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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.

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Prof david r jackson ece dept

ECE 2317

Applied Electricity and Magnetism

Spring 2014

Prof. David R. Jackson

ECE Dept.

Notes 12


Prof david r jackson ece dept

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


Prof david r jackson ece dept

Perfect Electric Conductors

Ohm’s law:

or

 [S/m]

PEC:

Also,

B

A

since


Prof david r jackson ece dept

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.


Prof david r jackson ece dept

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


Prof david r jackson ece dept

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


Prof david r jackson ece dept

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


Prof david r jackson ece dept

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).


Prof david r jackson ece dept

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.)


Prof david r jackson ece dept

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).


Prof david r jackson ece dept

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.)


Prof david r jackson ece dept

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).


Prof david r jackson ece dept

Faraday Cage Effect (cont.)

Faraday-cage effect

He is safe from the Tesla coil!


Prof david r jackson ece dept

Faraday Cage Effect (cont.)

Faraday-cage effect

She is safe from the Van de Graaff generator!

(Boston Science Museum)


Prof david r jackson ece dept

Faraday Cage Effect (cont.)

Faraday-cage effect

Faraday shield at Art Nouveau power plant in Heimbach, Germany

Entrance to a “Faraday room”


Prof david r jackson ece dept

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


Prof david r jackson ece dept

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


Prof david r jackson ece dept

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)


Prof david r jackson ece dept

Shielding and Grounding (cont.)

a

b

q

(a) r < a

(b) a < r < b

(PEC)

Neutral shell

(c) r > b


Prof david r jackson ece dept

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.)


Prof david r jackson ece dept

Shielding and Grounding (cont.)

QA

q

QB

Find QA, QB:

Neutral shell

+

+

+

+

+

+

+

-

-

-

+

+

q

-

-

QB

QA

Neutral shell:

-

-

-

+

+

+

+

+

+

+


Prof david r jackson ece dept

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)


Prof david r jackson ece dept

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.)


Prof david r jackson ece dept

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.”


Prof david r jackson ece dept

Shielding and Grounding (cont.)

Charge on outer surface:

r > b: E = 0

-

-

-

q

-

-

-

-

-q

-

PEC wire

Earth

Hence


Prof david r jackson ece dept

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.


Prof david r jackson ece dept

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.


Prof david r jackson ece dept

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.


Prof david r jackson ece dept

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.


Prof david r jackson ece dept

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


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