P Workshop: Using Visualization in Teaching Introductory E&M AAPT National Summer Meeting, Edmonton, Alberta, Canada. Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy

P MIT Class: Electromagnetic Waves

P Maxwell’s Equations 0 0 Solve in free space (no charge/current) to get…

P Electromagnetic Radiation

P A Question of Time…

P Electromagnetic Waves: Plane Waves

P Traveling Waves Consider f(x) = x=0 What is g(x,t) = f(x-vt)? x=0 t=0 x=vt 0 t=t 0 x=2vt 0 t=2t 0 f(x-vt) is traveling wave moving to the right!

P Traveling Sine Wave: Space Look at t = 0: g(x,0) = y = y 0 sin(kx): x Amplitude (y 0 ) What is g(x,t) = f(x+vt)? Travels to left at velocity v y = y 0 sin(k(x+vt)) = y 0 sin(kx+kvt)

P Traveling Sine Wave: Time Amplitude (y 0 ) Look at x=0:

P Traveling Sine Wave

P Electromagnetic Waves Remember: Hz

P Electromagnetic Radiation: Plane 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 E (gold) and B (blue)

P Traveling EM Wave: Space At a fixed time (e.g. t=0): x Amplitude (E 0 )

P Traveling EM Wave: Time Amplitude (E 0 ) At x=0, just a function of time:

P Traveling E & B Waves

P PRS Question: Wave

P PRS: Wave The graph shows a plot of the function y = cos(k x). The value of k is (m) :00 1.½ m -1 2.¼ m -1  m -1  /2 m -1 5.I don’t know

P PRS Answer: Wave = 4 m  k = 2  / =  /2 m -1 y = cos (  x /2) is 1 at x = –4 m, 0 m, 4 m, etc. Answer: 4. k =  /2 m -1 (m)

P Properties of EM Waves Travel (through vacuum) with speed of light At every point in the wave and any instant of time, E and B are in phase with one another, with E and B fields perpendicular to one another, and to the direction of propagation (they are transverse):

P Direction of Propagation

P PRS Question: Direction of Propagation

P PRS: Direction of Propagation The figure shows the E (yellow) and B (blue) fields of a plane wave. This wave is propagating in the 0 1.+x direction 2.–x direction 3.+z direction 4.–z direction 5.I don’t know

P PRS Answer: Propagation The propagation direction is given by the direction of E x B (Yellow x Blue) Answer: 4. The wave is moving in the –z direction

P PRS Questions: Traveling Wave

P PRS: Traveling Wave The B field of a plane EM wave is The electric field of this wave is given by I don’t know :20

P PRS Answer: Traveling Wave From the argument of the sin(ky -  t), we know the wave propagates in the +y direction. Answer: 4.

P 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?”

P Electromagnetic Radiation Both E & B travel like waves: But there are strict relations between them: Here, E y and B z are “the same,” traveling along x axis

P Amplitudes of E & B E y and B z are “the same,” just different amplitudes

P Electromagnetic Radiation: Plane Waves

P How Do Maxwell’s Equations Lead to EM Waves? Derive Wave Equation Optional

P Wave Equation Start with Ampere-Maxwell Eq:

P Wave Equation So in the limit that dx is very small: Apply it to red rectangle: Start with Ampere-Maxwell Eq:

P Wave Equation Now go to Faraday’s Law

P Wave Equation Faraday’s Law: So in the limit that dx is very small: Apply it to red rectangle:

P D Wave Equation for E Take x-derivative of 1st and use the 2nd equation

P D Wave Equation for E This is an equation for a wave. Let:

P D Wave Equation for B Take x-derivative of 1st and use the 2nd equation

P Electromagnetic Waves Both E & B travel like waves: But there are strict relations between them: Here, E y and B z are “the same,” traveling along x axis

P Energy in EM Waves Energy densities: Consider cylinder: What is rate of energy flow per unit area?

P Poynting Vector and Intensity units: Joules per square meter per sec Direction of energy flow = direction of wave propagation Intensity I:

P Momentum & Radiation Pressure EM waves transport energy: This is only for hitting an absorbing surface. For hitting a perfectly reflecting surface the values are doubled: They also transport momentum: And exert a pressure:

P Standing Waves

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

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

P Group Work: Standing Waves

P Generating Plane Electromagnetic Radiation

P30 - Shake a Sheet of Charge generating plane wave radiationgenerating plane wave radiation applet

P30 - 2) If sheet position is What is B(x,t)? What is E(x,t)? What Direction? Group Problem: B Field Generation Sheet (blue) has uniform charge density  Starting time T ago pulled down at velocity v 1) What is B field? (HINT: Change drawing perspective) sheet

P30 - You Made a Plane Wave! generating plane wave

P30 - How to Think About E-Field E-Field lines like strings tied to plane This is the field you calculated & that propagates

P Group Problem: Energy in Wave You Found: 1) What is total power per unit area radiated away? 2) Where is that energy coming from? 3) Calculate power generated to see efficiency

P Generating Electric Dipole Electromagnetic Waves

P30 - Quarter-Wavelength Antenna Accelerated charges are the source of EM waves. Most common example: Electric Dipole Radiation. t = 0t = T/4t = T/2t = T

P30 - Why are Radio Towers Tall? AM Radio stations have frequencies 535 – 1605 kHz. WLW 700 Cincinnati is at 700 kHz. Tower is 747 ft tall

P30 - Quarter-Wavelength Antenna

P30 - Quarter-Wavelength Antenna

P30 - Spark Gap Transmitter

P Spark Gap Generator: An LC Oscillator

P Our spark gap antenna 1) Charge gap (RC) 2) Breakdown! (LC) 3) Repeat

P30 - Spark Gap Transmitter

P30 - PRS Question: Spark Gap Antenna

P30 - Spark Gap Antenna

P30 - Spark Gap Antenna

P30 - Demonstration: Antenna

P30 - Polarization

P30 - Polarization of TV EM Waves Why oriented as shown? Why different lengths?

P30 - Demonstration: Microwave Polarization

P30 - Experiment 8: Microwaves

P30 - Standing Waves What happens if two waves headed in opposite directions are allowed to interfere?

P30 - PRS Questions: Angular Distribution & Polarization of Radiation