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

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

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

Maxwell’s Equations and the Wave Equation This is a 3-Dimensional Wave Equation v = 2.99… x 10 8 m/s = light velocity In vacuum (  = 0 and j = 0): }

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 )}

Plane waves (1): orientation of fields  -i k e z .E = 0  -i k e z .H = 0  -i k e z x E = -i  H  -i k e z x H =  E +i  E (1) div E = 0 (2) div B = 0 (3) rot E = - d B/ dt (4) rot H = j f + d D /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 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

Plane waves (2): complex wave vector e z x e z x E = -E i k 2 = (  + i  ).  Result: k complex: k = k Re + i k Im exp (-i kz ) = exp (-i k Re z ) . exp ( k Im z ) (1)  -i k e z .E = 0 (2)  -i k e z .H = 0 (3)  -i k e z x E = -i  H (4)  -i k e z x H =  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 } harmonic } k Im < 0 : absorption >0 : amplification (“laser”)

Plane waves (3): B-E correspondence Faraday: rot E = - d B/ dt Maxwell (for j =0): rot H = d D /dt: similar result 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

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