Electric Fields in Material Space
The charges considered up to this point have been assumed to be
stationary and located in free space (vacuum) or air. If we place charge
within a gas, solid or liquid material, the charge associated with the
material atoms will be affected. Also, under the influence of the applied
electric field, charges not bound by other forces (free charges) may be set
in 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 opposite
direction constitutes a positive component of the overall current flowing
in the ax direction.
Conductor - current carrying medium.
Insulator - non-conducting medium.
Material Classification Based on Conductivity
The conductivity (F) of a given material is a measure of the ability of
material to conduct current. Conductivity is measured in units of S/m or
®/m. The inverse of conductivity is resistivity (Dc = 1/F). 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 bonds
between the outer (valence) electrons and the atom nucleus.
Positive nucleus charge = Total negative electron charge
Centroid of the nucleus charge - atom center
Centroid of the overall electron charge - atom center
@
The atom is
electrically neutral.
(DV = 0, V = 0, E = 0)
If under the influence of an electric field, the bond between the
valence electron and the atom nucleus is broken, the electron becomes a
free electron or conduction electron. Materials are classified as
conductors, insulators, or semiconductors based on the strength of these
bonds between the valence electrons and the atom nucleus. The stronger
the bond between the valence electrons and the nucleus in a particular
material, the fewer free electrons are available for conduction.
The values of conductivity designated at the boundaries between
material types are defined differently by many authors. Since conductivity
is, in general, a function of temperature, comparisons of conductivity are
made at a constant temperature (reference temperature, usually To = 20oC).
The dependence of resistivity on temperature may be expressed as
where Dco is the material resistivity at the reference temperature To and "
is the temperature coefficient for the material. Certain conductors and
oxides 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
Porcelain (10!12)
Glass (10!12)
Mica (10!15)
Wax (10!17 )
Semiconductors
Silicon (4.4×10!4 )
Germanium (2.2)
Conductors
Silver (6.1×107 )
Copper (5.8×107 )
Gold (4.1×107 )
Aluminum (3.5×107 )
Carbon (3×104 )
Ideal Models
Perfect Insulator (F = 0)
Perfect conductor (F = 4)
Current Types
Currents that flow in conductors are only one of three different types
of currents. The three types of currents are:
(1)
(2)
(3)
Conduction current (current in a conductor)
Example - current in a copper wire.
Convection current (current through an insulator)
Example - electron beam in a CRT.
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 of
currents given the different mechanisms involved.
For a current density J (A/m2) associated with any type of current, the
total 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 the
component of the current normal to the surface . The scalar result of this
integral is the magnitude of the total current flowing in the direction of the
unit normal. For the special case when the current density is uniform over
the 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 the
surface per second. A total current of 1 mA means that a net charge of 1
mC is passing through the surface each second.
Convection Current
Convection current is a flow of charged particles through an
insulating medium (example: an electron beam in a cathode-ray tube).
Thus, the equation defining convection current density is independent of
the conductivity of the medium since the medium characteristics
(insulator) do not affect the current. The medium through which the
convection current flows is typically a very good insulator (very low
conductivity). Convection current is defined in terms of the free charge
density in the current (DV) and the vector drift velocity (u) of the charge in
the current. The drift velocity is the average velocity at which the charge
is moving.
The convection current density is defined as
The total convection current is found by integrating the current density
over the cross section of the convection current.
Conduction Current
Conduction current is different from convection current in that the
current medium is a conductor rather than an insulator. A simple example
of conduction current is the current flowing in a conducting wire. If a
voltage V is applied to a cylindrical conductor (conductivity = F, length =
l, cross-sectional area = A), a conduction current results.
The potential difference between the ends of the conductor means that an
electric field exists within the conductor (pointing from the region of
higher potential to the region of lower potential). The conduction current
can be defined in the same way as convection current using the free charge
density (DV) and the vector drift velocity (u).
In a conductor, there is an abundance of free electrons. The drift velocity
in a conductor may be written as the product of the electric field (E) and
the conductor mobility (:).
The mobility of a material is a measure of how efficiently free
carriers can move through the material. Since typical conductors (metals)
are dense materials, the free electrons accelerated under the influence of
the electric field frequently collide with atom nuclei and other electrons.
