Download ELE3310: Basic ElectroMagnetic Theory

Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
ELE3310: Basic ElectroMagnetic Theory
A summary for the final examination
Prof. Thierry Blu
EE Department
The Chinese University of Hong Kong
November 2008
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Outline
1
Mathematics
Vectors and products
Differential operators
Integrals
2
Electromagneto-Statics
Fundamental differential equations
Integral expressions
Electromagnetism in the matter
3
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Vectors in three dimensions
Definitions
Notation: bold w (typewriting) or arrowed letter w
~ (handwriting)
Definition: a collection of three scalars (real numbers)
w = (wx , wy , wz ) known as its Cartesian coordinates
Characterization
p
Amplitude: a scalar defined by |w| = wx2 + wy2 + wz2
Direction: a unit vector defined by aw = w/w
Orthonormal bases
Definition: a vector
basis is
a set of threeunit vectors u, v and w
v
u
u
such that u ⊥
,v⊥
and w ⊥
w
w
v
Example: the canonical basis of the 3D space ax , ay and az
Usage: any vector w can be expressed as w = wx ax + wy ay + wz az
Other coordinate systems: cylindrical and spherical
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Scalar and Vector products
Scalar (dot) product
Notation: u · v
this is a scalar number!
Definition: u · v = ux vx + uy vy + uz vz
Characterization
u · v = 0 is equivalent
` to u ⊥ v´
|u · v| = |u| |v| cos angle(u, v)
Vector (cross) product
Notation: u × v
this is a vector!
Definition:
u × v = (uy vz − uz vy )ax + (uz vx − ux vz )ay + (ux vy − uy vx )az
Characterization
u × v = 0 is equivalent to u // v
u × v is always ⊥ to` u and to v´
|u × v| = |u| |v| sin angle(u, v)
Prof. Thierry Blu
u×v
ELE3310: Basic ElectroMagnetic Theory
v
u
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Gradient
Notation: ∇f , using the “nabla” or “del” operator
Definition: ∇f =
∂f
∂x ax
+
∂f
∂y ay
+
∂f
∂z az
this operator acts on scalar functions!
∇f returns a vector function!
Characterization
Always orthogonal to the equisurfaces1 of f (x, y, z)
Indicates the direction of steepest descent
2x
2
2
2
Example: if f (x, y, z) = x + y + z , then ∇f = 2y
2z
1 i.e.,
f (x, y, z) = constant
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Divergence
Notation: div(u) (preferred) or ∇ · u
Definition: div(u) =
∂ux
∂x
+
∂uy
∂y
+
∂uz
∂z
div(u) returns a scalar function!
this operator acts on vector functions!
0.1
0.05
0
−0.05
−0.1
Interpretation: if u is a velocity field, div(u) indicates by how much
elementary volumes are expanded (div(u) > 0) or contracted
(div(u) < 0) in the motion
Example: if u(x, y, z) = xax + yay + zaz , then div(u) = 3
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Rotational or curl
Notation: ∇ × u
Definition: ∇ × u =
∂uz
∂y
+
−
x
ax + ∂u
−
∂z
x
− ∂u
∂y az
∂uy
∂z
∂uy
∂x
∂uz
∂x
ay
this operator acts on vector functions!
∇ × u returns a vector function!
20
18
16
14
12
10
8
6
4
2
0
−2
Interpretation: if u is understood as a velocity field, ∇ × u indicates
how much and around which direction, elementary volumes are
rotating in the motion
Example: if u(x, y, z) = |{z}
Ω ×(xax + yay + zaz ), then ∇ × u = 2Ω
constant vector
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Essential identities
Divergence and curl
It is always true that div(∇ × u) = 0
Conversely, if v is such that div(v) = 0, then there exists u such
that v = ∇ × u
Gradient and curl
It is always true that ∇ × (∇ϕ) = 0
Conversely, if v is such that ∇ × v = 0, then there exists ϕ such
that v = ∇ϕ
Laplace operator
Can be applied to both scalar fields and vector fields.
Notation: ∇2 u (vector) or ∇2 ϕ (scalar)
∂2ϕ ∂2ϕ ∂2ϕ
Scalar case: ∇2 ϕ =
+
+
= div(∇ϕ)
∂x2
∂y 2
∂z 2
Vector case: ∇2 u = ∇2 ux ax + ∇2 uy ay + ∇2 uz az
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Contours and line integrals
A contour is a collection of points indexed by one parameter only.

