ELE3310: Basic ElectroMagnetic Theory

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1 A summary for the final examination EE Department The Chinese University of Hong Kong November 2008

2 Outline Mathematics 1 Mathematics Vectors and products Differential operators Integrals 2 Integral expressions Electromagnetism in the matter 3 Phasors and Plane Waves

3 Vectors in three dimensions Vectors and products Differential operators Integrals Definitions Notation: bold w (typewriting) or arrowed letter w (handwriting) Definition: a collection of three scalars (real numbers) w = (w x, w y, w z ) known as its Cartesian coordinates Characterization Amplitude: a scalar defined by w = p w 2 x + w 2 y + w 2 z Direction: a unit vector defined by a w = w/w Orthonormal bases Definition: a vector { basis is { a set of three{ unit vectors u, v and w v u u such that u w, v w and w v Example: the canonical basis of the 3D space a x, a y and a z Usage: any vector w can be expressed as w = w x a x + w y a y + w z a z Other coordinate systems: cylindrical and spherical

4 Scalar and Vector products Vectors and products Differential operators Integrals Scalar (dot) product Notation: u v this is a scalar number! Definition: u v = u x v x + u y v y + u z v z 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 = (u y v z u z v y )a x + (u z v x u x v z )a y + (u x v y u y v x )a z Characterization u v 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) v u

5 Vectors and products Differential operators Integrals Gradient Notation: f, using the nabla or del operator Definition: f = f x a x + f y a y + f z a z this operator acts on scalar functions! f returns a vector function! Characterization Always orthogonal to the equisurfaces 1 of f(x, y, z) Indicates the direction of steepest descent 2x Example: if f(x, y, z) = x 2 + y 2 + z 2, then f = 2y 2z 1 i.e., f(x, y, z) = constant

6 Vectors and products Differential operators Integrals Divergence Notation: div(u) (preferred) or u Definition: div(u) = ux x + uy y + uz z this operator acts on vector functions! div(u) returns a scalar function! 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) = xa x + ya y + za z, then div(u) = 3

7 Vectors and products Differential operators Integrals Rotational or curl Notation: u Definition: u = ( u z y ( uy + x uy z ) a x + ux y ) a z this operator acts on vector functions! u returns a vector function! ( u x z ) uz x a y 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) = }{{} Ω (xa x + ya y + za z ), then u = 2Ω constant vector

8 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) Scalar case: 2 ϕ = 2 ϕ x ϕ y ϕ z 2 = div( ϕ) Vector case: 2 u = 2 u x a x + 2 u y a y + 2 u z a z

9 Vectors and products Differential operators Integrals Contours and line integrals A contour is a collection of points indexed by one parameter only. r = ( x(t), y(t), z(t) ) 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 contour u(x, y, z) dl, where ( dx dl = dt a x + dy dt a y + dz ) dt a z dt A closed contour integral is denoted by and is called the circulation of the vector field u around this contour. In electromagnetism exercises, u is often in the same direction (or othogonal) as dl, with constant modulus. Thus, u dl = u length(contour) contour

10 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) Example of a sphere: y(s, t) = sin(s) sin(t) z(s, t) = cos(s) A surface integral is an expression of the form surface u 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 this surface In electromagnetism exercises, u is often in the same direction (or othogonal) as ds, with constant modulus. Thus, u ds = u area(surface) surface

11 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: contour u dl = supported surface u ds Green s divergence theorem Transformation of a closed surface integral into a volume integral: surface u ds = div(u) dxdydz enclosed volume

12 Integral expressions Electromagnetism in the matter Static electric field E(x, y, z) satisfies two differential equations E = 0 and div(e) = ρ ε 0 if ρ(x, y, z) is the local density of charges. Equivalently, E(x, y, z) satisfies two integral equations C E dl = 0 and charges inside S E ds = S ε }{{ 0 } Gauss Law

13 Integral expressions Electromagnetism in the matter Static magnetic field The magnetic flux density B(x, y, z) satisfies two differential equations B = µ 0 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 B dl = µ 0 current through C and B ds = 0 C S }{{} Ampère s circuital law

14 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/ε 0. 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 = µ 0 J.

15 Integral expressions Electromagnetism in the matter Coulomb s Law Explicit expressions of V and E ρ(r ) V (r) = 4πε 0 r r dx dy dz ρ(r ) r r E(r) = 4πε 0 r r 3 dx dy dz Biot-Savart Law Explicit expressions of A and B for circuits (I = current intensity) A(r) = µ 0I dl 4π circuit r r B(r) = µ 0I 4π circuit dl (r r ) r r 3

16 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) = ρ free replaces div(e) = ρ/ε 0, where D = εe Magnetic field intensity: modification of E caused by magnetic dipoles H = J free replaces B = µ 0 J, where H = B/µ Ohm s law: resistance of the matter to the motion of charged particles J = σ E }{{} conductivity

17 Integral expressions Electromagnetism in the matter Boundary conditions Continuities/discontinuities of the electromagnetic field across the interface between different matters Perfect conductor (ρ s = surface charge density) E = 0 and ρ = 0, inside the conductor E = ρ s ε a n, on the surface of the conductor }{{} 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

18 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 {}}{ E = (µh) t div(εe) = ρ }{{} Gauss s law Ampère s circuital law {}}{ H = J + (εe) t div(µh) = 0 }{{} 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 and B = µh

19 Phasors 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 S J ds = d dt inside 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.

20 Phasors and Plane Waves Energy and power The electromagnetic field carries energy Energy density: W = εe2 2 + µh2 2 Power 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 is exactly balanced by the variation of energy density inside this surface.

21 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 V t = 0

22 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 2 A εµ 2 A t 2 = 0 2 V εµ 2 V t 2 = 0 The propagation velocity c is given by c = 1/ εµ.

23 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)e jωt} H(x, y, z, t) = R { H(x, y, z)e jωt} Maxwell s equations for phasors E = jωµh H = J + jωεe div(εe) = ρ div(µh) = 0 Wave equation for phasors (with ρ = 0, J = σe and ε, µ constant) 2 E + k 2 E = 0 where k 2 = εµω 2 jσµω

24 Phasors and Plane Waves Plane waves A particular solution of Maxwell s equations for phasors E(x, y, z) = E 0 e }{{} jk r constant vector where k is a (possibly complex) vector. E and H are orthogonal, and transverse to the direction of propagation a k Electric field: k E = 0 Magnetic field: H = k E µω = 1 η a k E where η is the wave 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

25 Phasors and Plane Waves Plane waves in lossy media Propagation constant: γ = jk = α Skin depth: δ = 1 α }{{} attenuation +j β }{{} phase Group velocity: u g = 1 dβ dω

26 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