Prof. David R. Jackson ECE Dept.

ECE 2317 Applied Electricity and Magnetism Spring 2014 Prof. David R. Jackson ECE Dept. Notes 19

Dielectrics H+ H+ Water + + - O- Single H2O molecule:

Dielectrics (cont.) H+ H+ Water + + - O- +q + d - -q “Vector dipole moment” of molecule p: “Dipole” model:

Dielectrics (cont.) - + - - - + + + + - The dipoles representing the water molecules are normally pointing in random directions. + -

Dielectrics (cont.) + + + + + - - - - - - V + The dipoles partially align under an applied electric field. + - Ex + - + - + - + - x

Dielectrics (cont.) - + - + Define: P = totaldipole moment/volume N molecules (dipoles) inside V We can also write this as p =average vector dipole moment Nv = number of molecules per unit volume Define the electric flux density vector D:

Dielectrics (cont.) Linear material: Note: e> 0 for most materials The term e is called the “electric susceptibility.” Define: Then we have or where

Typical Linear Materials Teflon Water Styrofoam Quartz (a very polar molecule, fairly free to rotate) Note: A negative value of e would mean that the dipoles align against the field. Note: r> 1for most materials.

Exotic Materials Artificial “metamaterials” that have been designed that have exotic permittivity and/or permeability performance. http://en.wikipedia.org/wiki/Mhttp://en.wikipedia.org/wiki/Metamaterialetamaterial Negative index metamaterial array configuration, which was constructed of copper split-ring resonators and wires mounted on interlocking sheets of fiberglass circuit board. The total array consists of 3 by 20×20 unit cells with overall dimensions of 10×100×100 mm. (over a certain bandwidth of operation)

Exotic Materials (cont.) The Duke cloaking device masks an object from one wavelength at microwaves. Image from Dr. David R. Smith. Cloaking of objects is one area of research in metamaterials.

Dielectrics: Bound Charge - - - - - - - - - + + + + + + + + + Assume (hypothetically) that Pxincreases as xincreases inside the material: Note: For simplicity, the dipoles are shown perfectly aligned, with the number of aligned dipoles increasing with x. A net volume charge density is created inside the material, called the “bound charge density” or “polarization charge density.” vb=“bound charge density”: it is bound to the molecules. v=“free charge density”: this is charge you place inside the material. You can freely place it wherever you want.

Dielectrics: Bound Charge (cont.) - - - - - - - - - + + + + + + + + + Bound charge densities Bound surface charge density Note: The total charge is zero since the object is assumed to be neutral. Total charge density inside the material: This total charge density may be viewed as being in free space (since there is no material left after the molecules are removed).

Dielectrics: Bound Charge (cont.) Formulas for the two types of bound charge + + + + • - - • - - - • - - - + - The unit normal points outward from the dielectric. - - The derivation of these formulas is included in the Appendix.

Dielectrics: Gauss’s Law Inside a material: This equation is valid, but we prefer to have only the free-charge density in the equation, since this is what is known. The goal is to calculate (and hopefully eliminate) the bound-charge density term on the right-hand side.

Dielectrics: Gauss’s Law (cont.) The free-space form of Gauss’s law is or Hence This is why the definition of D is so convenient!

Dielectrics: Gauss’s Law (Summary) Important conclusion: Gauss’ law works the same way inside a dielectric as it does in vacuum, with only the free charge density (i.e., the charge that is actually placed inside the material) being used on the right-hand side.

Example r r q a b S Point Charge:Find D, E Dielectric spherical shell with r (The point charge q is the only free charge in the problem.) Gauss’ law:

Example (cont.) r r q a b S Hence We then have

Homogeneous Dielectrics Practical case: Assume a homogeneous dielectric with no free charge density inside. Note: All of the interior bound charges cancel from one row of molecules to the next.

Summary of Dielectric Principles Important points: • A dielectric can always be modeled with a bound volume charge density and a bound surface charge density. • A homogeneous dielectric with no free charge inside can always be modeled with only a bound surface charge density. • All of the bound charge is automatically accounted for when using the D vector and the total permittivity . • When using the D vector, the problem is solved using only the (known) free-charge density on the right-hand side of Gauss’ law.

Example A point charge surrounded by a homogeneous dielectric shell. Dielectric shell Find the bound surface charges densities and verify that we get the correct answer for the electric field by using them. b r a From Gauss’s law: q

Example (cont.) + + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - Dielectric shell r q The alignment of the dipoles causes two layers of bound surface charge density.

Example (cont.) Free-space model + + + - - - sb + + - - q - - - Note: We use the electric field inside the material, at the boundaries. + + + Hence (free-space Gauss’s law)

Example (cont.) Free-space model + + + - - - + + - - q - - - From Gauss’s law: + + +

Example (cont.) Free-space model + + + - - - + + - - q - - - + + + From Gauss’s law:

Observations • We get the correct answer by accounting for the bound charges, and putting them in free space. • It is much easier to not worry about bound charges, and simply use the concept of r.

Appendix In this appendix we derive the formulas for the bound change densities: where

Appendix (cont.) - - - - - + + + + + Model: Dipoles are aligned in the x direction end-to-end. The number of dipoles that are aligned (per unit volume) changes with x. Qb V Observation point Nv = # of aligned dipoles per unit volume x x-d/2 x+d/2 d

Appendix (cont.) - - - - - + + + + + Qb V V V Hence x x-d/2 x+d/2 Nv = # of aligned dipoles per unit volume

Appendix (cont.) - - - - - + + + + + V Hence x x-d/2 x+d/2 Nv = # of aligned dipoles per unit volume or

Appendix (cont.) For dipole aligned in the x direction, In general, or

Appendix (cont.) After applying the divergence theorem, we have the integral form Applying this to a surface at a dielectric boundary, we have Denoting we have

Prof. David R. Jackson ECE Dept.