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Todays lecture is brought to you by the letter P.http://www.labinitio.com (nz288.jpg)

Announcements There is lots of nice math in chapter 32! This lecture calls your attention to those parts of the chapter that you need to know for exams. Keep this lecture in mind when you study chapter 32. Exam 3 is the two weeks from yesterday. I will need to know by the end of next Wednesdays lecture of any students who have special needs different than for exam 2. Exam 3 will cover material through the end of todays lecture. Material presented in lecture next week will be covered on the final exam.

Note!Note: All of these mean average: Saverage Sav SavgFor tomorrows homework, you will need to use this equation from Physics 23:You will also need to look up the mass of the sun.

Todays agenda:

Electromagnetic Waves.

Energy Carried by Electromagnetic Waves.

Momentum and Radiation Pressure of an Electromagnetic Wave.

rarely in the course of human events have so many starting equations been given in so little time

We began this course by studying fields that didnt vary with timethe electric field due to static charges, and the magnetic field due to a constant current.

In case you didnt noticeabout a half dozen lectures ago things started moving!We found that changing magnetic field gives rise to an electric field. Also a changing electric field gives rise to a magnetic field.These time-varying electric and magnetic fields can propagate through space.

Electromagnetic WavesThese four equations provide a complete description of electromagnetism.Maxwells Equations

Production of Electromagnetic WavesApply a sinusoidal voltage to an antenna.Charged particles in the antenna oscillate sinusoidally.The accelerated charges produce sinusoidally varying electric and magnetic fields, which extend throughout space.The fields do not instantaneously permeate all space, but propagate at the speed of light.direction ofpropagationyzx

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=35This static image doesnt show how the wave propagates.Here are animations, available on-line:direction ofpropagationyzxHere is a movie.http://phet.colorado.edu/en/simulation/radio-waves(shows electric field only)

Electromagnetic waves are transverse waves, but are not mechanical waves (they need no medium to vibrate in).direction ofpropagationTherefore, electromagnetic waves can propagate in free space.At any point, the magnitudes of E and B (of the wave shown) depend only upon x and t, and not on y or z. A collection of such waves is called a plane wave.yzx

Manipulation of Maxwells equations leads to the following plane wave equations for E and B:These equations have solutions:You can verify this by direct substitution.whereEmax and Bmax in these notes are sometimes written by others as E0 and B0.Emax and Bmax are the electric and magnetic field amplitudesEquations on this slide are for waves propagating along x-direction.

You can also show thatAt every instant, the ratio of the magnitude of the electric field to the magnitude of the magnetic field in an electromagnetic wave equals the speed of light.

direction ofpropagationyzxEmax (amplitude)E(x,t)Click here to skip 2 slide summary and 1 slide hints that are N/A this semester.

Summary of Important Properties of Electromagnetic WavesElectromagnetic waves travel through empty space with the speed of light c = 1/(00).Emax and Bmax are the electric and magnetic field amplitudes.

Summary of Important Properties of Electromagnetic WavesThe components of the electric and magnetic fields of plane EM waves are perpendicular to each other and perpendicular to the direction of wave propagation. The latter property says that EM waves are transverse waves.direction ofpropagationyzx

Possible Homework Hints (may not be needed every semester)The speed of light in a nonconducting medium other than a vacuum is less than c:where is the relative dielectric constant (remember it from capacitors?) and m is called the relative permeability of the medium.Because you can show that These equations are not on your equation sheet, but you have permission to use them for tomorrows homework (if needed): use v for the wave speed, and replace 0 by 0 and 0 by m0. Also

Todays agenda:

Electromagnetic Waves.

Energy Carried by Electromagnetic Waves.

Momentum and Radiation Pressure of an Electromagnetic Wave.

The magnitude S represents the rate at which energy flows through a unit surface area perpendicular to the direction of wave propagation.Energy Carried by Electromagnetic WavesThus, S represents power per unit area. The direction of S is along the direction of wave propagation. The units of S are J/(sm2) =W/m2. *J. H. Poynting, 1884.

zxycSEBBecause B = E/c we can writeThese equations for S apply at any instant of time and represent the instantaneous rate at which energy is passing through a unit area.

