Microwave Measurement and Beam Instrumentation Course
at Jefferson Laboratory, January 15-26th 2018
Lecture:
Maxwell’s Equations
F. Marhauser
Monday, January 15, 2018
This Lecture
－ This lecture provides theoretical basics useful for follow-up lectures
on resonators and waveguides
－ Introduction to Maxwell’s Equations
•
•
•
•
•
•
Sources of electromagnetic fields
Differential form of Maxwell’s equation
Stokes’ and Gauss’ law to derive integral form of Maxwell’s equation
Some clarifications on all four equations
Time-varying fields  wave equation
Example: Plane wave
－
－
Phase and Group Velocity
Wave impedance
2
Maxwell’s Equations
A dynamical theory of the electromagnetic field
James Clerk Maxwell, F. R. S.
Philosophical Transactions of the Royal Society of London, 1865 155, 459-512,
published 1 January 1865
Maxwell’s Equations
- Originally there were 20 equations
Taken from Longair, M. 2015 ‘...a paper ...I hold to be great guns’: a commentary on Maxwell (1865)
‘A dynamical theory of the electromagnetic field’. Phil. Trans. R. Soc. A 373: 20140473.
Sources of Electromagnetic Fields
－ Electromagnetic fields arise from 2 sources:
•
Electrical charge (Q)
•
Electrical current (𝐼 =
Stationary charge creates electric field
𝑑𝑄
)
𝑑𝑡
Moving charge creates magnetic field
－ Typically charge and current densities are utilized in Maxwell’s equations
to quantify the effects of fields:
•
𝑑𝑄
ρ = 𝑑𝑉 electric charge density – total electric charge per unit volume V
(or 𝑄 =
•
𝐽=
𝜌 𝑑𝑉)
𝑉
𝐼(𝑆)
lim 𝑆 electric
𝑆→0
(or 𝐼 =
𝑆
current density – total electric current per unit area S
𝐽 ∙ 𝑑 𝑆)
－ If either the magnetic or electrical fields vary in time, both fields are
coupled and the resulting fields follow Maxwell’s equations
5
Maxwell’s Equations
Differential Form
𝛻∙𝐸 =
𝜌
𝜖0
(1)
𝛻∙𝐷 =𝜌
(2)
𝛻∙𝐵 =0
Gauss’s law for magnetism
(3)
𝜕𝐵
𝛻×𝐸 = −
𝜕𝑡
Faraday’s law of induction
or
Gauss’s law
𝜕𝐸
𝜕𝐷
or
(4) 𝛻 × 𝐻 = 𝐽 +
𝛻 × 𝐵 = 𝜇0 𝐽 + 𝜇0 𝜖0
𝜕𝑡
𝜕𝑡
