. The three types of currents are:

(1) Conduction current (current in a conductor)Example - current in a copper wire.

(2) Convection current (current through an insulator)Example - electron beam in a CRT.

(3) Displacement current (time-varying effect to be studied later)Example - AC current in a capacitor.

Separate equations are necessary to define each of these three types ofcurrents given the different mechanisms involved.

For a current density J (A/m2) associated with any type of current, thetotal current I passing through a given surface S is defined as

where ds = dsan, an is the unit normal to the surface and Jn is thecomponent of the current normal to the surface . The scalar result of thisintegral is the magnitude of the total current flowing in the direction of theunit normal. For the special case when the current density is uniform overthe surface S,

where A is the total area of the surface S. The total current in Amperes(Coulomb/second) represents the amount of charge passing through thesurface per second. A total current of 1 mA means that a net charge of 1mC is passing through the surface each second.

Convection Current

Convection current is a flow of charged particles through aninsulating medium (example: an electron beam in a cathode-ray tube).Thus, the equation defining convection current density is independent ofthe conductivity of the medium since the medium characteristics(insulator) do not affect the current. The medium through which theconvection current flows is typically a very good insulator (very lowconductivity). Convection current is defined in terms of the free chargedensity in the current (�V) and the vector drift velocity (u) of the charge inthe current. The drift velocity is the average velocity at which the chargeis moving.

The convection current density is defined as

The total convection current is found by integrating the current densityover the cross section of the convection current.

Conduction Current

Conduction current is different from convection current in that thecurrent medium is a conductor rather than an insulator. A simple exampleof conduction current is the current flowing in a conducting wire. If avoltage V is applied to a cylindrical conductor (conductivity = �, length =l, cross-sectional area = A), a conduction current results.

The potential difference between the ends of the conductor means that anelectric field exists within the conductor (pointing from the region ofhigher potential to the region of lower potential). The conduction currentcan be defined in the same way as convection current using the free chargedensity (�V) and the vector drift velocity (u).

In a conductor, there is an abundance of free electrons. The drift velocityin a conductor may be written as the product of the electric field (E) andthe conductor mobility (�).

The mobility of a material is a measure of how efficiently freecarriers can move through the material. Since typical conductors (metals)are dense materials, the free electrons accelerated under the influence ofthe electric field frequently collide with atom nuclei and other electrons.The resulting particle motion looks somewhat random but has a netcomponent of motion in the direction opposite to the electric field (averagevelocity of all like carriers = drift velocity of that carrier). Inserting thedrift velocity formula into the current density equation yields theconduction current density in terms of the electric field:

such that the conductivity is

If the current density in the conductor is uniform, the correspondingelectric field is also uniform (J = �E). The voltage between the ends of thewire can be expressed as the line integral of the electric field.

Thus, the voltage and the uniform electric field may be written as

The uniform current density is then

where

Resistance of a cylinder (length = l, cross-sectional area = A, conductivity = �) carryinga uniform current density

If the current density is not uniform, the resistance formula becomes

The power density inside the conductor is found by forming the dotproduct of the vector electric field and the vector current density.

The total power dissipated in the conductor is found by integrating thepower density throughout the conductor.

Example (Conduction current)

A copper wire (� = 5.8 × 10�7 �/m, �V = �1.4 × 1010 C/m3, radius =1 mm, length = 20 cm) carries a current of 1 mA. Assuming a uniformcurrent density, determine

(a.) the wire resistance.(b.) the current density.(c.) the electric field within the wire.(d.) the drift velocity of the electrons in the wire.

Perfect Conductor (� = �)

R = 0Equipotential volume

E = 0

Perfect Insulator (� = 0)

R = �

J = 0

�

Polarization in Dielectrics

Nonconducting materials are commonly designated as insulators ordielectrics. When an electric field is applied to a dielectric atom, an effectknown as polarization results. With no electric field applied, the centroidof the (negative) electron charge is coincident with the centroid of the(positive) nucleus charge such that the atom is electrically neutral. Whenan electric field is applied to the atom, the positively charged nucleus isdisplaced in the direction of the electric field while the centroid of thenegative electron charge is displaced in the direction opposite to theelectric field. The dielectric atom is thus polarized and may be modeledas an equivalent electric dipole.

