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MIT 8 02T - Maxwell’s Equations

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1 1 W14D1: Maxwell’s Equations and Electromagnetic Waves Today’s Reading Course Notes: Sections 13.5-13.7 Announcements Math Review Tuesday May 6 from 9-11 pm in 26-152 PS 10 due Week 14 Tuesday May 6 at 9 pm in boxes outside 32-082 or 26-152 2 3 Outline Maxwell’s Equations and the Wave Equation Understanding Traveling Waves Electromagnetic Waves Plane Waves Energy Flow and the Poynting Vector2 4 Maxwell’s Equations in Vacua 1.E ⋅ dAS∫∫=Qinε0(Gauss's Law)2.B ⋅ dAS∫∫= 0 (Magnetic Gauss's Law)3.E ⋅ dsC∫= −dΦBdt(Faraday's Law)4.B ⋅ dsC∫=µ0Ienc+µ0ε0dΦEdt(Ampere - Maxwell Law)0 0 No charges or currents 5 Wave Equations: Summary Electric & magnetic fields travel like waves satisfying: ∂2Ey∂x2=1c2∂2Ey∂t2∂2Bz∂x2=1c2∂2Bz∂t2with speed But there are strict relations between them: ∂Bz∂t= −∂Ey∂x∂Bz∂x= −µ0ε0∂Ey∂t c =1µ0ε06 Understanding Traveling Wave Solutions to Wave Equation3 7 Example: Traveling Wave Consider The variables x and t appear together as x - vt At t = 0: At vt = 2 m: At vt = 4 m: is traveling in the positive x-direction y(x,t) = y0e−(x − vt )2/a2y(x − vt) = y0e−(x )2/a2y(x − vt) = y0e−( x −(2 m))2/a2y(x − vt) = y0e−( x −(4 m))2/a2y(x − vt) = y0e−(x − vt )2/a2y(x − vt)8 Direction of Traveling Waves Consider The variables x and t appear together as x + vt At t = 0: At vt = 2 m: At vt = 4 m: is traveling in the negative x-direction y(x,t) = y0e−(x + vt )2/a2y(x − vt) = y0e−(x )2/a2y(x + vt) = y0e−( x +(2 m))2/a2y(x + vt) = y0e−( x +(4 m))2/a2y(x + vt) = y0e−(x + vt )2/a2y(x + vt)9 General Sol. to One-Dim’l Wave Eq. Consider any function of a single variable, for example Change variables. Let then and Now take partial derivatives using the chain rule Similarly Therefore 22/0()uayu ye−=2222yyuy yf fuf yfxuxu x xuxuu∂∂∂∂ ∂ ∂∂∂∂∂==≡ ====∂∂∂∂ ∂ ∂∂∂∂∂ and u = x − vt ∂u∂x= 1 and ∂u∂t= −v ∂ y∂t=∂ y∂u∂u∂t= −v∂ y∂u≡ −vf and ∂2y∂t2= −v∂ f∂t= −v∂ f∂u∂u∂t= v2∂ f∂u=∂2y∂u2 ∂2y∂x2=1v2∂2y∂t2y(x,t) satisfies the wave equation! ( )22/0() (,)xvt ayu yxt ye−−==4 10 Generalization Take any function of a single variable , where Then or (or a linear combination) is a solution of the one-dimensional wave equation corresponds to a wave traveling in the positive x-direction with speed v and corresponds to a wave traveling in the negative x-direction with speed v y(x − vt)y(x + vt) 1v2∂2y(x,t)∂t2=∂2y(x,t)∂x2y(x − vt)y(x + vt)()yuu = x ± vt11 Group Problem: Traveling Sine Wave Let , where . Show that satisfies . 