- Together with the Lorentz force these equations
𝐹 = 𝑞(𝐸 + v × 𝐵)
form the basic of the classic electromagnetism
𝐷 = 𝜖0 𝐸
𝜖0 =permittivity of free space
𝐵 = 𝜇0 𝐻
µ0 =permeability of free space
Ampère’s law
Lorentz Force
ρ = electric charge density (As/m3)
J = electric current density (A/m2)
D = electric flux density/displacement field (Unit: As/m2)
E = electric field intensity (Unit: V/m)
H = magnetic field intensity (Unit: A/m)
B = magnetic flux density (Unit: Tesla=Vs/m2)
6
Divergence (Gauss’) Theorem
{
div
(𝛻 ∙ 𝐹) 𝑑𝑉 =
𝑉
(𝐹 ∙ 𝑛)𝑑𝑆 =
𝑆
𝐹 ∙ 𝑑𝑆
𝑆
Integral of divergence of vector field (𝐹) over volume V inside closed boundary S equals
outward flux of vector field (𝐹) through closed surface S
𝛻∙𝐹 =
𝜕 𝜕 𝜕
𝜕𝐹𝑥 𝜕𝐹𝑦 𝜕𝐹𝑧
, ,
∙ 𝐹𝑥 , 𝐹𝑦 , 𝐹𝑧 =
+
+
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
7
Curl (Stokes’) Theorem
{
curl
(𝛻 × 𝐹) · 𝑑𝑆 =
𝑆
((𝛻 × 𝐹) ∙ 𝑛)𝑑𝑆 =
𝑆
𝐹 ∙ 𝑑𝑙
𝜕𝑆
Integral of curl of vector field (𝐹) over surface S equals
line integral of vector field (𝐹) over closed boundary dS defined by surface S
𝛻×𝐹 =
𝑖
𝑗
𝑘
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝐹𝑥
𝐹𝑦
𝐹𝑧
=
𝜕𝐹𝑧
𝜕𝑦
−
𝜕𝐹𝑦
𝜕𝑧
𝑖+
𝜕𝐹𝑥
𝜕𝑧
−
𝜕𝐹𝑧
𝜕𝑥
𝑗+
𝜕𝐹𝑦
𝜕𝑧
−
𝜕𝐹𝑥
𝜕𝑦
𝑘
𝜕𝐹𝑦 (𝑥, 𝑦) 𝜕𝐹𝑥 (𝑥, 𝑦)
−
𝑘
𝒆. 𝒈. : 𝑭𝒛 = 𝟎 → 𝛻 × 𝐹 =
𝜕𝑥
𝜕𝑦
Curl vector is perpendicular to surface S ; 𝑘 = 𝑛
𝑆
𝜕𝐹𝑦 (𝑥, 𝑦) 𝜕𝐹𝑥 (𝑥, 𝑦)
(
−
) 𝑑𝑆 =
𝜕𝑥
𝜕𝑦
𝐹𝑥 𝑑𝑥 + 𝐹𝑦 𝑑𝑦
𝜕𝑆
8
Green’s Theorem
Example: Curl (Stokes’) Theorem
curl
{
(𝛻 × 𝐹) · 𝑑𝑆 =
𝑆
((𝛻 × 𝐹) ∙ 𝑛)𝑑𝑆 =
𝑆
𝐹 ∙ 𝑑𝑙
𝜕𝑆
Integral of curl of vector field (𝐹) over surface S equals
line integral of vector field (𝐹) over closed boundary dS defined by surface S
9
Example: Curl (Stokes) Theorem
curl
{
(𝛻 × 𝐹) · 𝑑𝑆 =
𝑆
((𝛻 × 𝐹) ∙ 𝑛)𝑑𝑆 =
𝑆
𝐹 ∙ 𝑑𝑙
𝜕𝑆
Integral of curl of vector field (𝐹) over surface S equals
line integral of vector field (𝐹) over closed boundary dS defined by surface S
Example: Closed line integrals of various vector fields
No curl
Some curl
Stronger curl
10
No net curl
Maxwell’s Equations
Integral Form
Differential Form
𝛻∙𝐷 =𝜌
}
(𝛻 ∙ 𝐹) 𝑑𝑉 =
