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Prof. David R. JacksonECE Dept.
Spring 2014
Notes 12
ECE 2317 Applied Electricity and Magnetism
1
Conductors
[S/m]
2
J EOhm’s law
Good electric conductor: >> 1 Perfect electric conductor:
Material [S/m]
Silver 6.3107
Copper 6.0107
Copper (annealed) 5.8107
Gold 4.1107
Aluminum 3.5107
Zinc 1.7107
Brass 1.6107
Nickel 1.4107
Iron 1.0107
Tin 9.2106
Steel (carbon) 7.0106
Steel (stainless) 1.5106
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
[S/m]
J E
AB
0B
AB
A
V E dr
PEC:
Also, constant
0E
since
Ohm’s law:
3
/E J or
Perfect Electric Conductors
4
Electric lines of flux must enter or leave a conductor perpendicular to it.
0B
AB
A
V E dr
0
0
0
tAB
t
t
V E r
E r
E
On surface of PEC:
Et = tangential electric field = 0
A B
PEC
Et
r
We 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.
Perfect Electric Conductors (cont.)
Perfect Electric Conductors (cont.)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).
ˆ encl
S
D n dS Q
Contradiction !
+ +
++ +
+ +
5
0enclQ
0 0enclQ Hence
Perfect Electric Conductors (cont.)
6
Only s on the surface is allowed for a PEC.
v = 0
v = 0
v = 0
Even if these conductors were solid, there would be no volume charge density inside them.
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
Hollow region (no charge inside)
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 = 0 -
-
-- q
+
-
+
+
+
+
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)
VAB 0 Contradiction!
+
+
+
+
+
+
+
+
+
+
+
+
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
Imperfect (Practical) Electric ConductorsAssume:
1) We are in static equilibrium (statics)*
2) No connections to the object are made from the outside.
( )
0 ( )
J E
J
Ohm'slaw
steady state
Inside metal:
/ 0E J inside metal
The proof that v = 0 then proceeds as before (based on Gauss’s law).v = 0
(finite)
E = 0
r
E = 0 v = 0
Then
* / Time constant :
Faraday Cage: Imperfect ConductorIn equilibrium, there is still no field inside the shell, and no charge density on the inner surface.
Conducting metal shell (finite conductivity)
11
Static electric field source
0E inside metal
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.)
0v
E = 0 -
-
-- q
(finite)
0s
High Frequency: Imperfect Conductor
There may be penetration of fields at high frequency:
Conducting metal shell (finite conductivity)
12
Radiating antenna sourceE 0
E 0
E
This situation is discussed in ECE 3317 (skin-depth effect).
(finite)
2
=
Skin depth:
To be a good shield, the thickness of the shield should be large compared to a skin depth.
Faraday-cage effect
He is safe from the Tesla coil!
13
Faraday Cage Effect (cont.)
Faraday-cage effect
She is safe from the Van de Graaff generator!
14
(Boston Science Museum)
Faraday Cage Effect (cont.)
Faraday-cage effect
Entrance to a “Faraday room”Faraday shield at Art Nouveau power plant in Heimbach, Germany
15
Faraday Cage Effect (cont.)
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
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.
Shielding and Grounding
q
Spherical PEC shell Neutral (no charge)
ab
Drill hole and insert point charge, then solder hole.
Neutral shell (no charge)Find E:q
ab
18
(a) r < a
(c) r > b
(b) a < r < b
2
20
ˆ
4
ˆ4
encl
S
r
D n dS Q
D r q
qE r
r
0E
2
20
4
ˆ4
rD r q
qE r
r
Neutral shell
(PEC)
Shielding and Grounding (cont.)
a bq
19
20
ˆ4
qE r
r
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
Find QA, QB:
q-
---
-
--
-
+
+
+
+
+
+
+
++
+
+
+
+
+
+
+
QA QB
q
QA
QB
Neutral shell:
Shielding and Grounding (cont.)
A BQ Q
21
Neutral shell
2
ˆ
4
0
encl
S
r encl
encl
D n dS Q
D r Q
Q
(neutral shell)
q-
---
-
--
-
+
+
+
+
+
+
+
++
+
+
+
+
+
+
+
QA QB
Sso
HenceA Gaussian surface is chosen inside the metal shell.
Shielding and Grounding (cont.)
22
AQ q
BQ qWe then also have
Flux picture showing charge on 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
Next, “ground” the shell: r > b : E = 0
0B
AB
A
V E dr
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
Charge on outer surface:
ˆ
0
0
encl
S
encl
Gencl B
GB
D n dS Q
Q
Q q q Q
Q
Hence
r > b: E = 0
Shielding and Grounding (cont.)
q-
---
-
--
--q
PEC wire
Earth
GBQ
25
Charge on outer surface (cont.):
The charge q on the outer surface has flowed down to ground.
- q
Neutral shell before grounding
Shielding and Grounding (cont.)
26
-q
+
+
++PEC wire
Earth
q
After grounding
0GBQ
-
---
-
--
-
q
E = 0
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
J E
In steady-state:
Ohms' law for object, wire, or earth:
0J
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
0GBQ
s
e
w
q-
---
-
--
-
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
Summary of Effects
E = 0 -
-
-- q
+
-
+
+
+
+
There is no field on the inside(does not require grounding).
qAn ungrounded shield does not block the
electric field of an inside source.
Effect # 1: Faraday cage effect
Effect # 2: Static field penetrationE 0
E = 0
30
Summary of Effects (cont.)
A grounded shield removes the electric field in the exterior region and removes charge from the outer surface.
PEC wire
Earth
-
---
-
--
-
q
Effect # 3: Grounding
E = 0
(This assumes that there are no charges in the outside region.)
