MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
ELE3310: Basic ElectroMagnetic TheoryA summary for the final examination
Prof. Thierry Blu
EE DepartmentThe Chinese University of Hong Kong
November 2008
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Outline
1 MathematicsVectors and productsDifferential operatorsIntegrals
2 Electromagneto-StaticsFundamental differential equationsIntegral expressionsElectromagnetism in the matter
3 Time-Varying ElectromagnetismFundamental differential equationsPhasors and Plane Waves
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
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
Amplitude: a scalar defined by |w| =pw2
x + w2y + w2
z
Direction: a unit vector defined by aw = w/w
Orthonormal bases
Definition: a vector basis is a set of three unit vectors u, v and w
such that u ⊥
vw , v ⊥
uw and w ⊥
uv
Example: the canonical basis of the 3D space ax, ay and azUsage: any vector w can be expressed as w = wxax +wyay +wzazOther coordinate systems: cylindrical and spherical
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
Scalar and Vector products
Scalar (dot) product
Notation: u · v this is a scalar number!
Definition: u · v = uxvx + uyvy + uzvz
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 = (uyvz − uzvy)ax + (uzvx − uxvz)ay + (uxvy − uyvx)az
vu
u×vCharacterization
u× v = 0 is equivalent to u //vu× v is always ⊥ to u and to v|u× v| = |u| |v| sin
`angle(u,v)
´Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
Gradient
Notation: ∇f , using the “nabla” or “del” operator
Definition: ∇f = ∂f∂xax + ∂f
∂yay + ∂f∂z az
this operator acts on scalar functions!
∇f returns a vector function!
Characterization
Always orthogonal to the equisurfaces1of f(x, y, z)Indicates the direction of steepest descent
Example: if f(x, y, z) = x2 + y2 + z2, then ∇f =
∣∣∣∣∣∣2x2y2z
1i.e., f(x, y, z) = constantProf. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
Divergence
Notation: div(u) (preferred) or ∇ · uDefinition: div(u) = ∂ux
∂x + ∂uy
∂y + ∂uz
∂z
−0.1
−0.05
0
0.05
0.1
this operator acts on vector functions!
div(u) returns a scalar function!
Interpretation: if u is a velocity field, div(u) indicates by how muchelementary 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
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
Rotational or curl
Notation: ∇× u
Definition: ∇× u =(∂uz
∂y −∂uy
∂z
)ax +
(∂ux
∂z −∂uz
∂x
)ay
+(∂uy
∂x −∂ux
∂y
)az
this operator acts on vector functions!
∇× u returns a vector function!
Interpretation: if u is understood as a velocity field, ∇× u indicateshow much and around which direction, elementary volumes arerotating in the motion
−2
0
2
4
6
8
10
12
14
16
18
20
Example: if u(x, y, z) = Ω︸︷︷︸constant vector
×(xax + yay + zaz), then ∇× u = 2Ω
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
Essential identities
Divergence and curl
It is always true that div(∇× u) = 0Conversely, if v is such that div(v) = 0, then there exists u suchthat v = ∇× u
Gradient and curl
It is always true that ∇× (∇ϕ) = 0Conversely, if v is such that ∇× v = 0, then there exists ϕ suchthat v = ∇ϕ
Laplace operator
Can be applied to both scalar fields and vector fields.
Notation: ∇2u (vector) or ∇2ϕ (scalar)
Scalar case: ∇2ϕ =∂2ϕ
∂x2+∂2ϕ
∂y2+∂2ϕ
∂z2= div(∇ϕ)
Vector case: ∇2u = ∇2ux ax +∇2uy ay +∇2uz az
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
Contours and line integrals
A contour is a collection of points indexed by one parameter only.
r =(x(t), y(t), z(t)
)×
Example: a helix is obtained by
x(t) = cos(t)y(t) = sin(t)z(t) = t
A line integral is an expression of the form
w
contouru(x, y, z) · d`, where d` =
(dxdt
ax +dydt
ay +dzdt
az)dt
A closed contour integral is denoted byu
and is called thecirculation of the vector field u around this contour.
In electromagnetism exercises, u is often in the same direction (orothogonal) as d`, with constant modulus. Thus,
z
contouru · d` = |u| × length(contour)
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
Surfaces and surface integrals
A surface is a collection of points indexed by two parameters.
Example of a sphere:
x(s, t) = sin(s) cos(t)y(s, t) = sin(s) sin(t)z(s, t) = cos(s)
A surface integral is an expression of the form
x
surfaceu · ds
where ds is the elementary surface vector orthogonal to the surface
A surface integral is called the flux of the vector field u across thissurface
In electromagnetism exercises, u is often in the same direction (orothogonal) as ds, with constant modulus. Thus,
x
surfaceu · ds = |u| × area(surface)
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Vectors and productsDifferential operatorsIntegrals
Stokes’ theorem
Transformation of a closed contour line integral into a surfaceintegral—i.e., the transformation of a circulation into a flux:
z
contouru · d` =
x
supported surface∇× u · ds
Green’s divergence theorem
Transformation of a closed surface integral into a volume integral:
surfaceu · ds =
y
enclosed volumediv(u) dxdydz
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsIntegral expressionsElectromagnetism 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
CE · d` = 0 and
SE · ds =
charges inside Sε0︸ ︷︷ ︸
Gauss’ Law
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsIntegral expressionsElectromagnetism in the matter
Static magnetic field
The magnetic flux density B(x, y, z) satisfies two differentialequations
∇×B = µ0J 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
CB · d` = µ0 × current through C︸ ︷︷ ︸
Ampere’s circuital law
and
SB · ds = 0
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsIntegral expressionsElectromagnetism in the matter
Static electric potential
E(x, y, z) is related to its potential V (x, y, z) by
E = −∇V
V satisfies ∇2V = −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 ∇2A = −µ0J.