If a voltage V is applied to a cylindrical insulator (conductivity = �,length = l, cross-sectional area = A), the insulator is polarized. If theelectric field is assumed to be uniform, then the electric field within theinsulator is E = V/l.

The polarization within the dielectric produces an additional electricflux density component which is included in the electric flux densityequation as the vector polarization P.

The polarization P is defined as the dipole moment per unit volume suchthat

where n is the number of dipoles in the volume v. Assuming that thepolarization vector P is proportional to the electric field E, we may write

where �e is defined as the electric susceptibility (unitless). Inserting thisdefinition of P into the electric flux equation gives

where

Note that the electric susceptibility �e and the relative permittivity �r areboth measures of the polarization within a given material. The larger thevalue of �e or �r for the material, the more polarization within the material.For free space (vacuum), there is no polarization such that

P = 0 � �e = 0 or �r = 1

The amount of polarization found in air is extremely small, so that wetypically model our atmosphere with the free space permittivity.

The magnitude of the polarization in a dielectric increases with themagnitude of the applied electric field (the equivalent dipole momentsgrow with the electric field magnitude). For a good insulator, the bondsbetween the atom nuclei and the valence electrons are very strong and canwithstand very large electric fields. The electric field level at which thesebonds are broken, and the insulator begins to conduct (breakdown), isdesignated as the dielectric strength. Some typical values of dielectricstrengths for some common insulators are:

Mica 70 MV/mGlass 35 MV/mAir 3 MV/m

The total charge density (�T) in an insulating material consists of thefree conduction charge density (�v) plus the bound polarization chargedensity (�vp).

From our previous definition of the differential form of Gauss’s law, wesee that the divergence of the electric flux density yields the free chargedensity.

If we insert the expression for the electric flux density in terms of thepolarization and the free charge density in terms of the total charge density,we find

Equating terms yields

The divergence of the polarization vector gives the negative of the boundpolarization charge density.

Media Classifications

The electrical properties of a given medium are defined by threeconstants: conductivity (�), permittivity (�), and permeability (�). Thepermeability will be defined later when we study magnetic fields. Thefollowing media classifications are made based on the characteristics of themedium constants.

Linear medium - electrical properties do not vary with field magnitude. Homogeneous medium - electrical properties do not vary with position. Isotropic medium - electrical properties do not vary with field direction.

Otherwise, the medium is nonlinear, inhomogeneous, or anisotropic.

Continuity Equation

The continuity equation defines the basic conservation of chargerelationship between current and charge. That is, a net current in or out ofa given volume must equal the net increase of decrease in the total chargein the volume. If we define a surface S enclosing a volume V, the netcurrent out of the volume (Iout) is defined by

where ds = dsan and an is theoutward pointing normal. If thecurrent I is a DC current, then thenet current out of the volume iszero (as much current flows out as flows in). For a time-varying current,the net current out of the volume may be non-zero and can be expressed interms of the change in the total charge within the volume (Q).

The previous equation is the integral form of the continuity equation. Thedifferential form of the continuity equation can be found by applying thedivergence theorem to the surface integral and expressing the total chargein terms of the charge density.

The second and last terms in the equation above yield integrals that arevalid for any volume V that we may choose.

Since the previous equation is valid for any volume V, we may equate theintegrands of the integrals (the only way for the integrals to yield the samevalue for any volume V is for the integrands to be equal). This yields thecontinuity equation.

The continuity equation is given in differential form and relates the currentdensity at a given point to the charge density at that point. For steadycurrents (DC currents), the charge density does not change with time sothat the divergence of the current density is always zero.

The continuity equation is the basis for Kirchhoff’s current law.Given a circuit node connecting a system of N wires (assuming DCcurrents) enclosed by a spherical surface S, the integral form of thecontinuity equation gives

The integral form of the continuity equation (and thus Kirchhoff’s currentlaw) also holds true for time-varying (AC) currents if we let the surface Sshrink to zero around the node.

