Download Notes 12 3318 Conductors

ECE 3318
Applied Electricity and Magnetism
Spring 2017
Prof. David R. Jackson
ECE Dept.
Notes 12
1
Conductors
J  E
Ohm’s law
Material
 [S/m]
Silver
6.3107
Copper
6.0107
Copper (annealed)
5.8107
Gold
4.1107
Aluminum
3.5107
Zinc
1.7107
Electric conductors:  >> 1
Brass
1.6107
Perfect electric conductor (PEC):   
Nickel
1.4107
Iron
1.0107
Tin
9.2106
Steel (carbon)
7.0106
Steel (stainless)
1.5106

Note:
Many of the properties derived for
PECs hold very accurately for good
conductors.
http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity
2
Perfect Electric Conductors
Inside a perfect conductor, the electric field must be zero.
Ohm’s law:
J  E
or
E  J /
PEC
PEC:
 
E0
3
Perfect Electric Conductors (cont.)
Inside and on a perfect conductor, the potential must be constant.
E 0
A
B
C
  x, y, z 
PEC
Hence
B
VAB   E  dr  0
A
  constant
4
Perfect Electric Conductors (cont.)
Electric lines of flux must enter or leave a conductor perpendicular to it.
In other words: Et = tangential electric field = 0
On surface of PEC:
B
VAB   E  dr  E   r  0
A
Hence, E is perpendicular to r for
any r along the surface.
A
B
r
We choose a small path r in a
tangential direction along the surface.
PEC
5
Perfect Electric Conductors (cont.)
Inside a PEC v = 0
Proof: Assume a point inside where v > 0
v > 0 inside small volume
(since v is a continuous function).
 Qencl  0
S
+
+ ++
+ +
+
PEC
 D  nˆ dS  Q
encl
S
Hence
0  Qencl  0
Contradiction !
6
Perfect Electric Conductors (cont.)
Only s on the surface is allowed for a PEC.
PEC
PEC
PEC
http://en.wikipedia.org/wiki/Electrostatics
7
Faraday Cage Effect
Inside of a hollow PEC shell in statics:
 There is no electric field.
 There is no s on the inner surface.
PEC
s
+
-
q
-
Static electric field source
-
+
E=0
Note:
The Faraday cage does
not have to be grounded.
+
+
+
This is a shielded region (no electric field).
8
Faraday Cage Effect (cont.)
Proof of Faraday cage effect
Hollow cavity
+
+
+
+
PEC
+
 The electric field is zero inside the
conductor.
+
+
 The potential on the boundary of the inner
hollow cavity is constant.
+
 The uniqueness theorem then says that the
potential must be a constant throughout the
hollow cavity region.
+
+
+
+
PEC shell
+
 The electric field is then zero inside the
hollow cavity as it comes from the gradient
of the electric field.
Note: The uniqueness theorem is discussed later.
9
Faraday Cage: Practical Conductor
In static equilibrium, the Faraday cage still works, even with
a practical good conductor (finite conductivity):
From Ohm's law : J  0 inside metal

E 0
Static electric field source
-
q
-

-
E=0
-
E  0 inside metal
Conducting metal shell (finite conductivity)
The rest of the proof
works the same way.
10
Faraday Cage: Note on Magnetic Field
A Faraday cage made of a good conductor does not block
a static magnetic field!

H 0
N
S
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 Shielding
At high frequencies, both electric and magnetic fields are
blocked by the skin effect.

E0
H 0
Radiating antenna source
Skin depth:
Thickness >> 
=
2

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).
12
Faraday Cage Effect (cont.)
Faraday-cage effect
They are safe from the Tesla coil!
13
Faraday Cage Effect (cont.)
Faraday-cage effect
She is safe from the Van de Graaff generator!
(Boston Science Museum)
14
Faraday Cage Effect (cont.)
Faraday Cage Shielded Room
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
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
16
Shielding and Grounding
a
b
Spherical PEC shell
Neutral (no charge)
PEC
Drill hole and
insert point charge,
then solder hole.
q
PEC
a
Find E:
b
q
Neutral shell (no charge)
PEC
17
Shielding and Grounding (cont.)
(a)
r<a
 D  nˆ dS  Q
encl
S
Dr  4 r 2   q
 q 
E  rˆ 
2 
 4 0 r 
(b)
a<r<b
E0
(PEC)
a
b
q
PEC
Neutral shell
(c)
r>b
Dr  4 r 2   q
 q 
E  rˆ 
2 
4

r
0


18
Shielding and Grounding (cont.)
Neutral shell
q
PEC
 q 
E  rˆ 
(outside metal)
2 
4

r
0


The neutral metal shell does not block the static electric field
coming from the inside !
(The shell would be a good shield for high-frequency fields coming from the inside, however.)
19
Shielding and Grounding (cont.)
Find QA, QB:
QA
q
QB
PEC
Neutral shell
QA
-
- q- - -
+
+
+
+
+
+
+
+
+
PEC
+
+
+
-
+
+
+
Neutral shell: QA  QB
QB
+
20
Shielding and Grounding (cont.)
+
PEC
+
+
QA -
+
-
- q
- - -
S
+
+
+
Dr  4 r 2   Qencl
+
-
QB
+
+
+
encl
+
+
+
 D  nˆ dS  Q
+
The electric field is zero inside of a PEC.
+
so
S
A Gaussian surface is chosen
inside the metal shell.
Qencl  0
or
QA  q  0
Hence
QA  q
We then also have
QB  q
(neutral shell)
21
Shielding and Grounding (cont.)
+
+
+
+
+
+ q
+
PEC
- - q
- - - q
+
+
+
+
+
+
+
+
+
Flux picture showing the charge on the surfaces
Note: Flux lines go from positive charges to negative charges.
(Also, they go from higher potential to lower potential.)
22
Shielding and Grounding (cont.)
Next, “ground” the shell:
r b: E  0
PEC
- q --q - -
Proof:
E=0
We can use the uniqueness theorem,
considering the outside of the shell, the earth,
and the wire to form the boundary of our region.
PEC wire
Earth
The earth is modeled as a big conductor.
23
Shielding and Grounding (cont.)
Charge on outer surface:
r b: E  0
QBG
 D  nˆ dS  Q
PEC
encl
0
- qq
S
Qencl  0
q   q   QBG
S
PEC wire
Hence
QBG  0
Earth
24
Shielding and Grounding (cont.)
PEC
Charge on outer surface (cont.):
-
-
-q -
q - - -
+
+
+
+
+
+ q
+
PEC
- - q
- - - q
+
+
+
PEC wire
-
E0
+
+ q
+
+
+
+
+
QBG  0
+
+
+
Neutral shell before grounding
Earth
After grounding
The charge q on the outer surface has flowed down to ground.
25
Summary of Shielding Effects in Statics
PEC
q
- -
+
Effect # 1: Faraday cage effect
-
+
E0
-
There is no field on the inside
(does not require grounding).
+
+
+
Effect # 2: Static field penetration
+
+
+
+
+ q
+
- - q+

q
- - +
An ungrounded shield does not block the
static electric field of an inside source.
PEC
+
+
+ + +
E0
+ + +
26
Summary of Shielding Effects in Statics (cont.)
PEC
-
-
- q
- - -
-
E0
Effect # 3: Grounding
Conducting 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
27