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Lecture 21-1 Maxwell’s Equations (so far) Gauss’s law Gauss’ law for magnetism S E dA S Qinside 0 B dA0 Faraday’s law dB C E d l dt Ampere’s law* C B d l 0 I Lecture 21-2 Parallel-Plate Capacitor Revisited S B d l 0 I S -Q For surface S1, Is = I, but for surface S2, Is = 0 E Wait, LHS is the same (because C is the same)! B=0 ? Not experimentally! Q E 0 dQ dV 0 A dV I C dt dt d dt dE dE 0 A 0 dt dt ?? You could make this work if a fictitious current Id is added to Is in such a way that Id is zero for S1 but is equal to I for S2. dE Id 0 dt will work. Lecture 21-3 Displacement Current James Clerk Maxwell proposed that a changing electric field induces a magnetic field, in analogy to Faraday’s law: A changing magnetic field induces an electric field. Ampere’s law is revised to become Ampere-Maxwell law dE C B d l 0 ( I I d ) 0 I 0 0 dt where dE Id 0 dt is the displacement current. Lecture 21-4 MAXWELL’S EQUATIONS “COMPLETED” S E dA Qinside 0 dB C E d l dt S B dA0 dE C B d l 0 I 0 0 dt Basis for electromagnetic waves! The equations are often written in slightly different (and more convenient) forms when dielectric and/or magnetic materials are present. Lecture 21-5 ©2008 by W.H. Freeman and Company Lecture 21-6 READING QUIZ 1 The extended Amperes law involves a displacement current which was added to the real currents as a source of the magnetic (B) fields. Which of the following statements is correct? A| the displacement current has a constant magnitude ie. not a frequency dependent magnitude. B| The displacement current does not involve the constant εO= 1/4πk. C| The displacement current is proportional to εO. D| The displacement current was introduced by Faraday. Lecture 21-7 7B-20 TRANSVERSE ELECTROMAGNETIC WAVE Lecture 21-8 Electromagnetic Waves From Faraday’s Law dB C E dl dt E y Bz x t B Bm sin(kx t ) z E Em sin(kx t ) y Em Bm k c EB k f c Lecture 21-9 ELECTOMAGNETIC WAVE FARADAY INDUCTION PART ©2008 by W.H. Freeman and Company Lecture 21-10 Electromagnetic Waves From Ampère’s Law dE C B dl 0 0 dt E y Bz 0 0 x t E Em sin(kx t ) y B Bm sin(kx t ) z Em k / c Bm 0 0 EB k f c Lecture 21-11 TRANSVERSE ELECTROMAGNETIC WAVE AMPERE LAW PART ©2008 by W.H. Freeman and Company Lecture 21-12 Electric Dipole Radiation Lecture 21-13 ©2008 by W.H. Freeman and Company Lecture 21-14 WARM UP QUIZ 2 Light is an electromagnetic wave where the wavelength λ in meters times the frequency f in Hz is equal to the velocity of light c = 3 x 108 meters/second in vacuum. Which of the following statements is correct? (a) λ = 100 meters, f = 1x106 Hz (b) λ = 105 meters, f = 3x104 Hz (c) λ = 10-6 meters, f = 3x1014 Hz (d) λ = 1 meter, f = 3/2 x 108 Hz Lecture 21-15 Electromagnetic Wave Propagation in Free Space So, again we have a traveling electromagnetic wave Em c Bm k c Em 1 Bm 0 0c 0 4 107 (T m / A) 1 0 0 speed of light in vacuum 0 8.85 1012 (C 2 / N m2 ) B 1 E 2 x c t E B x t Ampere’s Law Faraday’s Law 2 B 1 2 B Wave Equation 2 2 2 x c t c 3.00 108 (m / s) Speed of light in vacuum is currently defined rather than measured (thus defining meter and also the vacuum permittivity). Lecture 21-16 Plane Electromagnetic Waves 2B 1 2B 2 2 2 x c t 2E 1 2E 2 2 2 x c t where B Bm sin(kx t ) z E Em sin(kx t ) y c Em 1 c Bm 0 0 / k EB k • Transverse wave • Plane wave (points of given phase form a plane) x • Linearly polarized (fixed plane contains E) Lecture 21-17 Energy Density of Electromagnetic Waves • Electromagnetic waves contain energy. We know already expressions for the energy density stored in E and B fields: EM wave 2 1B 1 2 uE 0 E , uB 2 0 2 E2 1 2 uB E uE 0 2 2 0c 2 Bm Em / c B E/c • So Total energy density is EB 0 u uE uB 0 E EB 0 0c 0 B2 2 u 0 E 2 0E 2 rms B2 0 2 Brms 0 EB Erms Brms 0 c 0 c Lecture 21-18 Energy Propagation in Electromagnetic Waves • Energy flux density = Energy transmitted through unit time per unit area • Intensity I = Average energy flux density (W/m2) P A 1 0 u c EB Define Poynting vector Erms Brms 0 S 1 0 EB Direction is that of wave propagation average magnitude is the intensity S I c 0 E 2 1 1 Bm2 Em Bm 2 c 0 Em c 2 2 0 2 0 Lecture 21-19 Radiation Pressure Electromagnetic waves carry momentum as well as energy. In terms of total energy of a wave U, the momentum is U/c. During a time interval Δt , the energy flux through area A is ΔU =IA Δ t . If radiation is totally absorbed: p U / c IA / c t p IA F t c momentum imparted pr F / A I / c B /(2 0 ) 2 m radiation pressure EXERTED If radiation is totally reflected: x2 p 2U / c 2 IA / c t 2 IA F , pr 2 I / c c Lecture 21-20 Maxwell’s Rainbow Light is an Electromagnetic Wave f c Lecture 21-21 Physics 241 – 10:30 QUIZ 3, November 8, 2011 An electromagnetic wave is traveling in +y direction and the magnetic field at a particular point on the y-axis points in the +z direction at a certain instant in time. At this same point and instant, what is the direction of the electric field? z a) -z B b) -x y c) -y d) +x e) None of the above x The direction of travel is that of E x B. Lecture 21-22 Physics 241 -- 11:30 QUIZ 3, November 8, 2011 An electromagnetic wave is traveling in +x direction and the electric field at a particular point on the x-axis points in the +z direction at a certain instant in time. At this same point and instant, what is the direction of the magnetic field? a) -z z b) -x E c) -y y d) +y e) None of the above x The direction of travel is that of E x B. Lecture 21-23 Physics 241 -- 11:30 QUIZ 3, March 31, 2011 An electromagnetic wave is traveling through a particular point in space where the direction of the electric field is along the +z direction and that of the magnetic field is along the +y direction at a certain instant in time. Which direction is this wave traveling? z a) +x E y b) -x c) -y d) -z e) None of the above B x The direction of travel is that of E x B.