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

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w13d2 maxwell s equations and electromagnetic waves

W13D2:Maxwell’s Equations and Electromagnetic Waves

Today’s Reading Course Notes: Sections 13.5-13.7

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

Announcements

outline
Outline

Maxwell’s Equations and the Wave Equation

Understanding Traveling Waves

Electromagnetic Waves

Plane Waves

Energy Flow and the Poynting Vector

maxwell s equations in vacua

0

0

Maxwell’s Equations in Vacua

No charges or currents

wave equations summary
Wave Equations: Summary

Electric & magnetic fields travel like waves satisfying:

with speed

But there are strict relations between them:

example traveling wave
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

direction of traveling waves
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

general sol to one dim l wave eq
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!

generalization
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

group problem traveling sine wave
Group Problem: Traveling Sine Wave

Let ,

where .

Show that

satisfies .

concept question wave number
Concept Question: Wave Number

The graph shows a plot of the function

The value of k is

concept q answer wave number
Concept Q. Answer: Wave Number

Answer: 4.

Wavelength is 4 m so wave number is

do problem 1 in this java applet http web mit edu 8 02t www applets superposition htm

Do Problem 1In this Java Applet http://web.mit.edu/8.02t/www/applets/superposition.htm

traveling sinusoidal wave
Traveling Sinusoidal Wave

Alternative form:

plane electromagnetic waves
Plane Electromagnetic Waves

http://youtu.be/3IvZF_LXzcc

electromagnetic waves plane sinusoidal waves
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

electromagnetic spectrum

Hz

Electromagnetic Spectrum

Wavelength and frequency are related by:

traveling plane sinusoidal electromagnetic waves

Traveling Plane Sinusoidal Electromagnetic Waves

are special solutions to the 1-dim wave equations

where

group problem plane waves
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?”

slide25

Electromagnetic Radiation: Plane Waves

Magnetic field vector uniform on infinite plane.

http://youtu.be/3IvZF_LXzcc

direction of propagation
Direction of Propagation

Special case generalizes

concept question direction of propagation
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
concept question answer propagation
Concept Question Answer: Propagation

Answer: 4. The wave is moving in the –z direction

The propagation direction is given by the

(Yellow x Blue)

properties of 1 dim l em waves
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):

concept question traveling wave
Concept Question: Traveling Wave

The B field of a plane EM wave is

The electric field of this wave is given by

concept q ans traveling wave
Concept Q. Ans.: Traveling Wave

Answer: 4.

From the argument of the , we know the wave propagates in the positive y-direction.

concept question em wave
Concept Question EM Wave

The electric field of a plane wave is:

The magnetic field of this wave is given by:

concept q ans em wave
Concept Q. Ans.: EM Wave

Answer: 1.

From the argument of the , we know the wave propagates in the negative z-direction.

energy in em waves
Energy in EM Waves

Energy densities:

Consider cylinder:

What is rate of energy flow per unit area?

poynting vector and intensity
Poynting Vector and Intensity

Direction of energy flow = direction of wave propagation

units: Joules per square meter per sec

Intensity I:

standing waves
Standing Waves

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

standing waves2
Standing Waves

Most commonly seen in resonating systems:

Musical Instruments, Microwave Ovens

standing waves do problem 2 in the java applet http web mit edu 8 02t www applets superposition htm

Standing Waves Do Problem 2 In the Java Applet http://web.mit.edu/8.02t/www/applets/superposition.htm

momentum radiation pressure
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:

problem catchin rays
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

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