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

Announcements  There is lots of nice math in chapter 10! 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 10.  Exam 3 is the two weeks from yesterday. I will need to know by the end of next Wednesday’s lecture of any students who have special needs different than for exam 2.  Exam 3 will cover material through the end of today’s lecture. Material presented in lecture next week will be covered on the final exam.

Note! Note: All of these mean “average:” <S> Saverage Sav Savg

Electromagnetic Waves. Energy Carried by Electromagnetic Waves. Today’s 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 didn’t vary with time—the electric field due to static charges, and the magnetic field due to a constant current. In case you didn’t notice—about 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 Waves Maxwell’s Equations These four equations provide a complete description of electromagnetism.

Production of Electromagnetic Waves Apply 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. y direction of propagation x z

This static image doesn’t show how the wave propagates. y direction of propagation x z This static image doesn’t show how the wave propagates. Here are animations, available on-line: http://phet.colorado.edu/en/simulation/radio-waves (shows electric field only) http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=35 Here is a movie.

Therefore, electromagnetic waves can propagate in free space. Electromagnetic waves are transverse waves, but are not mechanical waves (they need no medium to vibrate in). Therefore, 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. y direction of propagation x z

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

You can also show that At 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.

y direction of propagation x z Emax (amplitude) E(x,t)

Today’s agenda: Electromagnetic Waves. Energy Carried by Electromagnetic Waves. Momentum and Radiation Pressure of an Electromagnetic Wave.

Energy Carried by Electromagnetic Waves Electromagnetic waves carry energy, and as they propagate through space they can transfer energy to objects in their path. The rate of flow of energy in an electromagnetic wave is described by a vector S, called the Poynting vector.* The magnitude S represents the rate at which energy flows through a unit surface area perpendicular to the direction of wave propagation (energy per time per area). Thus, S represents power per unit area. The direction of S is along the direction of wave propagation. The units of S are J/(s·m2) =W/m2. *J. H. Poynting, 1884.

Because B = E/c we can write c B For an EM wave so y E S x Because B = E/c we can write c B z These equations for S apply at any instant of time and represent the instantaneous rate at which energy is passing through a unit area.

EM waves are sinusoidal. EM wave propagating along x-direction The average of S over one or more cycles is called the wave intensity I. The time average of sin2(kx - t) is ½, so Notice the 2’s in this equation. This equation is the same as 10-29 in your text, using c = 1/(00)½.

Note: Saverage and <S> mean the same thing! The magnitude of S is the rate at which energy is transported by a wave across a unit area at any instant: Thus, Note: Saverage and <S> mean the same thing!

remember: E and B are sinusoidal functions of time Energy Density The energy densities (energy per unit volume) associated with electric and magnetic fields are: Using B = E/c and c = 1/(00)½ we can write remember: 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:

instantaneous energy densities (E and B vary with time) When we average this instantaneous energy density over one or more cycles of an electromagnetic wave, we again get a factor of ½ from the time average of sin2(kx - t). and Recall so we see that The intensity of an electromagnetic wave equals the average energy density multiplied by the speed of light.

Help! “These factors of ¼, ½, and 1 are making my brain hurt!” It’s really not that bad. These are the energy densities associated with E(t) and B(t) at some time t: It’s really not that bad. Add uB and uE to get the total energy density u(t) at time t:

Help! Again, these are the energy densities associated with E(t) and B(t) at some time t: If you average uB and uE over one or more cycles, you get an additional factor of ½ from the time average of sin2(kx-t). The Emax and Bmax come from writing E = Emax sin(kx-t) and B = Bmax sin(kx-t), and canceling the sine factors.

Help! These are the average energy densities associated with E(t) and B(t) over one or more complete cycles. Add uE and uB to get the total average energy density over one or more cycles:

Help! Summary: At time t: Average: At time t: Average: If you use a starting equation that is not valid for the problem scenario, you will get incorrect results!

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. Strategy: we want Emax, Bmax. We are given average power. From average power we can calculate intensity, and from intensity we can calculate Emax and Bmax. Satellite From the average power we can calculate intensity, and from intensity we can calculate Emax and Bmax. Station *In problems like this you need to ask whether the power is radiated into all space or into just part of space.

Today’s lecture is brought to you by the letter P. 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. Area=4R2/2 All the radiated power passes through the hemispherical surface* so the average power per unit area (the intensity) is All the radiated power passes through the hemispherical surface* so the average power per unit area (the intensity) is Satellite R Station Today’s lecture is brought to you by the letter P. *In problems like this you need to ask whether the power is radiated into all space or into just part of space.

R Station Satellite You could get Bmax from I = c Bmax2/20, but that’s a lot more work

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. If you are smart, you will write <uB> = <uE> = 1.33x10-15 J/m3 and be done with it.

Today’s agenda: Electromagnetic Waves. Energy Carried by Electromagnetic Waves. Momentum and Radiation Pressure of an Electromagnetic Wave.

Today’s lecture is brought to you by the letter P. Momentum and Radiation Pressure EM waves carry linear momentum as well as energy. The momentum density carried by an electromagnetic wave is This equation is not on your equation sheet, but you have permission to use it for tomorrow’s homework (if needed) (10.28) where dp is the momentum carried in the volume dV. Today’s lecture is brought to you by the letter P.

Maxwell showed that the momentum change of the object is then: When the momentum carried by an electromagnetic wave is absorbed at a surface, pressure is exerted on that surface. 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: incident The direction of the momentum change of the object is in the direction of the incident radiation.

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. incident reflected The direction of the momentum change of the object is again in the direction of the incident radiation.

(Equations on this slide involve magnitudes of vector quantities.) Radiation Pressure The radiation pressure on the object is the force per unit area: From Newton’s 2nd Law (F = dp/dt) we have: For total absorption, incident So (Equations on this slide involve magnitudes of vector quantities.)

Today’s lecture is brought to you by the letter P. This is the instantaneous radiation pressure in the case of total absorption: For the average radiation pressure, replace S by <S>=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 tomorrow’s homework, if necessary. Today’s lecture is brought to you by the letter P.

Using the arguments above it can also be shown that: incident absorbed Using the arguments above it can also be shown that: incident reflected 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.

That’s because today’s lecture is brought to you by the letter P. Example: a satellite orbiting the earth has solar energy collection panels with a total area of 4.0 m2. If the sun’s 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 earth’s 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! That’s because today’s lecture is brought to you by the letter P.

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