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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

Today’s Reading Course Notes: Sections 13.5-13.7

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

Maxwell’s Equations and the Wave Equation

Understanding Traveling Waves

Electromagnetic Waves

Plane Waves

Energy Flow and the Poynting Vector

0

Maxwell’s Equations in Vacua

No charges or currents

Electric & magnetic fields travel like waves satisfying:

with speed

But there are strict relations between them:

### Understanding Traveling Wave Solutions to Wave Equation

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

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.

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!

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

Let ,

where .

Show that

satisfies .

The graph shows a plot of the function

The value of k is

Wavelength is 4 m so wave number is

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

Two periodicities:

Alternative form:

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)

http://youtu.be/3IvZF_LXzcc

Electromagnetic Spectrum

Wavelength and frequency are related by:

### Traveling Plane Sinusoidal Electromagnetic Waves

are special solutions to the 1-dim wave equations

where

### Group Problem: 1 Dim’l Sinusoidal EM Waves

to satisfy either condition below

then

1) Plot E, B at each of the ten points pictured for t = 0

2) Why is this a “plane wave?”

Magnetic field vector uniform on infinite plane.

http://youtu.be/3IvZF_LXzcc

Special case generalizes

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

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

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):

The B field of a plane EM wave is

The electric field of this wave is given by

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

The electric field of a plane wave is:

The magnetic field of this wave is given by:

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

### Energy in EM Waves:The Poynting Vector

Energy densities:

Consider cylinder:

What is rate of energy flow per unit area?

Direction of energy flow = direction of wave propagation

units: Joules per square meter per sec

Intensity I:

Find the Poynting vector associated with this plane wave.

### Appendix AStanding Waves

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

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

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

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