Relaxation Time

If some amount of charge is placed inside a volume of conductingmaterial, the Coulomb forces on the individual charges cause them tomigrate away from each other (assuming the charge is all positive or allnegative). The end result is a surface charge on the outer surface of theconductor while the inside of the conductor remains charge-neutral. Thetime required for the conductor to reach this charge-neutral state is relatedto a time constant designated as the relaxation time. The relaxation timecan be determined by inserting the relationship for the current density interms of electric field

into the continuity equation

which yields

The divergence of the electric field is related to the charge density by

Inserting this result into previous equation yields

or

The solution to this homogeneous, first order PDE is

where Tr is the relaxation time given by.

The relaxation time is a time constant that describes the rate of decay of thecharge inside the conductor. After a time period of Tr, the charge hasdecayed to 36.8 percent (1/e) of its original value.

Example (Relaxation time)

Determine the relaxation time for copper (�r = 1, � = 5.8×107 �/m)and fused quartz (�r = 5, � = 10�17 �/m).

Copper

Fused Quartz

Electric Field Boundary Conditions

A knowledge of the behavior of electric fields at a media interfacebetween distinct materials is necessary to solve many common problemsin electromagnetics. The fundamental boundary conditions involvingelectric fields relate the tangential components of electric field and thenormal components of electric flux density on either side of the mediainterface.

Tangential Electric Field

In order to determine the boundary condition on the tangentialelectric field at a media interface, we evaluate the line integral of theelectric field along a closed incremental path that extends into both regionsas shown below.

The closed line integral of the electric field yields a result of zero such that

If we take the limit of this integral as �y = 0, the integral contributions onthe vertical paths vanish.

The integrals along the upper and lower paths on either side of the interfacereduce to

where the electric field components are assumed to be constant over thepaths of length �x. Dividing the result by �x gives

or

The tangential components of electric field are continuous across a media interface.

If region 1 is a dielectric and region 2 is a perfect conductor (�2 = �), thenEt 2 = 0 and

The tangential component of electric field onthe surface of a perfect conductor is zero.

Normal Electric Flux Density

In order to determine the boundary condition on the normal electricflux density at a media interface, we apply Gauss’s law to an incrementalvolume that extends into both regions as shown below.

The application of Gauss’s law to the closed surface above gives

If we take the limit as the height of the volume �z = 0, the integralcontributions on the four sides of the volume vanish.

The integrals over the upper and lower surfaces on either side of theinterface reduce to

where the electric flux density is assumed to be constant over the upper andlower incremental surfaces. Evaluation of the surface integrals yields

Dividing by �x�y gives

where the charge density �S is assumed to be uniform.

The difference in the normal component of electric flux density across the media interface is equal to the chargedensity on the interface.

On a charge-free interface (�S = 0), such that

The normal components of electric flux density are continuous across a charge-free media interface.

If region 1 is a dielectric and region 2 is a perfect conductor (�2 = �), thenDn2 = 0 and

The normal component of electric flux density onthe surface of a perfect conductor equals the surface charge density.

The following statements describe the characteristics of a perfectconductor under static conditions:

(1) E = 0 inside the conductor.(2) �v = 0 inside the conductor [free charge, if present, lies on the

outer surface of the conductor (�s)].(3) The conductor is an equipotential volume.(4) Tangential E on the surface of the conductor is zero.(5) Normal D on the surface of the conductor equals �s.(6) The electric field lines are normal to the surface of the

conductor.

Example (Polarization/Boundary conditions)

A dielectric cylinder (region 1) of radius �=3 and permittivity�r1=2.5 is surrounded by another dielectric(region 2) of permittivity �r2=10. Given anelectric field inside the cylinder of

determine (a.) P1 and �vp1 (b.) E2 and D2.

(a.)

(b.)

Example (Boundary conditions)

Determine E and D everywhere for the charge-free boundary shownbelow given E1 (or D1).

In general, we may determine the relationship between the electricfield and electric flux vectors in the two regions in terms of the two angles�1 and �2 measured with respect to the normal to the interface.

According to the geometry of the field and flux components, we seethat

Dividing the first equation by the second gives

The electric field and electric flux density boundary conditions on thecharge-free boundary are

such that

Given both media characteristics and the direction of the field in one of theregions, the direction of the field in the other region can be determinedusing this formula.