Electromagnetic Wave Theory Wei-Chih Wang ME557 Department of Mechanical Engineering w wang University of Washington 1 Refraction and reflection (wave equation) The incident beam is characterized by its wavelength λi, its frequency νi and its velocity c and n ε µ λ , f , C 0 n ε µ λ , f , C ,, r r, r r refracted beam is o,, ο o i ι o characterized by its wavelength λr, its frequency νr and its velocity c , the simple dispersion relation for vacuum. Co = fi λi C = fr λr w wang 2 The speed of light in a medium is related to the electric and magnetic properties of the medium, and the speed of light in vacuum can be expressed as o The speed of light in a material to the material "constants" εr and the corresponding magnetic permeability µ0 of vacuum and µr of the material is 1 c = µr µoεrεo w wang 3 The index of refraction n of a non-magnetic material µr = 1 is linked to the dielectric constant εr via a simple relation, which is a rather direct result of the Maxwell equations. c o 1 / µ o ε o = = ε r = n c 1 / µ r µ o ε r ε o Plug back into dispersion relation, c λ f o = i i = n c λr fr Since f = f i r, λ n = i w wang λr 4 Maxwell Equations Integral form in the absence of magnetic or polarized media: I. Faraday's law of induction dl II. Ampere's law III. Gauss' law for magnetism IV. Gauss' law for electricity E = Electric Field (V/m) ρ = charge density (c/m3) i = electric current (A) 2 2 B = Magnetic flux density(Web/m , T) ε0 = permittivity J = current density(A/m ) D = Electric flux density (c/m2) µ = permeability c = speed of light or electric displacement field 0 H = Magnetic Field (A/m) ΦB = Magnetic flux (Web) P = Polarization -19 -6 -12 q = chargew wang 1.6x10 coulombs, µo = 1.26x10 H/m, εo = 8.85x10 F/m 5 Electric flux For instance, Gauss's law states that the flux of the electric field out of a closed surface is proportional to the electric charge enclosed in the surface (regardless of how that charge is distributed). The constant of proportionality is the reciprocal of the permittivity of free space. Its integral form is: The electric flux in an unclosed surface: φ = E ⋅dA E ∫ Sometimes electric flux appears in φ = D ⋅dA = εE ⋅dA terms of flux density D as: E ∫ ∫ w wang 6 The electric elasticity equation (Displacement field) D= ε E Where E = electric field ε = permittivity (dielectric constant) -12 in air εo = 8.85x10 F/m w wang 7 Magnetic flux We know from Gauss's law for magnetism that in a close surface, Normally, the magnetic flux in an φB = B ⋅dA unclosed surface ∫ Where B = magnetic flux density w wang 8 But when the generated fields pass through magnetic materials which themselves contribute internal magnetic fields, ambiguities can arise about what part of the field comes from the external currents and what comes from the material itself. It has been common practice to define another magnetic field quantity, usually called the "magnetic field strength" designated by H. It can be defined by the relationship Β = µ H + M M = magnetization. Normally, the M = 0 for nonmagnetic material -6 If in air, µo = 1.26x10 H/m w wang 9 Faraday’s Law of Induction dl This line integral is equal to the generated voltage or emf in the loop, so Faraday's law is the basis for electric generators. It also forms the basis for inductors and transformers. w wang 10 Ampere's Law In the case of static electric field, the line integral of the magnetic field around a closed loop is proportional to the electric current flowing through the loop. This is useful for the calculation of magnetic field for simple geometries. w wang 11 Gauss’s Law for Magnetism The net magnetic flux out of any closed surface is zero. This amounts to a statement about the sources of magnetic field. For a magnetic dipole, any closed surface the magnetic flux directed inward toward the south pole will equal the flux outward from the north pole. The net flux will always be zero for dipole sources. If there were a magnetic monopole source, this would give a non-zero area integral. The divergence of a vector field is proportional to the point source density, so the form of Gauss' law for magnetic fields is then a statement that there are no magnetic monopoles. w wang 12 Gauss’s Law for Electricity The electric flux out of any closed surface is proportional to the total charge enclosed within the surface. The integral form of Gauss' Law finds application in calculating electric fields around charged objects. In applying Gauss' law to the electric field of a point charge, one can show that it is consistent with Coulomb's law. While the area integral of the electric field gives a measure of the net charge enclosed, the divergence of the electric field gives a measure of the density of sources. It also has implications for the conservation of charge. w wang 13 