Engineering Electromagnetics W.H. Hayt Jr. and J. A. Buck

Engineering ElectromagneticsW.H. Hayt Jr. and J. A. Buck Chapter 5: Conductors and Dielectrics

Current is a flux quantity and is defined as: Current density, J, measured in Amps/m2 , yields current in Amps when it is integrated over a cross-sectional area. The assumption would be that the direction of J is normal to the surface, and so we would write: Current and Current Density

In reality, the direction of current flow may not be normal to the surface in question, so we treat current density as a vector, and write the incremental flux through the small surface in the usual way: where S = n da n Then, the current through a large surface is found through the flux integral: Current Density as a Vector Field

Consider a charge Q, occupying volume v, moving in the positive x direction at velocity vx In terms of the volume charge density, we may write: Suppose that in time t, the charge moves through a distance x = L = vxt Then The motion of the charge represents a current given by: or Relation of Current to Charge Velocity

We now have The current density is then: So that in general: Relation of Current Density to Charge Velocity

Suppose that charge Qiis escaping from a volume through closed surface S, to form current density J. Then the total current is: Qi(t) where the minus sign is needed to produce positive outward flux, while the interior charge is decreasing with time. We now apply the divergence theorem: The integrands of the last expression must be equal, leading to the Equation of Continuity so that or Continuity of Current

Energy Band Structure in Three Material Types Conductors exhibit no energy gap between valence and conduction bands so electrons move freely Insulators show large energy gaps, requiring large amounts of energy to lift electrons into the conduction band When this occurs, the dielectric breaks down. Semiconductors have a relatively small energy gap, so modest amounts of energy (applied through heat, light, or an electric field) may lift electrons from valence to conduction bands.

Free electrons move under the influence of an electric field. The applied force on an electron of charge Q = -e will be When forced, the electron accelerates to an equilibrium velocity, known as the drift velocity: where eis the electron mobility, expressed in units of m2/V-s. The drift velocity is used to find the current density through: The expression: from which we identify the conductivity for the case of electron flow : S/m is Ohm’s Law in point form In a semiconductor, we have hole current as well, and Electron Flow in Conductors

Consider the cylindrical conductor shown here, with voltage V applied across the ends. Current flows down the length, and is assumed to be uniformly distributed over the cross-section, S. First, we can write the voltage and current in the cylinder in terms of field quantities: Using Ohm’s Law: We find the resistance of the cylinder: b a Resistance

Consider a conductor, on which excess charge has been placed + + + + + + + E + + s + + Electric field at the surface points in the normal direction + + + solid conductor + + E = 0 inside + + + + + • Charge can exist only on the surface as a surface charge density, s -- not in the interior. • Electric field cannot exist in the interior, nor can it possess a tangential component at the surface (as will be shown next slide). 3. It follows from condition 2 that the surface of a conductor is an equipotential. Electrostatic Properties of Conductors

or Over the rectangular integration path, we use To find: dielectric n These become negligible as h approaches zero. Therefore conductor More formally: Tangential Electric Field Boundary Condition

Gauss’ Law is applied to the cylindrical surface shown below: dielectric This reduces to: as h approaches zero n Therefore s conductor More formally: Boundary Condition for the Normal Component of D

Tangential E is zero At the surface: Normal D is equal to the surface charge density Summary

The Theorem of Uniqueness states that if we are given a configuration of charges and boundary conditions, there will exist only one potential and electric field solution. In the electric dipole, the surface along the plane of symmetry is an equipotential with V = 0. The same is true if a grounded conducting plane is located there. So the boundary conditions and charges are identical in the upper half spaces of both configurations (not in the lower half). In effect, the positive point charge images across the conducting plane, allowing the conductor to be replaced by the image. The field and potential distribution in the upper half space is now found much more easily! Method of Images

Each charge in a given configuration will have its own image Forms of Image Charges

In dielectric, charges are held in position (bound), and ideally there are no free charges that can move and form a current. Atoms and molecules may be polar (having separated positive and negative charges), or may be polarized by the application of an electric field. Consider such a polarized atom or molecule, which possesses a dipole moment, p, defined as the charge magnitude present, Q, times the positive and negative charge separation, d. Dipole moment is a vector that points from the negative to the positive charge. Q d p = Qd ax Electric Dipole and Dipole Moment

A dielectric can be modeled as an ensemble of bound charges in free space, associated with the atoms and molecules that make up the material. Some of these may have intrinsic dipole moments, others not. In some materials (such as liquids), dipole moments are in random directions. Model of a Dielectric

v  Polarization Field The number of dipoles is expressed as a density, n dipoles per unit volume. The Polarization Field of the medium is defined as: [dipole moment/vol] or [C/m2]

Introducing an electric field may increase the charge separation in each dipole, and possibly re-orient dipoles so that there is some aggregate alignment, as shown here. The effect is small, and is greatly exaggerated here! E The effect is to increase P. = np if all dipoles are identical Polarization Field (with Electric Field Applied)

Consider an electric field applied at an angle to a surface normal as shown. The resulting separation of bound charges (or re-orientation) leads to positive bound charge crossing upward through surface of area S, while negative bound charge crosses downward through the surface. Dipole centers (red dots) that lie within the range (1/2) d cos above or below the surface will transfer charge across the surface. E Migration of Bound Charge

The total bound charge that crosses the surface is given by: S volume P E Bound Charge Motion as a Polarization Flux

The accumulation of positive bound charge within a closed surface means that the polarization vector must be pointing inward. Therefore: + + + + S P qb - - - - Polarization Flux Through a Closed Surface

Now consider the charge within the closed surface consisting of bound charges, qb , and free charges, q. The total charge will be the sum of all bound and free charges. We write Gauss’ Law in terms of the total charge, QT as: E + + + + + + + + QT S q P qb where free charge QT = Qb + Q bound charge Bound and Free Charge

We now have: and QT = Qb + Q where combining these, we write: which we use in the familiar form of Gauss’ Law: we thus identify: Gauss Law for Free Charge

Bound Charge: Total Charge: Free Charge: Charge Densities Taking the previous results and using the divergence theorem, we find the point form expressions:

A stronger electric field results in a larger polarization in the medium. In a linear medium, the relation between P and E is linear, and is given by: where e is the electric susceptibility of the medium. We may now write: where the dielectric constant, or relative permittivity is defined as: Leading to the overall permittivity of the medium: where Electric Susceptibility and the Dielectric Constant

In an isotropic medium, the dielectric constant is invariant with direction of the applied electric field. This is not the case in an anisotropic medium (usually a crystal) in which the dielectric constant will vary as the electric field is rotated in certain directions. In this case, the electric flux density vector components must be evaluated separately through the dielectric tensor. The relation can be expressed in the form: Isotropic vs. Anisotropic Media

We use the fact that E is conservative: n So therefore: Region 1 Leading to: 1 Region 2 2 More formally: Boundary Condition for Tangential Electric Field

n Region 1 1 We apply Gauss’ Law to the cylindrical volume shown here, in which cylinder height is allowed to approach zero, and there is charge density son the surface: Region 2 2 s The electric flux enters and exits only through the bottom and top surfaces, respectively. More formally: From which: and if the charge density is zero: Boundary Condition for Normal Electric Flux Density

Engineering Electromagnetics W.H. Hayt Jr. and J. A. Buck