Download WHAT ARE ELECTROMAGNETIC WAVES?

WHAT ARE ELECTROMAGNETIC WAVES?
A “wave” is a periodic disturbance propagating through a medium
EM Waves Convey Undulations in EM Fields:
Electric field E
wave velocity
z
0
Magnetic field H
Electric energy density
null
Magnetic
z
energy density
Electric and Magnetic Fields are Useful Fictions:
Explain all classical electrical experiments with simple equations
4 Maxwell’s equations plus the Lorentz force law
(quantum effects are separate)
L2-1
WHAT ARE ELECTRIC AND MAGNETIC FIELDS?
f = q (E + v × µo H)[Newtons]
Lorentz Force Law:
E
Electric field
[volts/meter; V/m]
H
Magnetic field
[amperes/meter; A/m]
f
Mechanical force
q
Charge on a particle
v
Particle velocity vector
[newtons; N]
[coulombs; C]
[meters/second; m/sec ]
µo
Vacuum permeability
1.26 × 10−6 Henries 
Electric and Magnetic Fields are what Produce Force f:
f = qE when v = 0, defining E via an observable
f = qv × µo H when E = 0, defining H via an observable
+++
+++ force f
+++
-
L2-2
MAXWELL’S EQUATIONS
Differential Form:
1Tesla
= Webers/m2 = 104 Gauss
Gauss's Laws ∇ • D = ρ
Faraday's Law: ∇ × E = −∂B ∂t
Ampere's Law: ∇ × H = J + ∂D ∂t
∇ •B = 0
[volts/meter; V/m]
E
H
B
Electric field
Magnetic field
Magnetic flux density
D
Electric displacment
J
Electric current density
ρ
Electric charge density
[amperes/meter; A/m]
[Tesla; T ] 1
coulombs/m2 ; C/m2 
amperes/m2 ; A/m2 


coulombs/m3 ; C/m3 
Integral Form:
∫c E • ds = ∫A − (∂B ∂t ) • da
∫c H • ds = ∫A (J + ∂D ∂t ) • da
Stoke’s da
Theorem
A
ds
∫A D • da = ∫V ρ dv
∫A B • da = 0
Gauss’
da
Law
V
A
L2-3
VECTOR OPERATORS ∇, ×, •
“Del” (∇
∇) Operator:
Gradient of φ:
∇ = xˆ ∂ ∂x + yˆ ∂ ∂y + zˆ ∂ ∂z
∇φ = xˆ ∂φ ∂x + yˆ ∂φ ∂y + zˆ ∂φ ∂z
“Vector Cross Product”:
ˆ x + yA
ˆ y + zA
ˆ z
A = xA
xˆ
A × B = det A x
yˆ
Ay
zˆ
Az
Bx
By
Bz
“Vector Dot Product”:
= xˆ ( A yBz − A zB y ) + yˆ ( A zB x − A xBz ) +
ẑ ( A xB y − A yB x )
A • B = A x B x + A y B y + A zB z
“Divergence of A”:
∇ • A = ∂A x ∂x + ∂A y ∂y + ∂A z ∂z
“Curl of A”:
xˆ
yˆ
zˆ
∇ × A = det ∂ ∂x ∂ ∂y ∂ ∂z
Ax
Ay
Az
“Laplacian Operator”:
∇ • (∇φ ) = ∇2 φ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z2 φ
(
)
L2-4
MAXWELL’S EQUATIONS: VACUUM SOLUTION
Maxwell’s Equations:
Faraday's Law: ∇ × E = −∂B ∂t
Ampere's Law: ∇ × H = J + ∂D ∂t
0
Gauss's Laws ∇ • D = ρ
∇ •B = 0 0
B = µo H
EM Wave Equation:
D = εo E
(
)
(
)
Eliminate H : ∇ × ∇ × E = −µo ( ∂ ∂t ) ∇ × H
Use identity: ∇ × (∇ × A ) = ∇ (∇ • A ) − ∇2 A
Yields:
∇ (∇ • E ) − ∇2 E = −µo ( ∂ ∂t ) (∇ × H) = −µo εo ∂ 2 E ∂t 2
0
EM Wave Equation1
∇ 2 E − µo εo ∂ 2 E ∂t 2 = 0
Since:
Second derivative in space = const × second derivative in time,
Solution is any f(r,t) with identical space and time dependences
1Homogeneous
Vector Helmholz Equation
L2-5
WAVE EQUATION SOLUTION
Wave Equation has Many Solutions!
∇2 E − µo εo ∂ 2 E ∂t 2 = 0
Where
(
)
∇2 φ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z2 φ
Example:
Try:
ˆ y ( z,t ) [no x,y dependence, “UPW”]
E = yE
(∂2 ∂z2 )Ey − µoεo∂2Ey
∂t 2 = 0
E y ( z,t ) = E+ ( t − z c ), where E+ (.) is an arbitrary function
Test:
( −c )−2 E+" ( t − z c ) = µo εoE+" ( t − z c ) ⇒
c =1
( µo εo )
Generally: c ≅ 3 × 108 [m/s] in vacuum
E y ( z,t ) = E+ ( t − z c ) + E− ( t + z c )
z = t = 0, arg = 0
E+(t – z/c)
z = ct
more generally
propagation
arg = 0
z
The position where
arg = 0 moves at velocity c
L2-6
UNIFORM PLANE WAVE IN Z-DIRECTION
Electric Fields (Example):
More specifically, let:
Magnetic Fields:
∇×E
Ey(z,t) = E+(t - z/c) [V/m]
Ey(z,t) = E+ cos ω(t – z/c) = E+ cos (ωt – kz),
where k = ω/c = ω µo εo
Use Faraday’s Law: ∇ × E = −∂B ∂t
xˆ
yˆ
zˆ
= det ∂ ∂x ∂ ∂y ∂ ∂z = − xˆ ∂E+ cos (ωt − kz ) ∂z
0
0
Ex
Ey
Ez
0
0
= − x̂kE+ sin ( ωt − kz )
H ( z,t ) = xˆ (1 µo ) ∫ kE+ sin (ωt − kz ) dt [ A/m]
= − xˆ (kE+ ωµo ) cos (ωt − kz ) = − xˆ (E+ ηo ) cos ( ωt − kz )
k = 1 = µ ε ; η = µ ε 
o o
o
o o
 ω c

L2-7
UNIFORM PLANE WAVE EM FIELDS
EM Wave in z direction:
ˆ +( ωt − kz ) , H ( z,t ) = − xˆ (E+ ηo ) cos (ωt − kz )
E ( z,t ) = yE
x
H ( z,0 )
E ( z,0 )
z
y
Electric energy density
Magnetic energy density
z
L2-8
ACOUSTIC WAVES
A “wave” is a periodic disturbance propagating through a medium
Acoustic Waves:
Velocity
air molecules
Pressure
wave velocity
0
Particle velocity
Kinetic
energy
z
∆ Potential energy
z
Generally, two types of energy are interchanged
L2-9