W13D2: Maxwell’s Equations and Electromagnetic Waves Today’s Reading Course Notes: Sections 13.5-13.7 Class 18

Announcements No Math Review next week PS 10 due Week 14 Tuesday May 7 at 9 pm in boxes outside 32-082 or 26-152 Next Reading Assignment W13D3 Course Notes: Sections 13.9, 13.11, 13.12 Class 30

Outline Maxwell’s Equations and the Wave Equation Understanding Traveling Waves Electromagnetic Waves Plane Waves Energy Flow and the Poynting Vector Class 30

Maxwell’s Equations in Vacua No charges or currents Class 30

Wave Equations: Summary Electric & magnetic fields travel like waves satisfying: with speed But there are strict relations between them: Class 30

Understanding Traveling Wave Solutions to Wave Equation Class 30

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 Class 30

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 Class 30

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 y(x,t) satisfies the wave equation! Class 30

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 Class 30

Group Problem: Traveling Sine Wave Let , where . Show that satisfies . Class 30

Wavelength and Wave Number: Spatial Periodicity Class 30

Concept Question: Wave Number The graph shows a plot of the function The value of k is Class 30

Concept Q. Answer: Wave Number Wavelength is 4 m so wave number is Class 30

Period: Temporal Periodicity Class 30

Do Problem 1 In this Java Applet http://web. mit. edu/8 Class 30

Traveling Sinusoidal Wave: Summary Two periodicities: Class 30

Traveling Sinusoidal Wave Alternative form: Class 30

Plane Electromagnetic Waves http://youtu.be/3IvZF_LXzcc Class 30

Electromagnetic Waves: Plane Sinusoidal Waves 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) http://youtu.be/3IvZF_LXzcc Class 30

Electromagnetic Spectrum Hz Wavelength and frequency are related by: Class 30

Traveling Plane Sinusoidal Electromagnetic Waves are special solutions to the 1-dim wave equations where Class 32

Group Problem: 1 Dim’l Sinusoidal EM Waves Show that in order for the fields to satisfy either condition below then Class 32

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?” Class 30

Electromagnetic Radiation: Plane Waves Magnetic field vector uniform on infinite plane. http://youtu.be/3IvZF_LXzcc Class 30

Direction of Propagation Special case generalizes Class 30

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 +x direction –x direction +z direction –z direction Class 30

Concept Question Answer: Propagation Answer: 4. The wave is moving in the –z direction The propagation direction is given by the (Yellow x Blue) Class 30

Properties of 1 Dim’l EM Waves 1. 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 direction of propagation (they are transverse): Class 30

Concept Question: Traveling Wave The B field of a plane EM wave is The electric field of this wave is given by Class 32

Concept Q. Ans.: Traveling Wave Answer: 4. From the argument of the , we know the wave propagates in the positive y-direction. Class 32

Concept Question EM Wave The electric field of a plane wave is: The magnetic field of this wave is given by: Class 30

Concept Q. Ans.: EM Wave Answer: 1. Week 13, Day 2 Concept Q. Ans.: EM Wave Answer: 1. From the argument of the , we know the wave propagates in the negative z-direction. Class 31 33

Energy in EM Waves: The Poynting Vector Class 18

Energy in EM Waves Energy densities: Consider cylinder: Week 13, Day 1 Energy in EM Waves Energy densities: Consider cylinder: What is rate of energy flow per unit area? Class 30 35

Poynting Vector and Intensity Week 13, Day 1 Poynting Vector and Intensity Direction of energy flow = direction of wave propagation units: Joules per square meter per sec Intensity I: Class 30 36

Group Problem: Poynting Vector An electric field of a plane wave is given by the expression Find the Poynting vector associated with this plane wave. Class 30

Appendix A Standing Waves Class 30

Standing Waves What happens if two waves headed in opposite directions are allowed to interfere? Class 30

Standing Waves Class 30

Standing Waves Most commonly seen in resonating systems: Musical Instruments, Microwave Ovens Class 30

Standing Waves Do Problem 2 In the Java Applet http://web. mit. edu/8 Class 30

Appendix B Radiation Pressure Class 30

Momentum & Radiation Pressure Week 13, Day 1 Momentum & Radiation Pressure EM waves transport energy: They also transport momentum: And exert a pressure: This is only for hitting an absorbing surface. For hitting a perfectly reflecting surface the values are doubled, as follows: Class 30 44

Problem: Catchin’ Rays As you lie on a beach in the bright midday sun, approximately what force does the light exert on you? The sun: Total power output ~ 4 x 1026 Watts Distance from Earth 1 AU ~ 150 x 106 km Speed of light c = 3 x 108 m/s