Presentation on theme: "Applied Electricity and Magnetism"— Presentation transcript:
1Applied Electricity and Magnetism ECE 2317Applied Electricity and MagnetismSpring 2014Prof. David R. JacksonECE Dept.Notes 12
2Conductors Ohm’s law [S/m] Good electric conductor: >> 1 Material [S/m]Silver6.3107Copper6.0107Copper (annealed)5.8107Gold4.1107Aluminum3.5107Zinc1.7107Brass1.6107Nickel1.4107Iron1.0107Tin9.2106Steel (carbon)7.0106Steel (stainless)1.5106Good electric conductor: >> 1Perfect electric conductor: Perfect Electric Conductor: PECNote: Many of the properties derived for PECs hold very accurately for good conductors.
3Perfect Electric Conductors Ohm’s law:or [S/m]PEC:ABAlso,since
4Perfect Electric Conductors (cont.) Electric lines of flux must enter or leave a conductor perpendicular to it.Et = tangential electric field = 0On surface of PEC:ABPECEtWe assume the hypothetical existence of a tangential electric field (there may also be a normal component), and choose a small path r in this direction along the surface.
5Perfect Electric Conductors (cont.) Inside a PEC v = 0Proof: Assume a point inside where v > 0v > 0 inside small volume(since v is a continuous function).S+++++++HenceContradiction !
6Perfect 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 = 0v = 0v = 0
7Faraday Cage Effect s v = 0 Inside of a hollow PEC shell in statics: There is no electric field.Only s on the outer surface is allowed.PEC shell+E = 0sv = 0(from last slide)Hollow region (no charge inside)No surface charge density on the inner surface
8Faraday 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 sourceE = 0-q+This is a shielded region (no electric field).
9Faraday Cage Effect (cont.) Proof of Faraday cage effect (qualitative)Assume an electric field exits inside the shell at some point+E 0ABA 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 shellThere 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.)
10Imperfect (Practical) Electric Conductors Assume:We are in static equilibrium (statics)*2) No connections to the object are made from the outside.E = 0v = 0ThenInside metal:v = 0E = 0rThe proof that v = 0 then proceeds as before(based on Gauss’s law).
11Faraday Cage: Imperfect Conductor In equilibrium, there is still no field inside the shell, and no charge density on the inner surface.Static electric field sourceE = 0-qConducting 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.)
12High Frequency: Imperfect Conductor There may be penetration of fields at high frequency:E 0E 0ERadiating antenna sourceSkin 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).
13Faraday Cage Effect (cont.) He is safe from the Tesla coil!
14Faraday Cage Effect (cont.) She is safe from the Van de Graaff generator!(Boston Science Museum)
15Faraday Cage Effect (cont.) Faraday shield at Art Nouveau power plant in Heimbach, GermanyEntrance to a “Faraday room”
16Faraday Cage Effect (cont.) _ _ _+ + +_ _ _+ + +1) Occupants are pretty safe(but let the charge dissipate before you get out!)2) Occupants are not very safeEarthEarth
17Faraday Cage Effect (cont.) A car is not always a perfect Faraday cage.Friday, August 16, 2013HOUSTON – 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.
18Shielding and Grounding bSpherical PEC shellNeutral (no charge)Drill hole andinsert point charge, then solder hole.qabqFind E:Neutral shell (no charge)
19Shielding and Grounding (cont.) (a) r < aabq(b) a < r < b(PEC)Neutral shell(c) r > b
20Shielding and Grounding (cont.) qNeutral shell(outside metal)The neutral metal shell does not block the static electric field!(The shell would be a good shield for high-frequency fields.)
21Shielding and Grounding (cont.) qQAQBFind QA, QB:Neutral shellq-+QAQBNeutral shell:
22Shielding and Grounding (cont.) q-+QAQBSsoA Gaussian surface is chosen inside the metal shell.HenceWe then also have(neutral shell)
23Shielding and Grounding (cont.) q-q-+Flux picture showing charge on surfacesNote: Flux lines go from positive charges to negative charges.(Also, they go from higher potential to lower potential.)
24Shielding and Grounding (cont.) Next, “ground” the shell:r > b : E = 0q--qE = 0PEC wireEarthABFlux lineProof: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.This is a contradiction since all conductors are at the same potential.The earth is modeled as a “big fat conductor.”
25Shielding and Grounding (cont.) Charge on outer surface:r > b: E = 0q--qPEC wireEarthHence
26Shielding and Grounding (cont.) -q+PEC wireEarthqAfter grounding-E = 0Charge on outer surface (cont.):- qqNeutral shell before groundingThe charge q on the outer surface has flowed down to ground.
27Imperfect Conductor: Shielding and Grounding -qPractical wireEarthq-Note: In steady state (statics), the object, wire, and earth must each be at a constant potential, even if they are not perfect conductors.Ohms' law for object, wire, or earth:In steady-state:Hence, inside the object, wire, and earth we still have E = 0.Therefore, all of the previous conclusions still hold.
28Shielding and Grounding (cont.) Important Conclusions about GroundingIf 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.
29Summary of Effects Effect # 1: Faraday cage effect There is no field on the inside(does not require grounding).E = 0E 0Effect # 2: Static field penetrationAn ungrounded shield does not block the electric field of an inside source.
30Summary of Effects (cont.) PEC wireEarth-qE = 0Effect # 3: GroundingA grounded shield removes the electric field in the exterior regionand removes charge from the outer surface.(This assumes that there areno charges in the outside region.)