The resulting particle motion looks somewhat random but has a net
component of motion in the direction opposite to the electric field (average
velocity of all like carriers = drift velocity of that carrier). Inserting the
drift velocity formula into the current density equation yields the
conduction current density in terms of the electric field:
such that the conductivity is
If the current density in the conductor is uniform, the corresponding
electric field is also uniform (J = FE). The voltage between the ends of the
wire 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, crosssectional area = A, conductivity = F) carrying
a uniform current density
The power density inside the conductor is found by forming the dot product
of the vector electric field and the vector current density.
The total power dissipated in the conductor is found by integrating the
power density throughout the conductor.
Example (Conduction current)
A copper wire (F = 5.8 × 107 É/m, DV = !1.4 × 1010 C/m3, radius = 1
mm, length = 20 cm) carries a current of 1 mA. Assuming a uniform
current 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 (F = 4)
R=0
E=0
@
Equipotential volume
Perfect Insulator (F = 0)
R=4
J=0
Polarization in Dielectrics
Nonconducting materials are commonly designated as insulators or
dielectrics. When an electric field is applied to a dielectric atom, an effect
known as polarization results. With no electric field applied, the centroid
of the (negative) electron charge is coincident with the centroid of the
(positive) nucleus charge such that the atom is electrically neutral. When
an electric field is applied to the atom, the positively charged nucleus is
displaced in the direction of the electric field while the centroid of the
negative electron charge is displaced in the direction opposite to the
electric field. The dielectric atom is thus polarized and may be modeled
as an equivalent electric dipole.
If a voltage V is applied to a cylindrical insulator (conductivity = F,
length = l, cross-sectional area = A), the insulator is polarized. If the
electric field is assumed to be uniform, then the electric field within the
insulator is E = V/l.
The polarization within the dielectric produces an additional electric
flux density component which is included in the electric flux density
equation as the vector polarization P.
The polarization P is defined as the dipole moment per unit volume such
that
where n is the number of dipoles in the volume v. Assuming that the
polarization vector P is proportional to the electric field E, we may write
where Pe is defined as the electric susceptibility (unitless). Inserting this
definition of P into the electric flux equation gives
where
Note that the electric susceptibility Pe and the relative permittivity ,r are
both measures of the polarization within a given material. The larger the
value of Pe or ,r for the material, the more polarization within the material.
For free space (vacuum), there is no polarization such that
P=0
Y
Pe = 0 or ,r = 1
The amount of polarization found in air is extremely small, so that we
typically model our atmosphere with the free space permittivity.
The magnitude of the polarization in a dielectric increases with the
magnitude of the applied electric field (the equivalent dipole moments
grow with the electric field magnitude). For a good insulator, the bonds
between the atom nuclei and the valence electrons are very strong and can
withstand very large electric fields. The electric field level at which these
bonds are broken, and the insulator begins to conduct (breakdown), is
designated as the dielectric strength. Some typical values of dielectric
strengths for some common insulators are:
Mica
Glass
Air
70 MV/m
35 MV/m
3 MV/m
The total charge density (DT) in an insulating material consists of the
free conduction charge density (Dv) plus the bound polarization charge
density (Dvp).
From our previous definition of the differential form of Gauss’s law, we
see that the divergence of the electric flux density yields the free charge
density.
If we insert the expression for the electric flux density in terms of the
polarization 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 bound
polarization charge density.
Media Classifications
The electrical properties of a given medium are defined by three
constants: conductivity (F), permittivity (,), and permeability (:). The
permeability will be defined later when we study magnetic fields. The
following media classifications are made based on the characteristics of the
medium 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 charge
relationship between current and charge. That is, a net current in or out of
a given volume must equal the net increase of decrease in the total charge
in the volume. If we define a surface S enclosing a volume V, the net
current out of the volume (Iout) is defined by
where ds = dsan and an is the
outward pointing normal. If the
current I is a DC current, then the
net current out of the volume is
zero (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 in
terms of the change in the total charge within the volume (Q).
The previous equation is the integral form of the continuity equation. The
differential form of the continuity equation can be found by applying the
divergence theorem to the surface integral and expressing the total charge
in terms of the charge density.
