MIT OpenCourseWare http://ocw.mit.edu 6.013/ESD.013J Electromagnetics and Applications, Fall 2005
Please use the following citation format: Markus Zahn, Erich Ippen, and David Staelin, 6.013/ESD.013J Electromagnetics and Applications, Fall 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons AttributionNoncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
6.013 Final Exam Formula Sheets December 21, 2005 Cartesian Coordinates (x,y,z): ∂Ψ ∂Ψ ∂Ψ ∇Ψ = xˆ + yˆ + zˆ ∂x ∂y ∂z ∂A x ∂A y ∂A z ∇iA = + + ∂x ∂y ∂z ∂A y ⎞ ⎛ ∂A ∂A ⎞ ⎛ ∂A y ∂A x ⎞ ⎛ ∂A ∇ × A = xˆ ⎜ z − + yˆ ⎜ x − z ⎟ + zˆ ⎜ − ⎟ ∂z ⎠ ∂x ⎠ ⎝ ∂x ∂y ⎟⎠ ⎝ ∂z ⎝ ∂y 2 2 2 ∇2Ψ = ∂ Ψ + ∂ Ψ + ∂ Ψ ∂x 2 ∂y2 ∂z 2 Cylindrical coordinates (r,φ,z): ∇Ψ = rˆ ∂Ψ + φˆ 1 ∂Ψ + zˆ ∂Ψ ∂r r ∂φ ∂z ∂ ( rA r ) 1 ∂A φ ∂A z ∇iA = 1 + + ∂z r ∂r r ∂φ
rˆ r φˆ zˆ rA ∂ ⎛ ⎞ ⎛ 1 ∂A z ∂Aφ ⎞ ( ) A A A ∂ ∂ ∂ φ ⎛ ⎞ ∇ × A = rˆ ⎜ − + φˆ ⎜ r − z ⎟ + zˆ 1 ⎜ − r ⎟ = 1 det ∂ ∂r ∂ ∂φ ∂ ∂z r ⎝ ∂r ∂z ⎟⎠ ∂r ⎠ ∂φ ⎠ r ⎝ ∂z ⎝ r ∂φ A r rA φ A z
( )
2 2 ∇ 2 Ψ = 1 ∂ r ∂Ψ + 1 ∂ Ψ + ∂ Ψ r ∂r ∂r r 2 ∂φ2 ∂z 2 Spherical coordinates (r,θ,φ): ∇Ψ = rˆ ∂Ψ + θˆ 1 ∂Ψ + φˆ 1 ∂Ψ ∂r r ∂θ r sin θ ∂φ
(
)
∂Aφ ∂ r 2Ar ∂ ( sin θAθ ) ∇iA = 1 + 1 + 1 ∂r r sin θ ∂θ r sin θ ∂φ r2 ⎛ ∂ ( sin θA φ ) ∂Aθ ⎞ ⎛ 1 ∂A 1 ∂ ( rAφ ) ⎞ 1 ⎛ ∂ ( rAθ ) − ∂A r ⎞ r − ∇ × A = rˆ 1 ⎜ − ⎟ ⎟ + θˆ ⎜ ⎟ + φˆ ⎜ r sin θ ⎝ ∂θ ∂φ ⎠ r ⎝ ∂r ∂θ ⎠ ⎝ r sin θ ∂φ r ∂r ⎠ rˆ rθˆ r sin θ φˆ 1 det ∂ ∂r ∂ ∂θ = ∂ ∂φ 2 r sin θ A r rAθ r sin θA φ
(
)
(
)
∂ sin θ ∂Ψ + 1 ∂ 2Ψ ∇ 2 Ψ = 1 ∂ r 2 ∂Ψ + 1 ∂r ∂θ r 2 ∂r r 2 sin θ ∂θ r 2 sin 2 θ ∂φ2
Gauss’ Divergence Theorem: ∫ ∇iG dv = ∫ Ginˆ da
Vector Algebra: ∇ = xˆ ∂ ∂x + yˆ ∂ ∂y + zˆ ∂ ∂z A • B = A x Bx + A y By + Az Bz
Stokes’ Theorem:
∇ • ( ∇× A ) = 0 ∇× ( ∇× A ) = ∇ ( ∇ • A ) − ∇ 2 A
V
A
∫A ( ∇ × G )inˆ da = ∫ C Gid
Page 1 of 4
Basic Equations for Electromagnetics and Applications Fundamentals f = q ( E + v × μ o H ) [ N ] (Force on point charge)
E1// − E 2 // = 0
∇ × E = −∂ B ∂t
H1// − H 2 // = J s × nˆ B1⊥ − B2 ⊥ = 0
d ∫ c E • ds = − dt ∫A B • da ∇ × H = J + ∂ D ∂t
( D1⊥ − D2 ⊥ ) = ρs
∇ • D = ρ → ∫A D • da = ∫V ρdv
Electromagnetic Quasistatics E = −∇Φ ( r ) , Φ ( r ) = ∫V′ ( ρ ( r ) / 4πε | r ′ − r |) dv′
∇ • B = 0 → ∫A B • da = 0
−ρf ε C = Q/V = Aε/d [F] L = Λ/I i(t) = C dv(t)/dt v(t) = L di(t)/dt = dΛ/dt we = Cv2(t)/2; wm = Li2(t)/2 Lsolenoid = N2μA/W τ = RC, τ = L/R Λ = ∫A B • da (per turn)