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsIntegral expressionsElectromagnetism in the matter
Coulomb’s Law
Explicit expressions of V and E
V (r) =y ρ(r′)
4πε0|r− r′|dx′dy′dz′
E(r) =y ρ(r′)
4πε0r− r′
|r− r′|3dx′dy′dz′
Biot-Savart Law
Explicit expressions of A and B for circuits (I = current intensity)
A(r) =µ0I
4π
z
circuit
d`′
|r− r′|
B(r) =µ0I
4π
z
circuit
d`′ × (r− r′)|r− r′|3
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsIntegral expressionsElectromagnetism in the matter
Constitutive equations
Linear relations characterizing the reaction of the matter to theelectromagnetic 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 magneticdipoles
∇×H = Jfree replaces ∇×B = µ0J, where H = B/µ
Ohm’s law: resistance of the matter to the motion of chargedparticles
J = σ︸︷︷︸conductivity
E
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsIntegral expressionsElectromagnetism in the matter
Boundary conditions
Continuities/discontinuities of the electromagnetic field across theinterface between different matters
Perfect conductor (ρs = surface charge density)E = 0 and ρ = 0, inside the conductor
E =ρsε
an︸︷︷︸direction normal to the interface
, on the surface of the conductor
Perfect dielectric (no free charges/currents): continuity of
the tangential (to the interface) components of E and of Hthe 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
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors and Plane Waves
Maxwell’s Equations
Four differential equations coupling E(x, y, z, t) and H(x, y, z, t) andvalid in the matter
Faraday’s law︷ ︸︸ ︷∇×E = −∂(µH)
∂t
Ampere’s circuital law︷ ︸︸ ︷∇×H = J +
∂(εE)∂t
div(εE) = ρ︸ ︷︷ ︸Gauss’s law
div(µH) = 0︸ ︷︷ ︸no magnetic charges
Reminder: Displacement electric field D and magnetic flux density Bare related to E and H through the constitutive relations
D = εE and B = µH
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors and Plane Waves
The equation of continuity
States the conservation of the electric charge in a moving volume
Differential equation: div(J) +∂ρ
∂t= 0
Integral equation
SJ · ds = − d
dt
y
inside Sρ(x, y, z, t) dxdydz
i.e., the current flow through S is exactly balanced by the variationof electric charge inside S.
It is the inconsistency of the statics equations with the equation ofcontinuity that led J.C. Maxwell to state his famous relations.
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors and Plane Waves
Energy and power
The electromagnetic field carries energy
Energy density: W =εE2
2+µH2
2Power flow (Poynting vector): P = E×H
For a non-conductive dielectric medium
div P +∂W
∂t= 0
states that the electromagnetic power flux through a closed surface isexactly balanced by the variation of energy density inside this surface.
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors 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 vectorpotential is chosen so that
div(A) + εµ∂V
∂t= 0
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors and Plane Waves
The wave equation
In a medium with constant permittivity ε and permeability µ thepotentials satisfy a (second order) wave propagation equation
∇2A− εµ∂2A∂t2
= 0
∇2V − εµ∂2V
∂t2= 0
The propagation velocity c is given by c = 1/√εµ.
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors and Plane Waves
Phasors
Considering an electromagnetic field at frequency ω, its spatial variationsare characterized by a complex-valued vector
E(x, y, z, t) = RE(x, y, z)ejωt
H(x, y, z, t) = R
H(x, y, z)ejωt
Maxwell’s equations for phasors
∇×E = −jωµH ∇×H = J + jωεEdiv(εE) = ρ div(µH) = 0
Wave equation for phasors (with ρ = 0, J = σE and ε, µ constant)
∇2E + k2E = 0 where k2 = εµω2 − jσµω
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors and Plane Waves
Plane waves
A particular solution of Maxwell’s equations for phasors
E(x, y, z) = E0︸︷︷︸constant vector
e−jk·r
where k is a (possibly complex) vector.
E and H are orthogonal, and transverse to the direction ofpropagation ak
Electric field: k ·E = 0
Magnetic field: H =k×E
µω=
1
ηak ×E where η is the wave
impedance.
Polarizations
Linear: E and H stay parallel to a real vectorelliptical (left- or right-handed): E and H have a complex phasedifference between their components
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors and Plane Waves
Plane waves in lossy media
Propagation constant: γ = jk = α︸︷︷︸attenuation
+j β︸︷︷︸phase
Skin depth: δ =1α
Group velocity: ug =1dβdω
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory
MathematicsElectromagneto-Statics
Time-Varying Electromagnetism
Fundamental differential equationsPhasors and Plane Waves
Interfaces
Result of the incidence of a plane wave on a plane separating two mediawith different electromagnetic characteristics:
Reflection: a plane wave propagating in the direction symmetric toincidence with respect to the interface (Snell’s law of reflection)
Transmission: a plane wave propagating in a direction dependingon the relative propagation velocities between the two media (Snell’slaw of
Standing waves: interferences between the incident and reflectedwaves in the direction normal to the interface
Reflection/transmission coefficients: obtained by solving for thereflection and transmission EM fields using the boundary conditionsat the interface
Prof. Thierry Blu ELE3310: Basic ElectroMagnetic Theory