Maxwell Equations − ∂B Faraday’s Law ∇× E = ∂t ∂D Ampere’s Law ∇× H = J + ∂t Gauss’s Law for ∇ • B = 0 Magnetism Gauss’s Law for Electricity ∇ • D = ρ E = Electric Field (V/m) ρ = charge density (c/m3) i = electric current (A) 2 2 B = Magnetic flux density(Web/m , T) ε0 = permittivity J = current density(A/m ) 2 D = Electric flux density (c/m ) µ0 = permeability c = speed of light H = Magneticw wang Field (A/m) ΦB = Magnetic flux (Web) P = Polarization 14 Wave equation Maxwell's Equations contain the wave equation for electromagnetic waves. One approach to obtaining the wave equation: 1. Take the curl of Faraday's law: 2. Substitute Ampere's law for a charge and current-free region: This is the three-dimensional wave equation in vector form. It looks more familiar when reduced a plane wave with field in the x-direction only: w wang 15 Curl The curl of a vector function is the vector product of the del operator with a vector function: where i,j,k are unit vectors in the x, y, z directions. It can also be expressed in determinant form: w wang 16 Curl in Cylindrical Polar Coordinates The curl in cylindrical polar coordinates, expressed in determinant form is: w wang 17 Curl in Spherical Polar Coordinates The curl in spherical polar coordinates, expressed in determinant form is: w wang 18 Use ∇×(∇× E) = ∇(∇ • E) − ∇2E Wave equation 2 2 becomes ∇ E +ω µoεo E = 0 We consider the simple solution where E field is parallel to the x axis and its function of z coordinate only, the wave equation then becomes, ∂2E x +ω 2µ ε E = 0 ∂z 2 o o x A solution to the above differential equation is − jkz E = xˆEoe Substitute above equation into wave equation yields, 2 2 2 2 (dispersion relation) w wang(−k +ωµεµε)E = 0 k =ω 19 Let’s transform the solution for the wave equation into real space and time, (assume time harmonic field) E E(z,t) = Re{Ee jωt } = xˆE cos(ωt − kz) o x k H k = 2π/λ, where k = wave number z Image we riding along with the wave, we asked what Velocity shall we move in order to keep up with the wave, The answer is phase of the wave to be constant ωt - kz = a constant The velocity of propagation is therefore given by, dz ω 1 = v = = (phase velocity) dt k w wang µoεo 20 Poynting’s Theorem For a time –harmonic electromagnetic wave, the power density Per unit area associate with the wave is defined in complex Representation by vector S, S S= E x H* (W/m2) hyperphysics Time average pontying vector <s> is defined as average of the Time domain Poynting vector S over a period T=2π/ω. 1 2π < S >= d(ωt)E(x, y, z,t)× H (x, y, z,t) 2π ∫0 (or) 1 w wang < S >= Re{E × H} 21 2 Boundary Conditions y nˆ ε1 H 1 H w H3 4 H2 x ε2 l At an interface between two media, the file quantities must satisfy Certain conditions. Consider an interface between two dielectric media With dielectric constants ε1 and ε2, in the z component Ampere’s Law, we have, ∂H ∂H y − x = J + jwD ∂x ∂y z z or H 4 − H3 H1 − H 2 − = J z + jwDz w wang l w 22 Now let area shrink to a point where w goes to zero before l does. So Jz =Js ~ Jvw, then H 2 − H1 = J z Or in general, nˆ ×(H 2 − H1) = J s Applying the same above argument to Faraday’s Law and we get, nˆ ×(E1 − E2 ) = 0 The tangential electric field E is continuous across the boundary surface.The discontinuity in the tangential component of H is equal to the surface current density Js. w wang 23 Apply the divergence theory ∇ • B = 0 and ∇ • D = ρ for The pillbox volume shown y nˆ ε 1 w x ε2 l As w -> 0, we get (B1 − B2 ) • nˆ = 0 (D1 − D2 ) • nˆ = ρ The normal component of B is continuous across the boundary surface.The discontinuity in the normal component of D is equal to the surface charge density ρ w wang 24 Boundary Condition for Perfect Conductor On the surface of a perfect conductor, E2 = 0 and H2= 0 y nˆ ε 1 w x ε2 l w wang 25 Reflection and Transmission (TE, S wave) µ1, ε1, n1 µ2, ε2, n2 r Rl Eo − jkrxx+ jkrzz H = (xˆkrz + zˆkrx ) e x ωµ1 t − jktxx− jktzz E = yˆTl Eoe r − jkrxx+ jkrzz E = yˆRl Eoe θr θt z θi i − jkxx− jky y E = yˆEoe t Tl Eo − jktxx− jktzz H = (−xˆktz + zˆktx ) e ωµ2 E R =reflection coefficient H i = (−xˆk + zˆk ) o e− jkxx− jkzz l z x T =transmission coefficient ωµ1 l TE = transverse electric, perpendicularly polarized (E perpendicular to plan of incident) w wang 26 If neither two are perfect conductors, Js=0, then boundary conditions requires both the tangential electric-filed and magnetic-field components be continuous at z =0 thus, − jkxx − jkrxx − jktxx e + Rle = Tle (E component) − kz − jkxx krz − jkrxx − ktz − jktxx e + Rle = Tle (B component) ωµ1 ωµ1 ωµ2 For the above equations to hold at all x, all components must be the same, thus we get the phase matching condition: kx = k sinθi = krx = kr sinθr = ktx = kt sinθt From this we obtain law of reflection: Since k =k because k2 = k2