1v2∂2y(x,t)∂t2=∂2y(x,t)∂x2 y(x,t) = y(x − vt) = y0sin(k(x − vt)) y(u) = y0sin(ku) u = x − vt12 Wavelength and Wave Number: Spatial Periodicity Fix t = 0 : y(x,0) = y0sin(kx) When x =λ⇒ kλ= 2π⇒ k = 2π/λ Consider y(x,t) = y0sin(k(x − vt)) λis called the wavelength, k is called the wave number5 13 Concept Question: Wave Number The graph shows a plot of the function The value of k is 1. k = 2π/ (2 m)2. k = 2π/ (1 m)3. k = 2π/ (0.5 m)4. k = 2π/ (4 m) y(x,0) = cos(kx)14 Period: Temporal Periodicity Fix x = 0 : y(0,t) = y0sin(− kvt) = − y0sin(kvt) When t = T ⇒ kvT = 2π⇒ 2πvT /λ= 2π/ kv Consider y(x,t) = y0sin(k(x − vt)) T is called the period ⇒ T =λ/ v15 Do Problem 1 In this Java Applet http://web.mit.edu/8.02t/www/applets/superposition.htm6 16 Traveling Sinusoidal Wave: Summary y(x,t) = y0sin(k(x − vt)) Wave Number : k = 2π/λDispersion Relation : T =λ/ v Direction of Propagation : + x - direction Spatial period : Wavelength λ ; Temporal period T .Two periodicities: 17 Traveling Sinusoidal Wave Wave Number : k = 2π/λAngular Frequency : ω= 2π/ TDispersion Relation : λ= vT ⇔ω= kvFrequency : f = 1 / T ⇒ v =λf y(x,t) = y0sin(k(x − vt)) = y0sin(kx −ωt)Alternative form: 18 Plane Electromagnetic Waves  http://youtu.be/3IvZF_LXzcc7 19 Electromagnetic Waves: Plane Sinusoidal Waves  http://youtu.be/3IvZF_LXzcc Watch 2 Ways: 1) Sine wave traveling to right (+x) 2) Collection of out of phase oscillators (watch one position) Don’t confuse vectors with heights – they are magnitudes of electric field (gold) and magnetic field (blue) 20 Electromagnetic Spectrum Wavelength and frequency are related by: λf = cHz 21 Traveling Plane Sinusoidal Electromagnetic Waves E = E0sin(kx −ωt)ˆj B = B0sin(kx −ωt)ˆkare special solutions to the 1-dim wave equations ∂2Ey∂x2=1c2∂2Ey∂t2∂2Bz∂x2=1c2∂2Bz∂t2where k ≡ 2π/λ,ω≡ 2π/ T , c =λ/ T8 22 Group Problem: 1 Dim’l Sinusoidal EM Waves E = E0sin(kx −ωt)ˆj, B = B0sin(kx −ωt)ˆk Show that in order for the fields to satisfy either condition below ∂Bz∂t= −∂Ey∂x∂Bz∂x= −1c2∂Ey∂tthen B0= E0/ c23 Group Problem: Plane Waves 1) Plot E, B at each of the ten points pictured for t = 0 2) Why is this a “plane wave?” E(x, y, z,t) = Ey,0sin2πλ(x − ct)⎛⎝⎜⎞⎠⎟ˆj B(x, y, z,t) =1cEy,0sin2πλ(x − ct)⎛⎝⎜⎞⎠⎟ˆk24  Electromagnetic Radiation: Plane Waves  http://youtu.be/3IvZF_LXzcc Magnetic field vector uniform on infinite plane.9 25 Direction of Propagation dirE dirB dirE ×Bˆiˆjˆkˆjˆkˆiˆkˆiˆjˆjˆi −ˆkˆkˆj −ˆiˆiˆk −ˆj E = E0sin(kx −ωt)ˆj;B = B0sin(kx −ωt)ˆk ⇒ dir(E ×B) =ˆiSpecial case generalizes 26 Concept Question: Direction of Propagation The figure shows the E (yellow) and B (blue) fields of a plane wave. This wave is propagating in the 1. +x direction 2. –x direction 3. +z direction 4. –z direction Properties of 1 Dim’l EM Waves c =1µ0ε0= 3.0 × 108ms E0/ B0= c1. Travel (through vacuum) with speed of light 2. At every point in the wave and any instant of time, electric and magnetic fields are in phase with one another, amplitudes obey 3. Electric and magnetic fields are perpendicular to one another, and to the


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MIT 8 02T - Maxwell’s Equations

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