Prof. David R. Jackson ECE Dept.

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

Conductors Ohm’s law  [S/m] Good electric conductor:  >> 1 Perfect electric conductor:    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

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

Perfect Electric Conductors (cont.) Electric lines of flux must enter or leave a conductor perpendicular to it. Et= tangential electric field = 0 On surface of PEC: Et A B PEC We assume the hypothetical existence of a tangential electric field (there may also be a normal component), and choose a small path rin this direction along the surface.

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

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

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

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

Faraday Cage Effect (cont.) E0 + + + + + + + A + B + + + + + Proof of Faraday cage effect (qualitative) Assume an electric field exits inside the shell at some point A flux line must exist through this point. The flux line must start and end on the shell (otherwise it contradicts Gauss’s law) VAB 0 Contradiction! Hence, there is no electric field (and no flux lines) inside the cavity. PEC shell 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.)

Imperfect (Practical) Electric Conductors Assume: We are in static equilibrium (statics)* 2) No connections to the object are made from the outside. • E= 0 • v = 0 Then Inside metal: r v = 0 E= 0 The proof that v = 0 thenproceeds as before (based on Gauss’s law).

Faraday Cage: Imperfect Conductor In equilibrium, there is still no field inside the shell, and no charge density on the inner surface. Static electric field source - q - - - E=0 Conducting metal shell (finite conductivity) • 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.)

High Frequency: Imperfect Conductor There may be penetration of fields at high frequency: E0 E Radiating antenna source E0 Skin depth: Conducting metal shell (finite conductivity) To be a good shield, the thickness of the shield should be large compared to a skin depth. This situation is discussed in ECE 3317 (skin-depth effect).

Faraday Cage Effect (cont.) Faraday-cage effect He is safe from the Tesla coil!

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

Faraday Cage Effect (cont.) Faraday-cage effect Faraday shield at Art Nouveau power plant in Heimbach, Germany Entrance to a “Faraday room”

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

Faraday Cage Effect (cont.) A car is not always a perfect Faraday cage. Friday, August 16, 2013 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. http://www.khou.com

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

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

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

Shielding and Grounding (cont.) QA q QB Find QA, QB: Neutral shell + + + + + + + - - - + + q - - QB QA Neutral shell: - - - + + + + + + +

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

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

Shielding and Grounding (cont.) Next, “ground” the shell: r > b : E = 0 - E = 0 - Proof: - q - - - - 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 - A Flux line PEC wire B This is a contradiction since all conductors are at the same potential. Earth The earth is modeled as a “big fat conductor.”

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

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

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

Shielding and Grounding (cont.) Important Conclusions about Grounding • 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.

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

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

Prof. David R. Jackson ECE Dept.