1. Prof. David R. Jackson ECE Dept. Fall 2014 Notes 12 ECE 2317 Applied Electricity and Magnetism 1

2. Conductors  [S/m] 2 Ohm’s law Good electric conductor:  >> 1 Perfect electric conductor:    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 Perfect Electric Conductor: PEC 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 EtEt We choose a small path  r in a tangential direction along the surface, parallel to a hypothetical tangential electric field. Perfect Electric Conductors (cont.)

5. 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.

7. Faraday Cage Effect Inside of a hollow PEC shell in statics:  There is no electric field.  Only  s on the outer surface is allowed. 7  v = 0 (from last slide) No surface charge density on the inner surface PEC shell + + + + + + + + + + + + E = 0 + ss Hollow region (no charge inside)

8. Even with a source outside, there is still no field inside the shell, and no charge density on the inner surface. 8 Static electric field source Faraday Cage Effect (cont.) This is a shielded region (no electric field). E = 0E = 0 - - - - q + - + + + +

9. Proof of Faraday cage effect (qualitative) Assume an electric field exits inside the shell at some point PEC shell A flux line must exist through this point. The flux line must start and end on the shell (otherwise it contradicts Gauss’s law) V AB  0 Contradiction! + + + + + + + + + + + + Assume that E  0 A B + There is also no surface charge density on the inner surface. (Otherwise, there would be a flux line coming from the inner surface, and we would have the same type of situation.) Hence, there is no electric field (and no flux lines) inside the cavity. 9 Faraday Cage Effect (cont.)

10. 10 Imperfect (Practical) Electric Conductors Assume: 1)We are in static equilibrium (statics)* 2) No connections to the object are made from the outside. Inside metal: The proof that  v = 0 then proceeds as before (based on Gauss’s law).  v = 0 E = 0 r  E = 0   v = 0 Then Assumption: No DC current can flow in steady state, since there are no connections to the object.

11. Faraday Cage: Imperfect Conductor In equilibrium: Conducting metal shell (finite conductivity) 11 Static electric field source  There is no electric field inside the hollow shell.  There is no surface charge density on the inner surface of the shell. (The proof is the same as before, since the electric field is zero inside the metal and hence the shell is all at the same potential.) E = 0E = 0 - - - - q

12. High Frequency: Imperfect Conductor There may be penetration of fields at high frequency: Conducting metal shell (finite conductivity) 12 Radiating antenna source E  0E  0 E  0E  0 E 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.

13. Faraday-cage effect They are safe from the Tesla coil! 13 Faraday Cage Effect (cont.)

14. Faraday-cage effect She is safe from the Van de Graaff generator! 14 (Boston Science Museum) Faraday Cage Effect (cont.)

15. Faraday Cage Shielded Room 15 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

16. 1) Occupants are pretty safe (but let the charge dissipate before you get out!) 16 Earth _ _ _ + + + Earth _ _ _ + + + 2) Occupants are not very safe Faraday Cage Effect (cont.)

17. 17 Faraday Cage Effect (cont.) HOUSTON – A woman's car was struck by lightning while she was driving on Richey just south of FM 1960 on Friday. The incident happened just north of Intercontinental Airport. The driver of the vehicle was transported to the hospital, but her condition was unknown. The back window of the vehicle was blown out. Friday, August 16, 2013 http://www.khou.com A car is not always a perfect Faraday cage.

18. 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 18

19. (a) r < a (c) r > b (b) a < r < b Neutral shell (PEC) Shielding and Grounding (cont.) a b q 19

20. q (outside metal) The neutral metal shell does not block the static electric field! Shielding and Grounding (cont.) 20 (The shell would be a good shield for high-frequency fields.) Neutral shell

21. Find Q A, Q B : q - - - - - - - - + + + + + + + + + + + + + + + + QAQA QBQB q QAQA QBQB Neutral shell: Shielding and Grounding (cont.) 21 Neutral shell

22. (neutral shell) q - - - - - - - - + + + + + + + + + + + + + + + + QAQA QBQB S so Hence A Gaussian surface is chosen inside the metal shell. Shielding and Grounding (cont.) 22 We then also have

23. 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.) 23

24. Next, “ground” the shell: r > b : E = 0 Proof: The earth is modeled as a “big fat conductor.” If the electric field were not zero at a point, there would be a flux line through the point that would end on the conductors, which would correspond to a voltage drop between them. q - - - - - - - - -q E = 0 PEC wire Earth A B Flux line Shielding and Grounding (cont.) This is a contradiction since all conductors are at the same potential. 24

25. Charge on outer surface: Hence r > b: E = 0 Shielding and Grounding (cont.) q - - - - - - - - -q PEC wire Earth 25

26. 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.) 26 -q + + + + PEC wire Earth q After grounding - - - - - - - - q E = 0

27. Note: In steady state (statics), 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. 27 -q Practical wire Earth q - - - - - - - -

28. Important Conclusions about Grounding Shielding and Grounding (cont.) 28  If a system is not grounded, there may be a build up of charge on the object.  If a system is not grounded, metal cases to not block static fields from existing. Grounding removes all charge from the outer surface (metal case) and removes any static electric field surrounding the metal case. In steady-state (statics), these conclusions are true for both perfect conductors and practical conductors.

29. 29 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

30. 30 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.)