ele3310: basic electromagnetic theoryivp.ee.cuhk.edu.hk/~ele3310/data/summaryhandout.pdf ·...

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



Outline

1 MathematicsVectors and productsDifferential operatorsIntegrals

2 Electromagneto-StaticsFundamental differential equationsIntegral expressionsElectromagnetism in the matter

3 Time-Varying ElectromagnetismFundamental differential equationsPhasors and Plane Waves




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





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




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




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





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





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





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)





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)





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




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





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





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.





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





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





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




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





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.





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.





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





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.





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





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





Plane waves in lossy media

Propagation constant: γ = jk = α︸︷︷︸attenuation

+j β︸︷︷︸phase

Skin depth: δ =1α

Group velocity: ug =1dβdω





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


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