𝑉
𝐷 ∙ 𝑑𝑆 =
𝐹 ∙ 𝑑𝑆
𝑆
𝑆
𝜌 𝑑𝑉
Gauss’s law
𝑉
Gauss’ theorem
𝛻∙𝐵 =0
𝐵 ∙ 𝑑𝑆 = 0
Gauss’s law for magnetism
𝑆
𝛻×𝐸 = −
𝜕𝐵
𝜕𝑡
𝜕𝐷
𝛻×𝐻 =𝐽+
𝜕𝑡
}
𝐸 ∙ 𝑑𝑙 = −
(𝛻 × 𝐹) · 𝑑 𝑆 =
𝑆
𝐹 ∙ 𝑑𝑙
𝜕𝑆
𝜕𝑆
𝑆
𝜕𝐵
∙ 𝑑𝑆
𝜕𝑡
Stokes’ theorem
𝐻 ∙ 𝑑𝑙 =
𝜕𝑆
𝐷 = 𝜖0 𝐸
𝜖0 =permittivity of free space
𝐵 = 𝜇0 𝐻
µ0 =permeability of free space
𝐽 ∙ 𝑑𝑆 +
𝑆
𝑆
Faraday’s law of induction
𝜕𝐷
∙ 𝑑𝑆
𝜕𝑡
Ampère’s law
ρ = electric charge density (C/m3=As/m3)
J = electric current density (A/m2)
D = electric flux density/displacement field (Unit: As/m2)
E = electric field intensity (Unit: V/m)
H = magnetic field intensity (Unit: A/m)
B = magnetic flux density (Unit: Tesla=Vs/m2)
11
Electric Flux & 1st Maxwell Equation
Definition of Electric Flux
1. Uniform field
Gauss: Integration over closed surface
𝐸 = 𝐸 ∙ 𝑆 = 𝐸 ∙ 𝑛 𝑆 = 𝐸 ∙ 𝑆 ∙ 𝑐𝑜𝑠 𝜃 [𝑉𝑚]
- angle between field and normal vector
to surface matters
𝜖0 𝐸 ∙ 𝑑 𝑆 = 𝜖0  𝑬 =
𝑆
𝜌 𝑑𝑉 =
𝑖
𝑉
𝐸 =
𝑞𝑖
𝑖 𝑞𝑖
; 𝐷 = 𝜖0 𝐸
𝜖0
Example: Metallic plate,
assume only surface
charges on one side
2. Non-Uniform field
𝜖0 𝐸 ∙ 𝑑𝑆 = 𝜖0 𝐸 𝜋𝑅2 =
𝑑𝐸 = 𝐸 ∙ 𝑑𝑆 = 𝐸 ∙ 𝑛 𝑑𝑆
𝑖
𝑆
𝐸 =
𝑄𝑐𝑖𝑟𝑐𝑙𝑒
𝐸 =
𝜖0 𝜋𝑅2
𝐸 ∙ 𝑑𝑆
𝑆
12
𝑞𝑖 = 𝑄𝑐𝑖𝑟𝑐𝑙𝑒
; 𝑐𝑖𝑟𝑐𝑙𝑒 𝑆 = 𝜋𝑅 2
Electric Flux & 1st Maxwell Equation
Definition of Electric Flux
1. Uniform field
Gauss: Integration over closed surface
𝐸 = 𝐸 ∙ 𝑆 = 𝐸 ∙ 𝑛 𝑆 = 𝐸 ∙ 𝑆 ∙ 𝑐𝑜𝑠 𝜃 [𝑉𝑚]
- angle between field and normal vector
to surface matters
𝜖0 𝐸 ∙ 𝑑 𝑆 = 𝜖0  𝑬 =
𝑆
𝐸 =
𝜌 𝑑𝑉 =
𝑞𝑖
𝑖
𝑉
𝑖 𝑞𝑖
; 𝐷 = 𝜖0 𝐸
𝜖0
Example: Capacitor
2. Non-Uniform field
𝐸 =0
𝐸 =0
𝑑𝐸 = 𝐸 ∙ 𝑑𝑆 = 𝐸 ∙ 𝑛 𝑑𝑆
𝐸 =
𝐸 =
𝐸 ∙ 𝑑𝑆
𝑆
13
𝑄𝑐𝑖𝑟𝑐𝑙𝑒 −𝑄𝑐𝑖𝑟𝑐𝑙𝑒
+
=0
𝜖0
𝜖0
Electric Flux & 1st Maxwell Equation
Examples of non-uniform fields
Point charge Q
Add charges
𝜖0 𝐸 ∙ 𝑑 𝑆 =
𝑆
𝜌 𝑑𝑉 =
𝑞𝑖 = 𝑄𝑠𝑝ℎ𝑒𝑟𝑒
𝑖
𝑉
𝐸
𝐸
Integration of over closed spherical surface S
𝜖0 𝐸 ∙ 𝑑𝑆 = 𝜖0 𝑬(𝑟)4𝜋𝑟 2 = 𝑄
𝑆
; 𝑠𝑝ℎ𝑒𝑟𝑒 𝑆 = 4𝜋𝑟 2
𝑄
𝑬(𝑟) =
∙𝒓
𝜖0 4𝜋𝑟 2
pointing out radially
𝐸 =
𝑖 𝑞𝑖
𝜖0
=
𝑞 −𝑞
+
=0
𝜖0 𝜖0
𝐸 =
3𝑞 𝑄𝑠𝑝ℎ𝑒𝑟𝑒
=
𝜖0
𝜖0
Principle of Superposition holds:
𝐸(𝑟) =
14
1
𝜖0 4𝜋
𝑞1
𝑟𝑐1 − 𝑟
𝑟 +
2 𝑐1
𝑞2
𝑟𝑐2 − 𝑟
𝑟 +
2 𝑐2
𝑞3
𝑟𝑐3 − 𝑟
𝑟 +⋯
