FdM 1 Electromagnetism University of Twente Department Applied Physics First-year course on Part III: Electromagnetic Waves : Slides © F.F.M. de Mul

FdM 2 Electromagnetic Waves Gauss’ and Faraday’s Laws for E (D)-field Gauss’ and Maxwell’s Laws for B (H)-field Maxwell’s Equations and The Wave Equation Harmonic Solution of the Wave Equation Plane waves (1): orientation of field vectors Plane waves (2): complex wave vector Plane waves (3): the B - E correspondence The Poynting Vector

FdM 3 Gauss’ and Faraday’s Laws for E B c dS Faraday’s Law : S E S V dV Gauss’ Law : Div = micro-flux per unit of volume Rot = micro-circulation per unit of area

FdM 4 Gauss’ and Maxwell’s Law for B S V dV B Gauss’ Law : j S L Maxwell’s Fix for Ampere’s Law :

FdM 5 Maxwell’s Equations and the Wave Equation In vacuum (  = 0 and j = 0): This is a 4-Dimensional Wave Equation v = 2.99… x 10 8 m/s = light velocity }

FdM 6 Maxwell’s Equations and the Wave Equation 4-Dimensional Wave Equation : v = 2.99… x 10 8 m/s = light velocity The light velocity appears to be constant, independent of motion or not. This gave Lorentz and Einstein the idea of The Theory of Relativity

FdM 7 Harmonic Solution of the Wave Equation: Plane Waves E may have 3 components: E x E y E z Choose x-axis // E  E y = E z = 0 (Polarization direction = x-axis). Does a plane-wave expression for E x satisfy the wave equation? E x = E x0 exp{i (  t-kz)} : E x0 = amplitude + polarization vector +z-axis = direction of propagation Insertion into wave equation: k 2 =  0  0  2 =  2 / c 2 k =  / c = 2  / k = wave number ; = wavelength (in 3D-case: k = wave vector ) Analogously: B y = B y0 exp{i (  t-kz)}

FdM 8  -ik e z.E = 0  -ik e z.H = 0  -ik e z xE = -i  H  -ik e z xH =  E+i  E Plane waves (1): orientation of fields Suppose: E // x-axis;.. propagation // +z-axis: k // e z.. E x = E x0 exp i (  t-kz) x z y E Propag- ation (1) div E = 0 (2) div B = 0 (3) rot E = - dB/dt (4) rot H = j f + dD/dt j f =  E Consequences: (1)+(2): E and H  e z (3)+(4): E  H If E chosen // x-axis, then H // y-axis H

FdM 9 Plane waves (2): complex wave vector (1)  -ik e z.E = 0 (2)  -ik e z.H = 0 (3)  -ik e z xE = -i  H (4)  -ik e z xH =  E+i  E (1)+(2): E and H  e z (3)+(4): E  H Suppose: E // x-axis;.. propagation // +z-axis: k // e z.. E x = E x0 exp i (  t-kz) x z y E Propag- ation H e z x e z x E = -E ik 2 =(  +i  ).  Result: k complex: k = k Re + ik Im exp (-ikz) = exp (-ik Re z). exp (k Im z) } harmonic } k Im <0 : absorption >0 : amplification (“laser”)

FdM 10 Plane waves (3): B-E correspondence Suppose: E // x-axis; B // y-axis;.. propagation // +z-axis: k // e z.. E x = E x0 exp i (  t-kz).. B y = B y0 exp i (  t-kz) x z y E Propag- ation B Faraday: rot E = - dB/dt Maxwell (for j=0): rot H = dD/dt: similar result

FdM 11 The Poynting vector S Definition (for free space) : S = E x H  S =  (E  H) = H  (  E) - E  (  H) = = H  (-dB/dt) - E  (j f +dD/dt) Apply Divergence Theorem to integrate over wave surface A: { Change in Electro- magnetic field energy { Joule heating losses [J/s] { Outflux of energy [J/s] = [W] S = energy outflux per m 2 = Intensity [W/m 2 ] Direction of S : // k E H k S