∇ • J = −∂ρ ∂t
∇2 Φ =
E = electric field (Vm-1) H = magnetic field (Am-1) D = electric displacement (Cm-2) B = magnetic flux density (T) Tesla (T) = Weber m-2 = 10,000 gauss ρ = charge density (Cm-3) J = current density (Am-2)
σ = conductivity (Siemens m-1)
KCL : ∑ i Ii (t) = 0 at node
J s = surface current density (Am-1)
KVL : ∑ i Vi (t) = 0 around loop
-2
ρs = surface charge density (Cm ) εo = 8.85 × 10
Q = ω0 wT / Pdiss = ω0 / Δω
-1
Fm
ω0 = ( LC )
μo = 4π × 10-7 Hm-1
−0.5
V 2 ( t ) / R = kT
c = (εoμo)-0.5 ≅ 3 × 108 ms-1 e = -1.60 × 10-19 C ηo ≅ 377 ohms = (μo/εo)0.5 ( ∇ 2 − με∂ 2 ∂t 2 ) E = 0 [Wave Eqn.]
Electromagnetic Waves
( ∇ 2 − με∂ 2
∂t 2 ) E = 0 [Wave Eqn.]
Ey(z,t) = E+(z-ct) + E-(z+ct) = Re{Ey(z)ejωt}
( ∇ 2 + k 2 ) Eˆ = 0, Eˆ = Eˆ o e− jk ir
Hx(z,t) = ηo-1[E+(z-ct)-E-(z+ct)] [or(ωt-kz) or (t-z/c)]
k = ω(με)0.5 = ω/c = 2π/λ
∫A ( E × H ) • da + ( d dt ) ∫V ( ε E 2 + μ H 2 ) dv = − ∫V E • J dv (Poynting Theorem) 2
2
kx2 + ky2 + kz2 = ko2 = ω2με vp = ω/k, vg = (∂k/∂ω)-1
θ r = θi sin θt / sin θi = ki / kt = ni / nt
Media and Boundaries D = εo E + P
θc = sin −1 ( nt / ni )
∇ • D = ρf , τ = ε σ
θ B = tan −1 ( ε t / ε i )
∇ • ε o E = ρf + ρ p
θ > θ c ⇒ Eˆ t = Eˆ iTe +α x − jk z z
∇ • P = −ρp , J = σE
k = k ′ − jk ′′ Γ = T −1
B = μH = μo ( H + M )
(
1 2
0 = if σ = ∞
d
∫ c H • ds = ∫A J • da + dt ∫A D • da
-12
nˆ
)
ε ( ω ) = ε 1 − ωp 2 ω2 , ω p = ( Ne 2 mε )
(plasma)
for TM
TTE = 2 / (1 + [ηi cos θ t / ηt cos θi ]) TTM = 2 / (1 + [ηt cos θt / ηi cos θ i ])
ε eff = ε (1 − jσ / ωε ) Skin depth δ = ( 2 / ωμσ )
0.5
0.5
0.5
[m] Page 2 of 4
Radiating Waves ∇ 2 A − 12 c 1 ∇2Φ − 2 c
A = ∫V ′
Φ = ∫V ′
Wireless Communications and Radar
∂ A = −μ J f ∂t 2 ρf ∂2Φ =− 2 ε ∂t 2
G(θ,φ) = Pr/(PR/4πr2)
PR = ∫4π Pr ( θ, φ, r ) r 2 sin θ dθdφ
μ J f ( t − rQP / c ) dV ′
Prec = Pr(θ,φ)Ae(θ,φ) Ae (θ, φ) = G(θ, φ)λ 2 4π
4π rQP
ρ f ( t − rQP / c ) dV ′
G (θ , φ ) = 1.5sin 2 θ (Hertzian Dipole)
4πε rQP
E = −∇Φ −
∂A , B = ∇× A ∂t
i 2 (t) =
R r = PR
2π μ / ε ⎛ d eff ⎞ ⎜ λ ⎟ 3 ⎝ ⎠
2
jk x + jk y y E ff ( θ ≅ 0 ) = ( je jkr λr ) ∫ E t (x, y)e x dxdy A
ˆ + ω2μεΦ ˆ = −ρˆ ε , ∇2Φ ˆ ( x, y, z ) e jωt ⎤ Φ ( x, y, z, t ) = Re ⎡⎢ Φ ⎥
− jkr Eˆ z = ∑i a i Ee i = (element factor)(array f)
ˆ + ω2μεA ˆ = −μJˆ , ∇2 A
Ebit ≥ ~4 × 10-20 [J] Z12 = Z21 if reciprocity
⎣
⎦