 x(t) = cos(t)
Example: a helix is obtained by y(t) = sin(t)

z(t) = t
A line integral is an expression of the form
r = x(t), y(t), z(t)
×
w
u(x, y, z) · d`,
where d` =
dx
ax +
dy
dz ay +
az dt
dt
dt
dt
u
A closed contour integral is denoted by and is called the
circulation of the vector field u around this contour.
contour
In electromagnetism exercises, u is often in the same direction (or
othogonal) as d`, with constant modulus. Thus,
z
u · d` = |u| × length(contour)
contour
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Surfaces and surface integrals
A surface is a collection of points indexed by two parameters.

 x(s, t) = sin(s) cos(t)
y(s, t) = sin(s) sin(t)
Example of a sphere:

z(s, t) = cos(s)
A surface integral is an expression of the form
x
u · ds
surface
where ds is the elementary surface vector orthogonal to the surface
A surface integral is called the flux of the vector field u across this
surface
In electromagnetism exercises, u is often in the same direction (or
othogonal) as ds, with constant modulus. Thus,
x
u · ds = |u| × area(surface)
surface
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Vectors and products
Differential operators
Integrals
Stokes’ theorem
Transformation of a closed contour line integral into a surface
integral—i.e., the transformation of a circulation into a flux:
z
x
u · d` =
∇ × u · ds
contour
supported surface
Green’s divergence theorem
Transformation of a closed surface integral into a volume integral:
{
y
u · ds =
div(u) dxdydz
surface
enclosed volume
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Integral expressions
Electromagnetism in the matter
Static electric field
E(x, y, z) satisfies two differential equations
∇×E=0
and
div(E) =
ρ
ε0
if ρ(x, y, z) is the local density of charges.
Equivalently, E(x, y, z) satisfies two integral equations
z
C
E · d` = 0
and
{
S
|
Prof. Thierry Blu
E · ds =
charges inside S
ε0
{z
}
Gauss’ Law
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Integral expressions
Electromagnetism in the matter
Static magnetic field
The magnetic flux density B(x, y, z) satisfies two differential
equations
∇ × B = µ0 J
and div(B) = 0
if J(x, y, z) is the local density of currents.
Equivalently, B(x, y, z) satisfies two integral equations
{
z
B · d` = µ0 × current through C
and
B · ds = 0
S
|C
{z
}
Ampère’s circuital law
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Integral expressions
Electromagnetism in the matter
Static electric potential
E(x, y, z) is related to its potential V (x, y, z) by
E = −∇V
V satisfies ∇2 V = −ρ/ε0 .
Static magnetic potential
B(x, y, z) is related to its vector potential A(x, y, z) by
B=∇×A
A is chosen so that div(A) = 0 and satisfies ∇2 A = −µ0 J.
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Integral expressions
Electromagnetism in the matter
Coulomb’s Law
Explicit expressions of V and E
y
ρ(r0 )
V (r) =
dx0 dy 0 dz 0
4πε0 |r − r0 |
y ρ(r0 ) r − r0
E(r) =
dx0 dy 0 dz 0
4πε0 |r − r0 |3
Biot-Savart Law
Explicit expressions of A and B for circuits (I = current intensity)
µ0 I z
d`0
A(r) =
4π circuit |r − r0 |
µ0 I z
d`0 × (r − r0 )
B(r) =
4π circuit
|r − r0 |3
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Integral expressions
Electromagnetism in the matter
Constitutive equations
Linear relations characterizing the reaction of the matter to the
electromagnetic field
Displacement field: modification of E caused by electric dipoles
div(D) = ρfree replaces div(E) = ρ/ε0 , where D = εE
Magnetic field intensity: modification of E caused by magnetic
dipoles
∇ × H = Jfree replaces ∇ × B = µ0 J, where H = B/µ
Ohm’s law: resistance of the matter to the motion of charged
particles
J = |{z}
σ E
conductivity
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Integral expressions
Electromagnetism in the matter
Boundary conditions
Continuities/discontinuities of the electromagnetic field across the
interface between different matters
Perfect conductor (ρs = surface charge density)