2 𝑐3
Magnetic Flux & 2nd Maxwell Equation
Definition of Magnetic Flux
Uniform field
Gauss: Integration over closed surface
𝐵 = 𝐵 ∙ 𝑆 = 𝐵 ∙ 𝑛 𝑆 = 𝐵 ∙ 𝑆 𝑐𝑜𝑠 𝜃 [𝑊𝑏 = 𝑉𝑠]
𝐵 ∙ 𝑑𝑆 = 0
𝑀 = 0
𝑆
- There are no magnetic monopoles
- All magnetic field lines form loops
𝐵
𝐵
Non-Uniform field
𝑑𝐵 = 𝐵 ∙ 𝑑 𝑆 = 𝐵 ∙ 𝑛 𝑑𝑆
𝐵 =
𝐵 ∙ 𝑑𝑆
𝑆
Closed surface:
Flux lines out = flux lines in
What about this case?
Flux lines out > flux lines in ?
- No. In violation of 2nd Maxwell’s law, i.e.
integration over closed surface, no holes allowed
- Also: One cannot split magnets into separate
poles, i.e. there always will be a
15 North and South pole
Magnetic Flux & 3rd Maxwell Equation
If integration path is not changing in time
𝐸 ∙ 𝑑𝑙 = −
𝜕𝑆
𝑑
𝑑𝑡
𝐵 ∙ 𝑑𝑆 = −
𝑆
𝑑𝐵
𝑑𝑡
; 𝐵 =
𝑆
𝐵 ∙ 𝑑𝑆
- Change of magnetic flux induces an electric field along a closed loop
- Note: Integral of electrical field over closed loop may be non-zero,
when induced by a time-varying magnetic field
- Electromotive force (EMF) Ɛ:
𝐵(𝑡 = 𝑡1 )
𝐸 ∙ 𝑑𝑙 [𝑉]
Ɛ=
𝜕𝑆
- Ɛ equivalent to energy per unit
charge traveling once around loop
Faraday’s law of induction
16
Magnetic Flux & 3rd Maxwell Equation
If integration path is not changing in time
𝐸 ∙ 𝑑𝑙 = −
𝜕𝑆
𝑑
𝑑𝑡
𝐵 ∙ 𝑑𝑆 = −
𝑆
𝑑𝐵
𝑑𝑡
; 𝐵 =
𝑆
𝐵 ∙ 𝑑𝑆
- Change of magnetic flux induces an electric field along a closed loop
- Note: Integral of electrical field over closed loop may be non-zero,
when induced by a time-varying magnetic field
𝐵(𝑡)
- Electromotive force (EMF) Ɛ:
𝐸 ∙ 𝑑𝑙 [𝑉]
Ɛ=
𝜕𝑆
- Ɛ equivalent to energy per unit
charge traveling once around loop
- or voltage measured at end of open loop
Faraday’s law of induction
17
Ampère's (circuital) Law or 4th Maxwell Equation
Right hand side of equation:
𝐻 ∙ 𝑑𝑙 =
𝑆
𝑆
-
Note that 𝑆 𝐽 ∙ 𝑑 𝑆 is a surface integral, but S may
have arbitrary shape as long as ∂S is its closed
boundary
{
𝜕𝑆
𝐽 ∙ 𝑑𝑆 +
𝜕𝐷
∙ 𝑑𝑆
𝜕𝑡
𝐽 ∙ 𝑑𝑆 = 𝐼
conduction current I
Example: 𝐵
Left hand side of equation:
𝑆
-
What if there is a capacitor?
