12/23/02
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Basic Equations of Electrodynamics
Mathematical Identities
Electromagnetic Variables
Maxwell's Equations, Force
Constants
v(t) = Re{Vejωt}where V = |V|ejφ E = electric field (Vm-1)
∇ ×E = -∂ B/∂t
εo = 8.85×10-12 Fm-1
∇ = x̂ ∂/∂x + ŷ ∂/∂y + ẑ ∂/∂z
H = magnetic field (Am-1)
c
A
d
E ds
B da
dt
•
= −
•
∫
∫
!
µo = 4π×10-7 Hm-1
A• B = AxBx + AyBy + AzBz
D = electric displacement (Cm-2)
∇ ×H = J + ∂ D/∂t
c = (εoµo)-0.5 ≅ 3×108ms-1
∇ 2φ = (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2)φ
B = magnetic flux density (T)
c
A
A
d
H ds
J da
D da
dt
•
=
•
+
•
∫
∫
∫
!
h = 6.624×10-34 Js
sin2θ + cos2θ = 1
Tesla (T) = Weber m-2 = 10,000 gauss
∇ • D = ρ → ∫A D • d a = ∫V ρdv e = -1.60×10-19 C
∇ • (∇ ×A) = 0
ρ = charge density (Cm-3)
∇ • B = 0 → ∫A B • d a = 0
k = 1.38×10-23 JK-1
∇ × (∇ ×A) = ∇ (∇ • A) - ∇ 2 A J = current density (Am-2)
∇ • J = -∂ρ/∂t
ηo ≅ 377ohms =(µo/εo)0.5
∫V(∇ • G)dv = ∫A G • d a
σ = conductivity (Siemens m-1)
f = q( E + v × µo H) [N]
me = 9.1066×10-31 kg
(
)
A
c
G da
G ds
∇×
•
=
•
∫
∫!
Js = surface current density (Am-1) Waves
ejωt = cos ωt + j sin ωt
ρs = surface charge density (Cm-2) (∇ 2 - µε∂2/∂t2) E = 0 [Wave Eq.]
Media
cos α + cos β = 2 cos [(α+β)/2] cos[(α-β)/2]
(∇ 2 + k2) E = 0,
r
k
j
oe
E
E
•
−
=
D = εo E + P
H(f) = ∫-∞+∞ h(t)e-jωtdt
Boundary Conditions
k = ω(µε)0.5 = ω/c = 2π/λ
∇ • D = ρf, τ = ε/σ
ex = 1 + x + x2/2! + x3/3! + …
E1// - E2// = 0
kx2 + ky2 + kz2 = ko2 = ω2µε
∇ • εo E = ρf + ρp
sinα = (ejα – e-jα)/2j
H1// - H2// = n̂ ×Ks
vp = ω/k, vg = (∂k/∂ω)-1
∇ • P = -ρp, J = σ E
cosα = (ejα + e-jα)/2
B1⊥ - B2⊥ = 0
Ey(z,t) = E+(z-ct)+E-(z+ct) = Re{Ey(z)ejωt}
B = µ H = µo( H + M)
Planar Interfaces
D1⊥ - D2⊥ = ρs
Hx(z,t) = ηo-1[E+(z-ct)-E-(z+ct)] [or(ωt-kz) or (t-z/c)] ε = εo(1-ωp2/ω2)
θr = θi
0 = ↵ if σ=∞
∫A( E×H)• d a