 E = 0 and ρ = 0, inside the conductor
ρs
E=
an ,
on the surface of the conductor

ε |{z}
direction normal to the interface
Perfect dielectric (no free charges/currents): continuity of
the tangential (to the interface) components of E and of H
the normal (to the interface) components of εE and of µH
Conditions still valid for time-varying electromagnetic fields
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
Maxwell’s Equations
Four differential equations coupling E(x, y, z, t) and H(x, y, z, t) and
valid in the matter
Faraday’s law
z
}|
{
∂(µH)
∇×E=−
∂t
div(εE) = ρ
|
{z
}
Gauss’s law
Ampère’s circuital law
z
}|
{
∂(εE)
∇×H=J+
∂t
div(µH) = 0
|
{z
}
no magnetic charges
Reminder: Displacement electric field D and magnetic flux density B
are related to E and H through the constitutive relations
D = εE
Prof. Thierry Blu
and B = µH
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
The equation of continuity
States the conservation of the electric charge in a moving volume
∂ρ
=0
Differential equation: div(J) +
∂t
Integral equation
{
S
J · ds = −
d y
inside
dt
S
ρ(x, y, z, t) dxdydz
i.e., the current flow through S is exactly balanced by the variation
of electric charge inside S.
It is the inconsistency of the statics equations with the equation of
continuity that led J.C. Maxwell to state his famous relations.
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
Energy and power
The electromagnetic field carries energy
µH2
εE2
+
Energy density: W =
2
2
Power flow (Poynting vector): P = E × H
For a non-conductive dielectric medium
div P +
∂W
=0
∂t
states that the electromagnetic power flux through a closed surface is
exactly balanced by the variation of energy density inside this surface.
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
Potentials
µH = B(x, y, z) is still related to its vector potential A(x, y, z) by
B=∇×A
However, now A is not chosen anymore so that div(A) = 0.
E(x, y, z) is now related to the electric potential V (x, y, z) by
E = −∇V −
∂A
∂t
Additionally, when ε and µ are constant the magnetic vector
potential is chosen so that
div(A) + εµ
Prof. Thierry Blu
∂V
=0
∂t
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
The wave equation
In a medium with constant permittivity ε and permeability µ the
potentials satisfy a (second order) wave propagation equation
∂2A
=0
∂t2
∂2V
∇2 V − εµ 2 = 0
∂t
∇2 A − εµ
√
The propagation velocity c is given by c = 1/ εµ.
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
Phasors
Considering an electromagnetic field at frequency ω, its spatial variations
are characterized by a complex-valued vector
E(x, y, z, t) = R E(x, y, z)ejωt H(x, y, z, t) = R H(x, y, z)ejωt
Maxwell’s equations for phasors
∇ × E = −jωµH
div(εE) = ρ
∇ × H = J + jωεE
div(µH) = 0
Wave equation for phasors (with ρ = 0, J = σE and ε, µ constant)
∇2 E + k 2 E = 0
Prof. Thierry Blu
where
k 2 = εµω 2 − jσµω
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
Plane waves
A particular solution of Maxwell’s equations for phasors
E(x, y, z) = E0 e−jk·r
|{z}
constant vector
where k is a (possibly complex) vector.
E and H are orthogonal, and transverse to the direction of
propagation ak
Electric field: k · E = 0
k×E
1
Magnetic field: H =
= ak × E where η is the wave
µω
η
impedance.
Polarizations
Linear: E and H stay parallel to a real vector
elliptical (left- or right-handed): E and H have a complex phase
difference between their components
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
Plane waves in lossy media
Propagation constant: γ = jk = α +j β
|{z}
|{z}
attenuation
phase
1
Skin depth: δ =
α
Group velocity: ug =
1
dβ
dω
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory
Mathematics
Electromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equations
Phasors and Plane Waves
Interfaces
Result of the incidence of a plane wave on a plane separating two media
with different electromagnetic characteristics:
Reflection: a plane wave propagating in the direction symmetric to
incidence with respect to the interface (Snell’s law of reflection)
Transmission: a plane wave propagating in a direction depending
on the relative propagation velocities between the two media (Snell’s
law of
Standing waves: interferences between the incident and reflected
waves in the direction normal to the interface
Reflection/transmission coefficients: obtained by solving for the
reflection and transmission EM fields using the boundary conditions
at the interface
Prof. Thierry Blu
ELE3310: Basic ElectroMagnetic Theory