1. Prof. David R. Jackson ECE Dept. Spring 2016 Notes 20 ECE 3318 Applied Electricity and Magnetism 1

2. Dielectrics Water Single H 2 O molecule: + - + H+H+ O-O- H+H+ 2

3. Dielectrics (cont.) Water “Dipole” model: “Vector dipole moment” of molecule p : d +q -q + - + - + H+H+ O-O- H+H+ 3

4. Dielectrics (cont.) 4 The dipoles representing the water molecules are normally pointing in random directions. + + + - + - + - - - - + Note: The molecules are not floating in water – they make up the water.

5. Dielectrics (cont.) 5 The dipoles partially align under an applied electric field. + - V x + + + + - - - - ExEx + - + - + - + - + - + -

6. Define: P = total dipole moment/volume Define the electric flux density vector D: + - + - Dielectrics (cont.) We can also write this as p ave = average vector dipole moment N molecules (dipoles) inside  V 6 N v = number of molecules per unit volume

7. Linear material: Define: Then we have Note:  e > 0 for most materials Dielectrics (cont.) The term  e is called the “electric susceptibility.” 7 or where

8. Typical Linear Materials Teflon Water Styrofoam Quartz (a very polar molecule, fairly free to rotate) 8 Note:  r > 1 for most materials. Note: A negative value of  e would mean that the dipoles align against the field.

9. Exotic Materials Artificial “metamaterials” that have been designed that have exotic permittivity and/or permeability performance. 9 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. http://en.wikipedia.org/wiki/Mhttp://en.wikipedia.org/wiki/Metamaterialetamaterial (over a certain bandwidth of operation)

10. Exotic Materials (cont.) 10 Cloaking of objects is one area of research in metamaterials. The Duke cloaking device masks an object from one wavelength at microwaves. Image from Dr. David R. Smith.

11. Dielectrics: Bound Charge Assume (hypothetically) that P x increases as x increases inside the material: 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. Note: For simplicity, the dipoles are shown perfectly aligned, with the number of aligned dipoles increasing with x. 11 +- +- +- +- +- +- +- +- +-

12. Dielectrics: Bound Charge (cont.) 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). 12 Bound surface charge density Bound charge densities Note: The total charge is zero since the object is assumed to be neutral. +- +- +- +- +- +- +- +- +-

13. Formulas for the two types of bound charge 13 Dielectrics: Bound Charge (cont.) The derivation of these formulas is included in the Appendix. - - - - - + + + - - - + + The unit normal points outward from the dielectric. E = electric field inside material

14. Dielectrics: Gauss’s Law 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. 14 Inside a material:

15. The free-space form of Gauss’s law is Hence Dielectrics: Gauss’s Law (cont.) or This is why the definition of D is so convenient! 15

16. 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. 16

17. Example Point charge inside dielectric shell Dielectric spherical shell with  r S r q r a b Gauss’ law: (The point charge q is the only free charge in the problem.) 17 Find D, E

18. Example (cont.) 18 Hence We then have S r q r a b

19. Example (cont.) 19 Flux Plot Note that there are less flux lines inside the dielectric region (assuming that the flux lines represent the electric field).

20. Homogeneous Dielectrics 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. 20 Practical case: Hence

21. Homogeneous Dielectrics (cont.) 21 +- +- +- +- +- +- +- +- +- Blown-up view of molecules Practical case (homogeneous)

22. Summary of Dielectric Principles  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. Important points: 22

23. Example A point charge surrounded by a homogeneous dielectric shell. 23 Dielectric shell q rr a b Find the bound surface charges densities and verify that we get the correct answer for the electric field by using them in free space. From Gauss’s law:

24. Example (cont.) The alignment of the dipoles causes two layers of bound surface charge density. 24 Dielectric shell q rr + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + -

25. Free-space model 25 q + + + + ++ + + - - - - - - - -  sb Hence Example (cont.) Note: We use the electric field inside the material, at the boundaries.

26. Free-space model 26 From Gauss’s law: Example (cont.) q + + + + ++ + + - - - - - - - -

27. Free-space model 27 From Gauss’s law: Example (cont.) q + + + + ++ + + - - - - - - - -

28. 28Observations  We get the correct answer by accounting for the bound charges, and putting them in free space.  However, it is much easier to not worry about bound charges, and simply use the concept of  r.

29. Appendix In this appendix we derive the formulas for the bound change densities: 29 where

30. Model: Dipoles are aligned in the x direction end-to-end. The number of dipoles that are aligned (per unit volume) changes with x. N v = # of aligned dipoles per unit volume 30 VV x+d/2x-d/2 x +- +- +- +- +- d QbQb Observation point Appendix (cont.)

31. Hence N v = # of aligned dipoles per unit volume VV x+d/2x-d/2 x +- +- +- +- +- VV VV QbQb 31 Appendix (cont.)

32. Hence N v = # of aligned dipoles per unit volume VV x+d/2 x-d/2 x +- +- +- +- +- or 32 Appendix (cont.)

33. In general, or 33 For dipole aligned in the x direction, Appendix (cont.)

34. 34 After applying the divergence theorem, we have the integral form Applying this to a surface at a dielectric boundary, we have Denoting we have Appendix (cont.)