The second and last terms in the equation above yield integrals that are
valid for any volume V that we may choose.
Since the previous equation is valid for any volume V, we may equate the
integrands of the integrals (the only way for the integrals to yield the same
value for any volume V is for the integrands to be equal). This yields the
continuity equation.
The continuity equation is given in differential form and relates the current
density at a given point to the charge density at that point. For steady
currents (DC currents), the charge density does not change with time so
that
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 DC
currents) enclosed by a spherical surface S, the integral form of the
continuity equation gives
The integral form of the continuity equation (and thus Kirchhoff’s current
law) also holds true for time-varying (AC) currents if we let the surface S
shrink to zero around the node.
Relaxation Time
If some amount of charge is placed inside a volume of conducting
material, the Coulomb forces on the individual charges cause them to
migrate away from each other (assuming the charge is all positive or all
negative). The end result is a surface charge on the outer surface of the
conductor while the inside of the conductor remains charge-neutral. The
time required for the conductor to reach this charge-neutral state is related
to a time constant designated as the relaxation time. The relaxation time
can be determined by inserting the relationship for the current density in
terms 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 the
charge inside the conductor. After a time period of Tr, the charge has
decayed to 36.8 percent (1/e) of its original value.
Example (Relaxation time)
Determine the relaxation time for copper (,r = 1, F = 5.8×107 ®/m)
and fused quartz (,r = 5, F = 10!17 ®/m).
Copper
Fused Quartz
Electric Field Boundary Conditions
A knowledge of the behavior of electric fields at a media interface
between distinct materials is necessary to solve many common problems
in electromagnetics. The fundamental boundary conditions involving
electric fields relate the tangential components of electric field and the
normal components of electric flux density on either side of the media
interface.
Tangential Electric Field
In order to determine the boundary condition on the tangential
electric field at a media interface, we evaluate the line integral of the
electric field along a closed incremental path that extends into both regions
as 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 on
the vertical paths vanish.
The integrals along the upper and lower paths on either side of the interface
reduce to
where the electric field components are assumed to be constant over the
paths 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 (F2 = 4), then
Et 2 = 0 and
The tangential component of electric field on
the surface of a perfect conductor is zero.
Normal Electric Flux Density
In order to determine the boundary condition on the normal electric
flux density at a media interface, we apply Gauss’s law to an incremental
volume 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 integral
contributions on the four sides of the volume vanish.
The integrals over the upper and lower surfaces on either side of the
interface reduce to
where the electric flux density is assumed to be constant over the upper and
lower incremental surfaces. Evaluation of the surface integrals yields
Dividing by )x )y gives
where the charge density DS is assumed to be uniform.
The difference in the normal component of electric flux
density across the media interface is equal to the charge
density on the interface.
On a charge-free interface (DS = 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 (F2 = 4), then
Dn2 = 0 and
The normal component of electric flux density on
the surface of a perfect conductor equals the
surface charge density.
The following statements describe the characteristics of a perfect
conductor under static conditions:
(1)
(2)
(3)
(4)
(5)
(6)
E = 0 inside the conductor.
Dv = 0 inside the conductor [free charge, if present, lies on the
outer surface of the conductor (Ds)].
The conductor is an equipotential volume.
Tangential E on the surface of the conductor is zero.
Normal D on the surface of the conductor equals Ds.
The electric field lines are normal to the surface of the
conductor.
Example (Polarization/Boundary conditions)
A dielectric cylinder (region 1) of radius D =3 and permittivity
,r1=2.5 is surrounded by another dielectric
(region 2) of permittivity ,r2 =10. Given an
electric field inside the cylinder of
determine (a.) P1 and Dvp1 (b.) E2 and D2.
(a.)
(b.)
Example (Boundary conditions)
Determine E and D everywhere for the charge-free boundary shown
below given E1 (or D1).
In general, we may determine the relationship between the electric
field and electric flux vectors in the two regions in terms of the two angles
21 and 22 measured with respect to the normal to the interface.
According to the geometry of the field and flux components, we see
that
Dividing the first equation by the second gives
The electric field and electric flux density boundary conditions on the
charge-free boundary are
such that
Given both media characteristics and the direction of the field in one of the
regions, the direction of the field in the other region can be determined
using this formula.