MEDIA AND BOUNDARY CONDITIONS
Media: conductivity σ, permittivity ε, permeability μ
Media are the only tools we have to create or sense EM fields
Conductivity (σ):
⎯J (current density, A/m2) = nq v = σE
n = #q’s/m3, q = charge (Coulombs), v = average velocity (m/s)
∞
Semiconductors:
+
P{escape} ∝ e-W/kT
σ
Metal
⎯E
Metals:
Superconductor
+- +
+
-+ +
J ∝ v = at =
Semiconductor
0
f t = qE t , ∴ σ ∝ q t ;
m
m
m
(t = time before collisions reset v ≅ 0)
300K
Temperature T
t = f(Temp )
L2-1
DIELECTRICS
Vacuum:
D = εoE
∫∫SD • n̂da = ∫∫∫ V ρf dv
nucleus
electron
cloud
-
ρf = free charge density
d
+q Atom
⎯E
⎯E
Dielectric
Materials:
D = εE = εo E + P
+
polarized
- +ε - +
atoms
- + - +
- + - +
- + - +
medium
ˆ = ∫∫∫ (ρf +ρp ) dv
∫∫S εo E • nda
V
ˆ = − ∫∫∫ ρpdv
∫∫S P • nda
V
ρp is polarization (surface)
charge density
+
+
+
+
+
Metal plates
E
P
- ρv = 0
+
+
+
+
+
-
ρp
P = “Polarization Vector”
L2-2
MAGNETIC MATERIALS
Basic Equations:
∫∫ S B • nˆda = 0
e
+
+
e
B = μ o H in vacuum
B = μH = μ o (H + M )
B
e
magnetic
domain
domain
wall
e
+
M = “Magnetization Vector”
μ = permeability
atoms
B
⎯H = 0
magnetic
domains
⎯M
L2-3
SATURATION AND HYSTERISIS
⎯H = 0
magnetic
domains
⎯H > 0
⎯M
1 B • H = W [J/m3 ] Magnetic energy density
m
2
⎯B
slope ≅ μo
Residual flux ⎯B
density Br
slope ≅ μ
0
-Hc magnetic 0
⎯H
coercive force
saturation
⎯H
Area = hysterisis
loss [J/m3]
L2-4
MEDIA PARAMETERS
Conductivity σ
[Siemens/m]
Dielectric constant
(ε/εo)
Paraffin
~10-15
Glass
10-12
Dry earth
10-4-10-5
Distilled water 2x10-4
Sea water
3-5
Iron
107
Copper
5.8x107
Silver
6.1x107
Vacuum
1.0
Wood (fir)
1.8-2.0
Teflon, petroleum 2.1
Vaseline
2.2
Paper
2-3
Polystyrene
2.6
Sandy soil
2.6
Fused quartz
3.8
Ice
4.15
Pyrex glass
5.1
Aluminum oxide 8.8
Ethyl alcohol
24.5
Water
81
Titanium dioxide 100
Relative permeabilities
μ/μo
Bismuth
Silver
Copper
Water
Vacuum
Air
Aluminum
Cobalt
Nickel
Mild steel
Iron
Mu metal
Supermalloy
0.99983
0.99998
0.999991
0.999991
1.000000
1.0000004
1.00002
250
600
2000
5000
100,000
1,000,000
L2-5
INTEGRAL MAXWELL’S EQUATIONS
Graphical Equations:
∫∫S D • n̂da = ∫∫∫ V ρ dv
Gauss
ρ
∫∫S B • n̂da = 0
Gauss
0
∂
D
B
∫ E • ds = − ∂t ∫∫A B • n̂da
Faraday
E
c
B
H
∂
∫ H • ds = ∫∫A J • n̂da + ∂t ∫∫A D • n̂da Ampere
c
J
D = εE,
B = μH
D
Constitutive relations
L2-6
FIELDS PERPENDICULAR
TO BOUNDARIES
Using Gauss’s Law: ( ∫∫ S D • nˆ da = ∫∫∫v ρ dv ) :
ˆ
→ (D1⊥ − D2⊥ ) A = ρs A
∫∫ S D • nda
(Lim A → 0,
δ2
n̂
A
<<A)
surface charge density ρs
D1
δ
D2
surface S
Therefore:
D1⊥ − D2⊥ = ρs yields:
ˆ = (B1⊥ − B2⊥ ) A = 0 yields:
∫∫ A B • nda
n̂ • (D1 − D2 ) = ρs
n̂ • (B1 − B2 ) = 0
L2-7
FIELDS PARALLEL TO BOUNDARIES
Using Faraday’s Law: ∫c E • ds = - ∂ ∫∫A B • nda
ˆ
∂t
∫ Eids → (E1// − E2 // ) L = − dtd ∫∫ ABida → 0
(Lim δ→0)
C
Therefore:
E1//
⎯E1// =⎯E2//
n̂ × (E1 − E2 ) = 0
n̂
L
δ
E1
da,nˆ a
Using Ampere’s Law:
E2
ds
A
-E2//
∫c H • ds = ∫∫A (J + ∂D/∂t) • da
∂
H
i
ds
→
H
−
H
L
=
J
i
da
−
Dida
(
)
∫c
1//
2 //
∫∫ A
∂t ∫∫ A
→0
→ ( Js • n̂a ) L
Therefore: n̂ × (H1 − H2 ) = Js
L2-8
PERFECT CONDUCTORS
Electric Fields:
{if σ→∞ and⎯E ≠ 0} ⇒ { J = σE → ∞ } ⇒ { H → ∞ since
∫c Hids = ∫∫A (J + ∂D/∂t)ida }⇒ {Wm = μH2/2 [J/m3] → ∞, and wm → ∞}
Therefore:
⎯E = 0 inside perfect conductors
ρ = 0 (since ∫ ρ dv = ∫∫ εEinˆ da
v
Magnetic Fields:
{If E = 0 and
Therefore:
s
∂
Bin̂da = − ∫ c Eids} ⇒ { ∂B ∂t = 0}
∂t ∫∫A
)
⎯H = 0 inside perfect conductors
(if σ = ∞, and H(t=0) = 0)
Superconductors (Cooper pairs don’t impact lattice):
B ≅ 0 inside because σ = ∞
Cooper pairs of electrons disassociate and superconductivity
fails when the external B(T) is above a critical threshold
L2-9
SUMMARY: BOUNDARY CONDITIONS
General Boundary Conditions:
n̂ • (D1 − D2 ) = ρs
E1,D1
ρs
H1,B1
n̂ • (B1 − B2 ) = 0
⎯Js
ρs
n̂ × (E1 − E2 ) = 0
n̂ × (H1 − H2 ) = Js
H2 ,B2
E2 ,D2
Inside Perfect Conductors:
D2 = B2 = E2 = 0
n̂ • D1 = ρs
n̂ • B1 = 0
⇒⎯B is parallel to perfect conductors
n̂ × E1 = 0
⇒⎯E is perpendicular to perfect conductors
n̂ × H1 = Js
L2-10
MIT OpenCourseWare
http://ocw.mit.edu
6.013 Electromagnetics and Applications
Spring 2009
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