The time average of sin2(kx - t) is , soEM waves are sinusoidal.The average of S over one or more cycles is called the wave intensity I.This equation is the same as 10-29 in your text, using c = 1/(00).Notice the 2s in this equation.EM wave propagating along x-direction

Thus,Note: Saverage and mean the same thing!

The energy densities (energy per unit volume) associated with electric and magnetic fields are:Using B = E/c and c = 1/(00) we can write Energy Densityremember: E and B are sinusoidal functions of time

For an electromagnetic wave, the instantaneous energy density associated with the magnetic field equals the instantaneous energy density associated with the electric field.Hence, in a given volume the energy is equally shared by the two fields. The total energy density is equal to the sum of the energy densities associated with the electric and magnetic fields:

When we average this instantaneous energy density over one or more cycles of an electromagnetic wave, we again get a factor of from the time average of sin2(kx - t).so we see thatRecallThe intensity of an electromagnetic wave equals the average energy density multiplied by the speed of light.and

Quiz 8.

Example: a radio station on the surface of the earth radiates a sinusoidal wave with an average total power of 50 kW.* Assuming the wave is radiated equally in all directions above the ground, find the amplitude of the electric and magnetic fields detected by a satellite 100 km from the antenna.All the radiated power passes through the hemispherical surface* so the average power per unit area (the intensity) is*In problems like this you need to ask whether the power is radiated into all space or into just part of space.Todays lecture is brought to you by the letter P.

Example: for the radio station in the example on the previous two slides, calculate the average energy densities associated with the electric and magnetic field.

Todays agenda:

Electromagnetic Waves.

Energy Carried by Electromagnetic Waves.

Momentum and Radiation Pressure of an Electromagnetic Wave.

Momentum and Radiation PressureEM waves carry linear momentum as well as energy.The momentum density carried by an electromagnetic wave is Todays lecture is brought to you by the letter P.where dp is the momentum carried in the volume dV. This equation is not on your equation sheet, but you have permission to use them for tomorrows homework (if needed)

If we assume that EM radiation is incident on an object for a time t and that the radiation is entirely absorbed by the object, then the object gains energy U in time t.Maxwell showed that the momentum change of the object is then: The direction of the momentum change of the object is in the direction of the incident radiation.When the momentum carried by an electromagnetic wave is absorbed at a surface, pressure is exerted on that surface.

If instead of being totally absorbed the radiation is totally reflected by the object, and the reflection is along the incident path, then the magnitude of the momentum change of the object is twice that for total absorption.The direction of the momentum change of the object is again in the direction of the incident radiation.

The radiation pressure on the object is defined as the force per unit area:From Newtons 2nd Law (F = dp/dt) we have:For total absorption,SoRadiation Pressure(Equations on this slide involve magnitudes of vector quantities.)

This is the instantaneous radiation pressure in the case of total absorption:For the average radiation pressure, replace S by =Savg=I:Electromagnetic waves also carry momentum through space with a momentum density of Saverage/c2=I/c2. This is not on your equation sheet but you have special permission to use it in tomorrows homework, if necessary.Todays lecture is brought to you by the letter P.

incidentincidentreflectedabsorbedUsing the arguments above it can also be shown that:If an electromagnetic wave does not strike a surface, it still carries momentum away from its emitter, and exerts Prad=I/c on the emitter.

Example: a satellite orbiting the earth has solar energy collection panels with a total area of 4.0 m2. If the suns radiation is incident perpendicular to the panels and is completely absorbed find the average solar power absorbed and the average force associated with the radiation pressure. The intensity (I or Saverage) of sunlight prior to passing through the earths atmosphere is 1.4 kW/m2.Assuming total absorption of the radiation:Caution! The letter P (or p) has been used in this lecture for power, pressure, and momentum! Thats because todays lecture is brought to you by the letter P.

I know you are mostly engineers, and think applications are important

So I found you a revolutionarynew application that uses electro- magnetic waves.Revolutionary Application of Electromagnetic WavesThe UFO Detector. Only $48.54 at amazon.com (just search for ufo detector).The UFO detector continually monitors its surrounding area for any magnetic and electromagnetic anomalies.

New starting equations from this lecture:There are even more on your starting equation sheet; they are derived from the above!