-
While current is still be flowing (charging capacitor):
𝐵 ∙ 𝑑 𝑙 = 𝑩 𝑟 2𝜋𝑟 = 𝜇0 𝐼
𝜕𝑆
; 𝐵 = 𝜇0 𝐻
tangential to a circle at any
radius r of integration
𝑆
𝑑𝑄
𝐽 ∙ 𝑑𝑆 = 𝐼 =
𝑑𝑡
; 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝐶 = 2𝜋𝑟
𝜇0 𝐼
|𝑩 𝑟 | =
2𝜋𝑟
18
Ampère's (circuital) Law or 4th Maxwell Equation
-
𝐻 ∙ 𝑑𝑙 =
𝑆
𝑆
𝜕𝐷
∙ 𝑑𝑆
𝜕𝑡
𝐽 ∙ 𝑑𝑆 = 0
{
𝜕𝑆
𝐽 ∙ 𝑑𝑆 +
But one may also place integration surface S between
plates  current does not flow through surface here
𝑆
conduction current I
Example: 𝐵
𝑤ℎ𝑖𝑙𝑒 𝐵 ≠ 0 ?
- This is when the displacement field is required as a
corrective 2nd source term for the magnetic fields
𝑆
𝐵 ∙ 𝑑 𝑙 = 𝑩 𝑟 2𝜋𝑟 = 𝜇0 𝐼
𝜕𝑆
; 𝐵 = 𝜇0 𝐻
tangential to a circle at any
radius r of integration
; 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝐶 = 2𝜋𝑟
𝜇0 𝐼
|𝑩 𝑟 | =
2𝜋𝑟
𝜕𝑆
𝑑𝑄
𝑑𝐸
𝐷 ∙ 𝑑𝑆 =
= 𝐼𝐷 = 𝜖0
𝑑𝑡
𝑑𝑡
𝑆
; Gauss’s law
displacement current I
𝑑𝐸
𝐻 ∙ 𝑑 𝑙 = 𝐼 + 𝜖0
𝑑𝑡
{
Left hand side of equation:
𝜕𝐷
𝑑
∙ 𝑑𝑆 =
𝜕𝑡
𝑑𝑡
𝐸
19
𝐵
|𝑩 𝑟 | =
𝜇0 𝐼𝐷
2𝜋𝑟
Presence of Resistive Material
𝐽 ∙ 𝑑𝑆
𝑆
𝑆
{
𝜕𝑆
conduction current
-
displacement current
In resistive materials the current density J is proportional to the electric field
𝐽 = 𝐸 =
-
𝜕𝐷
∙ 𝑑𝑆
𝜕𝑡
+
{
𝐻 ∙ 𝑑𝑙 =
1
𝐸
𝜌
with  the electric conductivity (1/(Ω·m) or S/m), respectively
=1/ the electric resistivity (Ω·m)
Generally (ω, T) is a function of frequency and temperature
20
𝐵
Time-Varying E-Field in Free Space
-
Assume charge-free, homogeneous, linear, and isotropic medium
-
We can derive a wave equation:
𝛻×𝐸 = −
𝜕𝐵
𝜕𝑡
;Faraday’s law of induction
𝜕𝐵
𝜕
𝛻 × 𝛻 × 𝐸 = −𝛻 ×
=−
𝛻×𝐵
𝜕𝑡
𝜕𝑡
; || curl
𝛻 2 = Δ = Laplace operator
𝛻 𝛻 ∙ 𝐸 − 𝛻 2 𝐸 = −𝜇
𝜕
𝜕𝐷
𝐽+
𝜕𝑡
𝜕𝑡
;𝛻×𝐻 =𝐽+
𝛻
; curl of curl 𝛻 × 𝛻 × 𝐴 = 𝛻 𝛻 ∙ 𝐴 − 𝛻 2 𝐴
𝜕𝐷
𝜕𝑡
; Ampère’s law
𝜌
𝜕
𝜕𝐸
− 𝛻 2 𝐸 = −𝜇
𝐸 + 𝜖
𝜖0
𝜕𝑡
𝜕𝑡
; Gauss’s law
𝛻2𝐸
𝜕2𝐸
− 𝜇𝜖 2 = 0
𝜕𝑡
; 𝐽 = 𝐸
