Conductors and Dielectrics Conductors – Current, current density, drift velocity, continuity – Energy bands in materials – Mobility, micro/macro Ohm’s Law – Boundary conditions on conductors – Methods of Images Dielectrics – Polarization, displacement, electric field – Permittivity, susceptibility, relative permittivity – Dielectrics research – Boundary conditions on dielectrics

Conductors and Dielectrics Polarization – Static alignment of charge in material – Charge aligns when voltage applied, moves no further – Charge proportional to voltage Conduction – Continuous motion of charge through material – Enters one side, exits another – Current proportional to voltage Real-world materials – Plastics, ceramics, glasses -> dielectrics (maybe some conductivity) – Metals -> conductors, semiconductors, superconductors – Cement, Biosystems -> Both (water high dielectric, salt conductivity)

Current and current density Basic definition of current C/s = Amps Basic current density (J perp. surface) Vector current density n

Current density and charge velocity

Charge and current continuity Current leaving any closed surface is time rate of change of charge within that surface Using divergence theorem on left Taking time derivative inside integral Equating integrands Qi(t)Qi(t)

Example – charge and current continuity <<Some central repulsive force! ^^ Why is current increasing ?

Energy Band Structure in Three Material Types Discrete quantum states broaden into energy bands in condensed materials with overlapping potentials Valence band – outermost filled band Conduction band – higher energy unfilled band Band structure determines type of material a)Insulators show large energy gaps, requiring large amounts of energy to lift electrons into the conduction band. When this occurs, the dielectric breaks down. b)Conductors exhibit no energy gap between valence and conduction bands so electrons move freely c)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.

Ohm’s Law (microscopic form) Free electrons are accelerated by an electric field. The applied force on an electron of charge Q = -e is But in reality the electrons are constantly bumping into things (like a terminal velocity) so they attain an equilibrium or drift velocity: where  e is the electron mobility, expressed in units of m 2 /V-s. The drift velocity is used in the current density through: With the conductivity given as: So Ohm’s Law in point form (material property) S/m (electrons/holes) S/m (electrons)

Ohm’s Law (macroscopic form)

Ohm’s Law example 1

Ohm’s Law example 2

Boundary conditions for conductors

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

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

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

Example - Boundary Conditions for Conductors

Example – Streamlines of Electric Field

Boundary condition example (from my phone)* *

Method of Images 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!

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

Example of the Image Method Want to find surface charge density on conducting plane at the point (2,5,0). A 30-nC line of charge lies parallel to the y axis at x=0, z = 3. First step is to replace conducting plane with image line of charge -30 nC at z = -3.

Example of the Image Method (continued) Add both fields to get: (x component cancels) Vectors from each line charge to observation point: Electric Fields from each line charge -

Example of the Image Method (continued) Electric Field at P is thus: Displacement is thus n D Charge density is

Image Method using Potentials

Image Method using Potentials (cont)

Dielectrics Material has random oriented dipoles Applied field aligns dipoles (negative at (+) terminal, positive at (-) terminal Effect is to cancel applied field, lower voltage OR, increase charge to maintain voltage Either increases capacitance C= Q/V

Review Dipole Moment

Polarization as sum of dipole moments (per volume) E 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! The effect is to increase P. n = charge/volume p = polarization of individual dipole P = polarization/volume

Polarization near electrodes neutral positive negative E

Combining total, free, and bound charge neutral positive negative E

D, P, and E in Dielectric

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

Electric Susceptibility and the Dielectric Constant 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

Isotropic vs. Anisotropic Media 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:

Permittivity of Materials Typical permittivity for various solids and liquids. –Teflon – 2 –Plastics –Ceramics 8-10 –Titanates>100 –Acetone 2 1 –Water 78 Actual dielectric “constant” varies with: –Temperature –Direction –Field Strength –Frequency –Real & Imaginary components

Variation with frequency Charge polarization due to: –Ionic (low frequency) –Orientation (medium, microwave) –Atomic (IR) –Electronic (Visible, UV) Dielectric relaxation –As frequency is raised, molecule can no longer “track”. –Real permittivity decreases and imaginary permittivity peaks In medium and microwave range –Rotation, reorientation, etc >> Modeling: –Permittivity & impedance diagrams. –Statistical relaxation functions (Debye, Cole Davidson).

IEEE – March Application to Polymer Composites Dielectric Permittivity in Epoxy Resin 10Hz -10 MHz Polar-group rotation in epoxy resin. Low-frequency range 10 Hz – 10 MHz. Permittivity-loss transition at 1 MHz, at –4°C. Transition frequency increases with temperature.

IEEE – March Dielectric Permittivity in Epoxy Resin 1 MHz -1 GHz Aerospace resin Hexcel High frequency range 1 MHz – 1 GHz. Temperature constant 125°C, transition decreases with cure. TDR measurement method.

IEEE – March Permittivity in Epoxy Resin during Complete Cure Cycle

IEEE – March Application to cement hydration Cement Conductivity - Variation with Cure Imaginary counterpart of real permittivity (  ’’). Multiply by  to remove power law (  o  ’’). Decrease in ion conductivity, growth of intermediate feature with cure Frequency of intermediate feature does not match permittivity

IEEE – March Cement Cure -Dielectric Relaxation Model Requirements: Provide free-relaxation, two intermediate-frequency relaxations Provide conductivity and electrode polarization Debye for free & medium. Cole-Davidson for low. (literature, biosystems) Combined 9 variables fit over entire range, real & imaginary, 2-stage fit,  f = 8.2 ps

IEEE – March Cement Cure - Model Fitting Fits permittivity – both low and free relaxation. Fits conductivity – both medium and free relaxation. Fits permittivity polarization. Fits conductivity baseline.

IEEE – March 2005 Other applications Other Applications –Bio –Liquid Crystal –Composite polymers –Titanates –Wireless characterization –MRI dyes –Ground water monitoring –Oil Drilling fluid characterization (GPR)

Boundary Condition for Tangential Electric Field E Since E is conservative, we setup line integral straddling both dielectrics: Left and right sides cancel, so Leading to Continuity for tangential E E same, D higher in high permittivity material And Discontinuity for tangential D

Boundary Condition for Normal Displacement D n Apply Gauss’ Law to the cylindrical volume straddling both dielectrics Flux enters and exits only through top and bottom surfaces, zero on sides ss D same. E lower in high permittivity material Leading to Continuity for normal D (for ρ S = 0) And Discontinuity for normal E

Bending of D at boundary high low

Example

Example (continued) Polarization up, E field down, D maintains continuity

Example

Quiz 2 – Problem 4.21