A ( x, y, z, t ) = Re ⎡⎢ Aˆ ( x, y, z ) e jωt ⎤⎥ ⎣
⎦
(
)
ˆ (r ) = ∫ ′ ρˆ (r )e − jk⏐r ′− r⏐ / 4πε r '− r dV′ Φ V ˆ A(r) = ∫V ' ( μ Jˆ ( r ) e − jk r '− r 4π r '− r ) dV ' μˆ ˆ 4πr ) e − jkr sin θ Eˆ ffθ = H = ( jηkId ε ffφ
Forces, Motors, and Generators J = σ ( E + v × B)
F = I × B [ Nm -1 ] (force per unit length) E = −v × B inside perfectly conducting wire (σ → ∞ )
At ωo , w e = w m ⎛
2
⎞
w e = ∫V ⎜⎝ ε Eˆ 4 ⎟⎠ dv wm = ∫
V
(μ Hˆ 2 4) dv
Qn = ωn w Tn Pn = ωn 2αn
(
2 f mnp = ( c 2 ) ⎡⎣ m a ⎦⎤ + ⎣⎡ n b ⎦⎤ + ⎡⎣ p d ⎤⎦ sn = jωn - αn 2
dw T + f dz dt dt f = ma = d(mv)/dt
∇ • u = − (1 ρo cs 2 ) ∂p ∂t
-2
vi =
0.5
P = Po + p, U = 0 + u ∇p = −ρo ∂ u ∂t
2
)
Acoustics
Max f/A = B /2μ, D /2ε [Nm ]
2
2
( ∇2 − k 2 ∂ 2 ∂t 2 ) p = 0 2
k 2 = ω2 cs = ω2 ρo γPo
P = fv = Tω (Watts) T = I dω/dt I = ∑ i mi ri 2
ηs = p/u = ρocs = (ρoγPo)0.5 or ρ 0 RT gases
FE = λ E ⎡⎣ Nm −1 ⎤⎦ Force per unit length on line charge λ
ηs = (ρoK)0.5 solids, liquids
WM ( λ , x ) =
1 λ2 1 q2 ; WE ( q, x ) = 2 L ( x) 2 C ( x)
fM (λ, x ) = −
∂WM ∂x
∂W f E ( q, x ) = − E ∂x
q
λ
1 1 dL ( x ) d = − λ 2 (1/ L ( x ) ) = I 2 2 dx 2 dx
d 1 1 dC ( x ) = − q 2 (1/ C ( x ) ) = v 2 dx 2 dx 2
cs = v p = v g = ( γPo ρo )
0.5
or ( K ρo )
0.5
RT
or
p, u ⊥ continuous at boundaries
ˆ = pˆ + e- jkz + pˆ - e + jkz , p(z, t) = Re ⎡ p(z) ˆ e jω t ⎤ p(z) ⎣ ⎦ uˆ z =ηs -1 (pˆ + e-jkz - pˆ - e+jkz ) , u z (z, t) = Re ⎡⎣ uˆ z (z) e jω t ⎤⎦
∫A up • da + ( d dt )∫V ( ρo
u
2
)
2 + p2 2γPo dV Page 3 of 4
Optical Communications E = hf, photons or phonons hf/c = momentum [kg ms-1] dn 2 dt = − ⎡⎣ An 2 + B ( n 2 − n1 )⎤⎦
Transmission Lines Time Domain ∂v(z,t)/∂z = -L∂i(z,t)/∂t ∂i(z,t)/∂z = -C∂v(z,t)/∂t ∂2v/∂z2 = LC ∂2v/∂t2 v(z,t) = V+(t – z/c) + V-(t + z/c) i(z,t) = Yo[V+(t – z/c) – V-(t + z/c)] c = (LC)-0.5 = (με)-0.5 Zo = Yo-1 = (L/C)0.5 ΓL = V-/V+ = (RL – Zo)/(RL + Zo) Frequency Domain
ˆ (d 2 /dz 2 +ω 2 LC)V(z) =0 ˆ ˆ e-jkz + V ˆ e +jkz , v( z, t ) = Re ⎡Vˆ ( z )e jωt ⎤ V(z) =V + ⎣ ⎦ ˆI(z) = Y [V ˆ e-jkz - V ˆ e +jkz ], i ( z , t ) = Re ⎡ ˆI( z )e jωt ⎤ 0 + ⎣ ⎦
k = 2π/λ = ω/c = ω(με)0.5 ˆ ˆI(z) = Zo Zn (z) Z(z) = V(z) Zn (z) = [1 + Γ (z) ] [1 − Γ (z) ] = R n + jX n
Γ(z) = ( V− V+ ) e 2 jkz = [ Zn (z) − 1] [ Zn (z) + 1]
Z(z) = Zo ( ZL − jZo tan kz ) ( Zo − jZL tan kz ) VSWR = Vmax Vmin
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