VECTOR OPERATORS ∇, ×, •
Vector:
Vector Dot Product:
ˆ y + ẑA z
A = xˆ A x + yA
A • B = A xB x + A yB y + A zBz = |A||B|cos θ
xˆ
Vector Cross Product: A × B = det A x
yˆ
Ay
A z = |A||B|sin θ
Bx
By
Bz
ẑ
A
B
θ
= xˆ ( A yBz − A zBy ) + yˆ ( A zBx − A xBz ) + ẑ ( A xBy − A yBx )
“Del” (∇) Operator:
Gradient of φ:
Divergence of⎯A:
∇ = xˆ ∂ + yˆ ∂ + ẑ ∂
∂x
∂y
∂z
∂φ
∂φ
∂φ
∇φ = xˆ
+ yˆ
+ ẑ
∂x
∂y
∂z
∂A x ∂A y ∂A z
∇•A =
+
+
∂x
∂y
∂z
xˆ
“Curl of A”:
yˆ
ẑ
∇ × A = det ∂ ∂x ∂ ∂y ∂ ∂z
Ax
Ay
Az
L3-1
PHYSICAL SIGNIFICANCE OF ∇•, ∇×
∇i D is the “divergence of the vector field⎯D”
∫v (∇i A)dv = ∫∫s (Ain̂)da
Gauss’s divergence theorem:
Gauss’s Law, Differential Form: ∇ • D = ρ
∫v
(∇iD)dv =
∫∫s
ˆ da = ∫ ρ dv
(Din)
v
ρ
D
∇ × E is the “curl of the vector field⎯E”
Stokes’s theorem:
∫c Eids = ∫∫A (∇ × E)in̂ da
Faraday’s Law, Differential Form: ∇ × E = −
∂B
∂t
E
B
∂B
E
i
d
s
=
(
∇
×
E
)
n̂
da
=
−
i
∫c
∫∫A
∫∫A ∂t inˆ da = ∫c Eids
L3-2
MAXWELL’S EQUATIONS
Integral Form:
Differential Form:
∫∫ S D • n̂da = ∫∫∫ V ρ dv
ρ
∫∫ S B • nˆ da = 0
0
D = εE, B = μH
0
∂
∫ E • ds = − ∂t ∫∫A B • n̂da
c
0
∂
ˆ
H
•
d
s
=
J
•
n̂
da
+
∫
∫∫A
∫∫A D • nda
t
∂
c
⎯E
⎯H
⎯B
⎯D
⎯J
ρ
Electric field
Magnetic field
Magnetic flux density
Electric displacement
Electric current density
Electric charge density
D
E
B
H
B
∇ •D= ρ
∇•B=0
∇ × E = - ∂B
∂t
∇ × H = J + ∂D
∂t
J D
[volts/meter, V m-1]
[amperes/meter, A m-1]
[Tesla, T]
[ampere sec/m2, A s m-2]
[amperes/m2, A m-2]
[coulombs/m3, C m-3]
L3-3
MAXWELL’S EQUATIONS: VACUUM SOLUTION
Constitutive
Gauss‘s Law Relations
∇•D = ρ
D = εoE
∇•B = 0 0
B=μ H
∂B
Faraday’s Law: ∇ × E = − ∂t
Ampere’s Law: ∇ × H = J + ∂D
0 ∂t
o
EM Wave Equation:
Eliminate H : ∇ × ( ∇ × E ) = −μo ∂ ( ∇ × H)
∂t
Use identity: ∇ × ( ∇ × A ) = ∇ ( ∇ • A ) − ∇ 2 A
Yields:
2
∂
∂
∇ ( ∇ • E ) − ∇ E = −μo ( ∇ × H) = −μoεo E
∂t
0
∂t 2
2
1
EM Wave Equation
2E
∂
∇ E − μ o εo
=0
2
∂t
2
Second derivative in space ∝ second derivative in time,
therefore solution is any f(r,t) with identical dependencies on r,t
2
2
2
1
∂
∂
∂
2
Laplacian Operator: ∇ • ( ∇φ ) = ∇ φ = (
+
+
)φ
2
2
2
∂x
∂y
∂z
L3-4
WAVE EQUATION SOLUTION
Many are possible ⇒ Try Uniform Plane Wave (UPW), ≠ f(x,y)
2
∂
E =0
Example:
Try: E = ŷ Ey(z) in ∇ E − μ o ε o
2
∂t
0
0
2
2
2
∂
∂
∂
2
⇒ ∇ Ey = ( 2 +
+
)E y
2
2
∂x
∂y
∂z
2
2
∂ Ey
∂ Ey
Yields:
−
μ
ε
=0
o o
2
2
∂z
∂t
Trial solution: Ey(z,t) = E+(t – z/c); E+(arg) = arb. function of (arg)
2
Test solution: c-2 E”+ (t – z/c) - μoεo E”+(t – z/c) = 0 iff:
1
c=
≈ 3×108 [m s-1] in vacuum (velocity of light)
μo εo
E+(t – z/c)
0 z1
z = t = 0 ⇒ arg = 0
propagation
The position where arg = 0
moves at velocity c
z
arg = 0 at t1 = z1/c
L3-5
UNIFORM PLANE WAVE IN Z-DIRECTION
Example:
Ey(z,t) = E+(t - z/c) [V/m]
Func(arg) = Func*[(-c)(arg)] = Func*(z – ct)
E.G.:
Ey(z,t) = E+ cos[ω(t – z/c)] = E+ cos(ωt – kz),
where k = ω/c = ω μo εo
To find magnetic fields:
Faraday’s Law: ∇ × E = − ∂B
⇒⎯H = - ∫(∇ ×⎯E)μo-1 dt
∂t
xˆ
yˆ
ẑ
∇ × E = det ∂ ∂x ∂ ∂y ∂ ∂z = −x̂∂E+ cos ( ωt − kz ) ∂z
Ex
0
0
Ey
0
Ez
0
= - x̂ kE+sin(ωt – kz)
⎯H = x̂∫(k/μo)E+sin(ωt – kz) dt = - xˆ(E+/ηo)cos(ωt – kz)
k = ω μ oεo ,
ηo = μ o / εo
L3-6
UNIFORM PLANE WAVE: EM FIELDS
EM Wave in z direction:
E ( z,t ) = ŷE+ cos ( ωt − kz ) , H ( z,t ) = −xˆ (E+ ηo ) cos ( ωt − kz )
x
H ( z,0 )
E ( z,0 )
z
y
Electric energy density
Magnetic energy density
z
Linearity implies superposition of n→∞ waves, all θ,φ
L3-7
ELECTROMAGNETIC AND OTHER WAVES
A “wave” is a fixed disturbance propagating through a medium
A,B
B
wave velocity
0
z
A
A,B energy density
null
0
Medium
String
Acoustic
Ocean
Electromagnetic
z
A
B
A energy
B energy
stretch
pressure
height
H
velocity
velocity
velocity
E
potential
potential
potential
magnetic
kinetic
kinetic
kinetic
electric
L3-8
Role of Maxwell’s Equations and Fields
Sources
q
Observable Reality
⎯J(x,y,z) [A/m2]
ρ(x,y,z) [C/m3]
Maxwell’s
Equations
Observer
⎯E,⎯H⎯v
⎯f [N]
Lorentz Force Law:
q
f = q(E + v ×μoH)
The fields⎯E,⎯H and the displacement and flux densities⎯D,⎯B permit
division of electromagnetics into the Maxwell and Lorentz equations
L3-9
MIT OpenCourseWare
http://ocw.mit.edu
6.013 Electromagnetics and Applications
Spring 2009
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