W13d2 maxwell s equations and electromagnetic waves
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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

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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:


Understanding traveling wave solutions to wave equation

Understanding Traveling Wave Solutions to Wave Equation


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 .


Wavelength and wave number spatial periodicity

Wavelength and Wave Number: Spatial Periodicity


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


Period temporal periodicity

Period: Temporal Periodicity


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 summary

Traveling Sinusoidal Wave: Summary

Two periodicities:


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 1 dim l sinusoidal em waves

Show that in order for the fields

Group Problem: 1 Dim’l Sinusoidal EM Waves

to satisfy either condition below

then


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?”


W13d2 maxwell s equations and electromagnetic waves

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 the poynting vector

Energy in EM Waves:The Poynting Vector


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:


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.

Group Problem: Poynting Vector


Appendix a standing waves

Appendix AStanding Waves


Standing waves

Standing Waves

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


Standing waves1

Standing Waves


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


Appendix b radiation pressure

Appendix BRadiation Pressure


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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