Download Lecture 25: Electromagnetic Waves and Maxwell's Equations and more Study notes Physics in PDF only on Docsity! Lecture 25 - Electromagnetic Waves Chapters 38/39 - Thursday April 17th •Review of electromagnetic waves •Refractive index and refraction (Snell’s law) •Review of LRC circuits Final exam is Tuesday May 1st at 10am-noon (cumulative) If you have a conflict, you must come and see me a.s.a.p. Review session on Tues. next week at 5:30 pm in here One final WebAssign set (assorted problems on EM waves) The questions will be posted after the exam Assignment due next Wed. (26th), 11:59pm Solutions will be posted on Friday morning Exam 3 on Thursday (in class) - Chapters 33 to 38 WebAssign from chapter 38: E5, E28, E30, P3 Practice problems: E7, E9, E19, E23, E29, E39, P9, P11 This assignment is due tonight, 11:59pm. Maxwell’s equations in vacuum 0 0d ⎡ ⎤⋅ = ∇ ⋅ =⎣ ⎦∫ E A E 0 0d ⎡ ⎤⋅ = ∇ ⋅ =⎣ ⎦∫ B A B Bd dd dt dt ⎡ ⎤Φ ⎢ ⎥⋅ =− ∇× =−⎢ ⎥⎢ ⎥⎣ ⎦ ∫ BE s E 0 0 0 0 Ed dd dt dt μ ε μ ε ⎡ ⎤Φ ⎢ ⎥⋅ = ∇× =⎢ ⎥⎢ ⎥⎣ ⎦ ∫ EB s B •There is symmetry in these equations. •Furthermore, they predict the existence of EM-waves •They are also consistent with special relativity (1) (2) (3) (4) Waves I - wavelength and frequency Wavelength (consider wave at t = 0): ( ) ( ) ( ,0) sin sin sin m m m y x y kx y k x y kx k λ λ = = + = + You can always add 2π to the phase of a wave without changing its displacement, i.e. 22 ork k πλ π λ = = Tr an sv er se s in us oi da l w av e Tr an sv er se s in us oi da l w av e Waves I - wavelength and frequency Wavelength (consider wave at t = 0): 22 ork k πλ π λ = = We call k the angular wavenumber. The SI unit is radian per meter, or meter-1. This k is NOT the same as spring constant. Tr an sv er se s in us oi da l w av e Tr an sv er se s in us oi da l w av e Waves I - wavelength and frequency Period and frequency (consider wave at x = 0): ( ) ( ) (0, ) sin sin sin m m m y t y t y t y t T ω ω ω = − = − = − + Again, we can add 2π to the phase, 22 orT T πω π ω = = Tr an sv er se s in us oi da l w av e Tr an sv er se s in us oi da l w av e The speed of a traveling wave •For a wave traveling in the opposite direction, we simply set time to run backwards, i.e. replace t with −t. constantkx tω+ = 0 ordx dxk v dt dt k ωω⇒ + = = = − ( )( , ) sinmy x t y kx tω= + •So, general sinusoidal solution is: ( )( , ) sinmy x t y kx tω= ± •In fact, any function of the form ( )( , ) my x t y f kx tω= ± is a solution. Tr an sv er se s in us oi da l w av e Tr an sv er se s in us oi da l w av e The electromagnetic wave equation 2 2 2 2 1 o o E E x tμ ε ∂ ∂ = ∂ ∂ •General solution: ( ) ( )( , ) sin or ( , )m mE x t E kx t E x t E f kx tω ω= ± = ± ( ) ( ) 2 2 2 2 2 2, , E Ek y x t E x t x t ω∂ ∂= − = − ∂ ∂ 2 2 2 2 2 1 1or o o o o k v k ωω μ ε μ ε ⇒ − = − = = 1. . o o i e v c μ ε = = Refractive index 1 1 1 , e m o o e m cv c nκ κ μ ε κ κ = = = In a medium, the constants εo and μo are modified by their permeabilities κe and κm, thus the speed of an electromagnetic wave in a medium is given by: where n = (κeκm)1/2 is called the refractive index of the material. Poynting vector and light intensity Direction of motion 1 oμ = ×S E B This is the energy ‘flux’ associated with the EM wave - like an ‘energy current density’ or energy crossing unit area perpendicular to the flow, per unit time. 2 2 21 1 o o o o cS EB E cE B c ε μ μ μ = = = = Intensity (average rate of energy incidence per unit area): ( )2 2 2av 1 1sin 2m mo o I S S E kx t E c c ω μ μ = = = − = Refraction and Snell’s law •The law of reflection: θ1’ = θ1 •The law of refraction: n1sinθ1 = n2sinθ2 Refraction and total internal
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