Homogeneous wave equation
; we presumed no charge
21
Time-Varying B-Field in Free Space
-
Assume charge-free, homogeneous, linear, and isotropic medium
-
We can derive a wave equation:
𝜕𝐸
𝛻 × 𝐵 = 𝜇𝐽 + 𝜇𝜖
𝜕𝑡
; Ampère’s law
𝛻 × 𝛻 × 𝐵 = 𝜇 𝛻 × 𝐽 + 𝜇𝜖 𝛻 ×
𝛻 𝛻∙𝐵 −
𝛻 2𝐵
𝜕𝐸
𝜕𝑡
; || curl
𝜕2𝐵
= 𝜇 𝛻 × 𝐽 − 𝜇𝜖 2
𝜕𝑡
; curl of curl 𝛻 × 𝛻 × 𝐴 = 𝛻 𝛻 ∙ 𝐴 − 𝛻 2 𝐴
; Faraday’s law
𝛻2𝐵
𝜕2𝐵
− 𝜇𝜖 2 = 0
𝜕𝑡
Similar homogeneous wave equation as for E-Field
; Gauss’s law for magnetism 𝛻 ∙ 𝐵 = 0
; no moving charge (𝐽=0)
22
Time-Harmonic Fields
- In many cases one has to deal with purely harmonic fields (~𝑒 𝑖𝜔𝑡 )
𝜕2𝐵
− 𝜇𝜖 2 = 0
𝜕𝑡
𝛻 2 𝐵 = −𝜇𝜖𝜔2 𝐵
𝜕2𝐸
𝛻 𝐸 − 𝜇𝜖 2 = 0
𝜕𝑡
𝛻 2 𝐸 = −𝜇𝜖𝜔2 𝐸
𝛻2𝐵
2
23
Example: Plane Wave in Free Space
𝐴(𝑟, 𝑡) = 𝐴0 ∙ 𝑒 −𝑖 𝑘∙𝑟 ∙ 𝑒 𝑖𝜔𝑡 = 𝐴0 ∙ 𝑒 𝑖(𝜔𝑡−𝑘∙𝑟)
-
𝐴(𝑟, 𝑡) = 𝑅𝑒 𝐴
k is a wave vector pointing in direction of wave propagation
Wave is unconstrained in plane orthogonal to wave direction, i.e. has surfaces of constant phase
(wavefronts), wave vector k is perpendicular to the wavefront
-
Magnitude of field (whether it is E or B) is constant everywhere on plane, but varies with time and
in direction of propagation
- One may align propagation of wave (k) with z-direction, which simplifies the equation
- In Cartesian coordinates: 𝐴(𝑥, 𝑦, 𝑧, 𝑡) = 𝐴0 ∙ 𝑒 −𝑖𝑘𝑧 ∙ 𝑒 𝑖𝜔𝑡
-
Applying homogeneous wave equation 𝛻 2 𝐴 = −𝜇𝜖𝜔2 𝐴 (with 𝛻 2 𝐴 =
𝛻2𝐴
-
=
−𝑘 2 𝐴
=
−𝜇𝜖𝜔2 𝐴
We know speed of light in linear medium:
𝑘2
=
𝘷=
𝜇𝜖𝜔2
1
𝜇𝜖
𝜕𝐴𝑥
𝜕𝑥 2
+
𝜕𝐴𝑦
𝜕𝑦
2 +
𝜕𝐴𝑧
𝜕𝑧 2
2
𝜔
𝑘 2 = 𝜇𝜖𝜔2 = 2
𝘷
Example: Plane Wave in Free Space
Wikipedia CC BY-SA 2.0
Example: Plane Wave in Free Space
- Acknowledging that k is generally a vector: 𝑘 = 𝑘 ∙ 𝑘𝑧
𝑘2
𝜔2
𝘷2
- Inserting the just derived equation
= , i.e. a dependency with the angular frequency,
we can denote the relation of k with the wavelength
2𝜋𝑓
2𝜋
𝑘=
𝑘 =
𝑘
𝘷 𝑧
