1. Prof. David R. Jackson ECE Dept. Spring 2016 Notes 12 ECE 3318 Applied Electricity and Magnetism 1

2. Conductors  [S/m] 2 Ohm’s law Good electric conductor:  >> 1 Perfect electric conductor(PEC):    Material  [S/m] Silver 6.3  10 7 Copper 6.0  10 7 Copper (annealed) 5.8  10 7 Gold 4.1  10 7 Aluminum 3.5  10 7 Zinc 1.7  10 7 Brass 1.6  10 7 Nickel 1.4  10 7 Iron 1.0  10 7 Tin 9.2  10 6 Steel (carbon) 7.0  10 6 Steel (stainless) 1.5  10 6 Note: Many of the properties derived for PECs hold very accurately for good conductors. http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity

3.  [S/m] A B PEC: Also, Ohm’s law: 3 or Perfect Electric Conductors

4. 4 Electric lines of flux must enter or leave a conductor perpendicular to it. On surface of PEC: E t = tangential electric field = 0 A B PEC We choose a small path  r in a tangential direction along the surface. Perfect Electric Conductors (cont.) Hence, E cannot have any component that is parallel to the surface.

5. Perfect Electric Conductors (cont.) Inside a PEC  v = 0 Proof: Assume a point inside where  v > 0 S  v > 0 inside small volume (since  v is a continuous function). Contradiction ! + + + + + + + 5 Hence

6. Perfect Electric Conductors (cont.) 6 Only  s on the surface is allowed for a PEC.  v = 0 Even if these conductors were solid, there would be no volume charge density inside them. http://en.wikipedia.org/wiki/Electrostatics

7. Faraday Cage Effect Inside of a hollow PEC shell in statics:  There is no electric field.  No  s on the inner surface. 7 Static electric field source This is a shielded region (no electric field). E = 0E = 0 - - - - q + - + + + + ss PEC Note: The cage does not have to be grounded.

8. Proof of Faraday cage effect PEC shell  The electric field is zero inside the conductor.  The potential on the boundary of the cavity is constant.  The uniqueness theorem then says that the potential must be a constant throughout the cavity region.  The electric field is then zero inside the cavity as it comes from the gradient of the electric field. + + + + + + + + + + + + Hollow cavity + 8 Faraday Cage Effect (cont.) Note: The uniqueness theorem is discussed later.

9. Faraday Cage: Imperfect Conductor In static equilibrium, the Faraday cage still works, even with a practical conductor (finite conductivity): Conducting metal shell (finite conductivity) 9 Static electric field source E = 0E = 0 - - - - q The rest of the proof works the same way.

10. Faraday Cage: Note on Magnetic Field A Faraday cage does not block a static magnetic field! 10 B 0B 0 S N A high permeability material will act as a good shield for a static magnetic field (e.g., Mu-metal,  r = 100,000 ).

11. High Frequency: Imperfect Conductor At high frequencies, both electric and magnetic fields are blocked by the skin effect. Thickness >>  11 Radiating antenna source E  0E  0 This situation is discussed in ECE 3317 (skin-depth effect). Skin depth: To be a good shield, the thickness of the shield should be large compared to a skin depth. B  0B  0

12. Faraday-cage effect They are safe from the Tesla coil! 12 Faraday Cage Effect (cont.)

13. Faraday-cage effect She is safe from the Van de Graaff generator! 13 (Boston Science Museum) Faraday Cage Effect (cont.)

14. Faraday Cage Shielded Room 14 Faraday Cage Effect (cont.) The modular Faraday cage is designed to meet or even exceed the vast majority of shielding requirements. The system is constructed of shielded modular panels, available in either standard-sized or custom-designed panels to meet exact specifications in government, industry, research and development, university or hospital use. http://www.directindustry.com/prod/holland-shielding-systems-bv/faraday-cages-modular-35048-1031291.html

15. 1) Occupants are pretty safe (but let the charge dissipate before you get out!) 15 Earth _ _ _ + + + Earth _ _ _ + + + 2) Occupants are not very safe Faraday Cage Effect (cont.)

16. Shielding and Grounding q Spherical PEC shell Neutral (no charge) a b Drill hole and insert point charge, then solder hole. Neutral shell (no charge) Find E : q a b 16

17. (a) r < a (c) r > b (b) a < r < b Neutral shell (PEC) Shielding and Grounding (cont.) a b q 17

18. q (outside metal) The neutral metal shell does not block the static electric field coming from the inside ! Shielding and Grounding (cont.) 18 (The shell would be a good shield for high-frequency fields.) Neutral shell

19. Find Q A, Q B : q - - - - - - - - + + + + + + + + + + + + + + + + QAQA QBQB q QAQA QBQB Neutral shell: Shielding and Grounding (cont.) 19 Neutral shell

20. (neutral shell) so Hence A Gaussian surface is chosen inside the metal shell. Shielding and Grounding (cont.) 20 We then also have q - - - - - - - - + + + + + + + + + + + + + + + + QAQA QBQB S or

21. Flux picture showing the charge on the surfaces Shielding and Grounding (cont.) q q -q - - - - - - - - + + + + + + + + + + + + + + + + Note: Flux lines go from positive charges to negative charges. (Also, they go from higher potential to lower potential.) 21

22. Next, “ground” the shell: r > b : E = 0 Proof: The earth is modeled as a “big fat conductor.” We can use the uniqueness theorem, considering the outside of the shell, the earth, and the wire to form the boundary of our region. q - - - - - - - - -q E = 0 PEC wire Earth Shielding and Grounding (cont.) 22

23. Charge on outer surface: Hence r > b: E = 0 Shielding and Grounding (cont.) 23 q - - - - - - - - -q PEC wire Earth S

24. Charge on outer surface (cont.): The charge q on the outer surface has flowed down to ground. - q q q Neutral shell before grounding Shielding and Grounding (cont.) 24 -q + + + + PEC wire Earth q After grounding - - - - - - - - q E = 0

25. Note: In static equilibrium, the object, wire, and earth must each be at a constant potential, even if they are not perfect conductors. Imperfect Conductor: Shielding and Grounding In steady-state: Ohms' law for object, wire, or earth:  Hence, inside the object, wire, and earth we still have E = 0.  Therefore, all of the previous conclusions still hold. 25 -q Practical wire Earth q - - - - - - - -

26. 26 Summary of Shielding Effects in Statics There is no field on the inside (does not require grounding). An ungrounded shield does not block the static electric field of an inside source. Effect # 1: Faraday cage effect Effect # 2: Static field penetration E  0E  0 E = 0

27. 27 A grounded shield removes the electric field in the exterior region and removes charge from the outer surface. Conducting wire Earth - - - - - - - - q Effect # 3: Grounding E = 0 (This assumes that there are no charges in the outside region.) Summary of Shielding Effects in Statics (cont.)