Lecture 5 Electric Flux Density and Dielectric Constant Boundary Conditions

1. Lecture 5Electric Flux Density and Dielectric ConstantBoundary Conditions Electromagnetics Prof. Viviana Vladutescu

2. Electric Flux Density Gauss’s Law: The total outward flux of the dielectric displacement (or simply the outward flux) over any closed surface is equal to the total free charge enclosed in the surface

3. Where- ε is the absolute permittivity (F/m) -εr is the relative permittivity or the dielectric constant of the medium -ε0 is the permittivity of free space -χe is the electric susceptibility (dimensionless)

4. Material Dielectric Constants Vacuum 1 Glass 5-10 Mica 3-6 Mylar 3.1 Neoprene 6.70 Plexiglas 3.40 Polyethylene 2.25 Polyvinyl chloride 3.18 Teflon 2.1 Germanium 16 Strontiun titanate 310 Titanium dioxide (rutile) 173 perp 86 para Water 80.4 Glycerin 42.5 Liquid ammonia(-78°C 25 Benzene 2.284 Air(1 atm) 1.00059 Air(100 atm) 1.0548

5. Homogeneous -εr independent of position Anisotropic εr is different for different of the electric field

6. Biaxial, Uniaxial and Isotropic Medium -biaxial -uniaxial -isotropic

7. KDP ADP crystals– Electric field along optic axis Uniaxial crystal becomes biaxial with applied field!

8. GaAs CdTe crystals – Electric field along optic axis Isotropic crystal become biaxial with applied field!

9. Dielectric Strength The maximum electric field intensity that a dielectric material can stand without breakdown

10. Boundary Conditions

11. Medium 1&2 are dielectrics CONDITION I (tangential components)

14. Condition I (tangential components)

15. CONDITION II (normal components)

17. Condition II (normal components) The normal component of D field is discontinuous across an interface where a surface charge exists, the amount of discontinuity being equal to the surface charge density

18. Example: Two dielectric media with permittivityε1 and respectively ε2 , are separated by a charge free boundary. The electric field intensity in medium 1 at point P1 has a magnitude E1 and makes an angle α1. Determine the magnitude and direction of the electric field intensity at point P2 in medium 2. α1 P1 P2 α2

20. The magnitude of E2 :

21. Boundary conditions at a Dielectric/Conductor Interface -inside a good conductor E=0 ET=0 D=0 Dn=ρs

23. Practice Problems

24. Divergence Theorem and Gauss’s Law Suppose D = 6rcosfaf C/m2. (a) Determine the charge density at the point (3m, 90, -2m). Find the total flux through the surface of a quartered-cylinder defined by 0 ≤ r ≤ 4m, 0 ≤ f ≤ 90, and -4m ≤z ≤ 0 by evaluating (b) the left side of the divergence theorem and (c) the right side of the divergence theorem. (a) (b)

25. note that the top, bottom and outside integrals yield zero since there is no component of D in the these dS directions. So, (c)

26. Electric Potential The potential field in a material with εr= 10.2 is V = 12 xy2 (V). Find E, P and D.

27. Boundary Conditions For z ≤ 0, er1 = 9.0 and for z > 0, er2= 4.0. If E1 makes a 30 angle with a normal to the surface, what angle does E2 make with a normal to the surface?

28. Therefore also and after routine math we find Using this formula we obtain for this problem q2 = 14°.