λ 𝑧
2𝜋 2𝜋𝑓
𝑘=
=
λ
𝘷
k is the wavenumber [1/m]
𝘷𝑝ℎ
𝜔
1
1
≡ =
= 𝑐0
𝑘
𝜇𝜖
𝜇𝑟 𝜖𝑟
𝘷𝑔𝑟
𝑑𝜔
1
≡
=
= 𝘷𝑝ℎ
𝑑𝑘
𝜇𝜖
Phase velocity
Group velocity = Phase velocity = speed of light
26
Wave Impedance
- Furthermore for plane wave, due to 3rd Maxwell equation we know that magnetic field is
orthogonal to electrical field and can derive for time-harmonic field:
𝜕𝐻
= −𝜇𝜔𝐻
𝜕𝑡
- Considering the absence of charges in free space and 4th Maxwell equation, we find:
𝛻 × 𝐸 = −𝜇
𝜕𝐸
= 𝜀𝜔𝐸
𝜕𝑡
- We then can find for the electrical field components considering
𝛻×𝐻 =𝜀
;𝛻 × 𝐹 =
−
𝜕𝐹𝑧
𝜕𝑦
𝜕𝐸𝑦
𝜕𝑧
−
𝜕𝐹𝑦
𝜕𝑧
𝑖+
𝜕𝐹𝑥
𝜕𝑧
−
𝜕𝐹𝑧
𝜕𝑥
𝑗+
𝜕𝐹𝑦
𝜕𝑧
−
𝜕𝐹𝑥
𝜕𝑦
𝑘
𝜕𝐸
𝑖 + 𝜕𝑧𝑥 𝑗 = −𝑖𝜇𝜔𝐻 = −𝑖𝜇𝜔𝐻𝑥 𝑖 − 𝑖𝜇𝜔𝐻𝑦 𝑗
- Similarly for the magnetic field considering
−
-
𝜕𝐻𝑦
𝜕𝑧
𝑖
𝜕𝐻
+ 𝜕𝑧𝑥 𝑗
= 𝑖𝜀𝜔𝐸 = 𝑖𝜀𝜔𝐸𝑥 𝑖 + 𝑖𝜀𝜔𝐸𝑦 𝑗
𝜕𝐸𝑦
= 𝑖𝜇𝜔𝐻𝑥
𝜕𝑧
𝜕𝐸𝑥
= −𝑖𝜇𝜔𝐻𝑦
𝜕𝑧
𝜕𝐻𝑦
= −𝑖𝜀𝜔𝐸𝑥
𝜕𝑧
𝜕𝐻𝑥
= 𝑖𝜀𝜔𝐸𝑦
𝜕𝑧
All field components are orthogonal to propagation direction
 this means that the plane wave is a Transverse-Electric-Magnetic (TEM) wave
27
Wave Impedance
- We obtained two sets of independent equations, that lead to two linearly independent solutions
1a)
1b)
2a)
2b)
𝜕𝐻𝑦
= −𝑖𝜀𝜔𝐸𝑥
𝜕𝑧
𝜕𝐸𝑥
= −𝑖𝜇𝜔𝐻𝑦
𝜕𝑧
𝜕𝐸𝑦
= 𝑖𝜇𝜔𝐻𝑥
𝜕𝑧
𝜕𝐻𝑥
= 𝑖𝜀𝜔𝐸𝑦
𝜕𝑧
- The wave equation for the electric field components yields:
𝜕 2 𝐸𝑥
= −𝑘 2 𝐸𝑥
2
𝜕𝑥
𝜕 2 𝐸𝑦
= −𝑘 2 𝐸𝑦
2
𝜕𝑥
- Utilizing the Ansatz:
;𝛻 2 𝐴 =
𝜕𝐴𝑥
𝜕𝑥 2
𝜕𝐴𝑦
𝜕𝐴
+ 𝜕𝑦2 + 𝜕𝑧 2𝑧
𝐸𝑥 = 𝐸𝑥,𝑝 𝑒 −𝑖𝑘𝑧 + 𝐸𝑥,𝑟 𝑒 +𝑖𝑘𝑧
𝐸𝑦 = 𝐸𝑦,𝑝 𝑒 −𝑖𝑘𝑧 + 𝐸𝑦,𝑟 𝑒 +𝑖𝑘𝑧
we can derive the corresponding magnetic field components:
𝑘
𝐻𝑦 =
𝐸𝑥,𝑝 𝑒 −𝑖𝑘𝑧 − 𝐸𝑥,𝑟 𝑒 +𝑖𝑘𝑧
𝜇𝜔
𝑘
𝐻𝑥 = −
𝐸𝑦,𝑝 𝑒 −𝑖𝑘𝑧 − 𝐸𝑦,𝑟 𝑒 +𝑖𝑘𝑧
𝜇𝜔
;1a)
𝑘
- Using the substitution 𝑍 = 𝜇𝜔 :
𝐻𝑦 =
1
𝐸𝑥,𝑝 𝑒 −𝑖𝑘𝑧 − 𝐸𝑥,𝑟 𝑒 +𝑖𝑘𝑧
𝑍
𝜇𝜔
𝜇𝜔
𝑍=
=
=
𝑘
𝜇𝜖𝜔
𝜇
≈
𝜖
𝜇0
𝜀0
Z is the wave impedance in Ohms
1
𝐻𝑥 = − 𝐸𝑦,𝑝 𝑒 −𝑖𝑘𝑧 − 𝐸𝑦,𝑟 𝑒 +𝑖𝑘𝑧
𝑍
𝜇𝑟
𝜀𝑟
;
𝑘2
=
28
𝜇𝜖𝜔2
𝑍0 =
𝜇0
≈ 120𝜋 Ω ≈ 376.73 Ω
𝜀0
vacuum impedance
;2a)
Appendix
Presence of Dielectric Material
- For linear materials
𝜖 = 𝜖𝑟 𝜖0
r is relative permittivity
𝜇 = 𝜖𝑟 𝜖0
r is relative permeability
- Particularly, the displacement current was conceived by Maxwell as the separation (movement)
of the (bound) charges due to the polarization of the medium (bound charges slightly separate
inducing electric dipole moment)
P is polarization density (‘polarization’) is the density of
𝐷 = 𝜖𝐸 = 𝜖0 𝐸 + 𝑃
permanent and induced electric dipole moments
- For homogeneous, linear isotropic dielectric material
𝑃 = 𝜖0 (𝜖𝑟 −1)𝐸
- For anisotropic dielectric material
(𝜖𝑟 −1) = c𝑒
c𝑒 = electric susceptibility
𝑃=
𝜖0 c𝑖,𝑗 𝐸𝑗
𝑗
- Material may be non-linear, i.e. P is not proportional to E( hysteresis in ferroelectric materials)
- Generally P(ω) is a function of frequency, since the bound charges cannot act immediately to
the applied field (c𝑒 (ω)  this gives rise to losses
30
Similar Expressions for Magnetization
- For magnetic fields the presence of magnetic material can give rise to a magnetization by
microscopic electric currents or the spin of electrons
- Example: If a ferromagnet (e.g. iron) is exposed to a magnetic field, the microscopic dipoles
align with the field and remain aligned to some extent when the magnetic field vanishes
(magnetization vector M)  a non-linear dependency between H and M occurs
- Magnetization may occur in directions other than that of the applied magnetic field
- The magnetization vector describes the density of the permanent or induced magnetic dipole
moments in a magnetic material
𝐵 = 𝜇0 ∙ 𝐻 = 𝜇0 𝐻 + 𝑀 = 𝜇0 1 + 𝒳𝑣 𝐻
- Herein 𝒳𝑣 is the magnetic susceptibility, which described whether is material if appealed or
retracted by the presence of a magnetic field
- The relative permeability of the material can then be denoted as:
𝜇𝑟 = 1 + 𝒳𝑣