[ Assignment View ] [ Print ] Eðlisfræði 2, vor 2007 32 Electromagnetic Waves Assignment is due at 2:00am on Wednesday, March 28, 2007 Credit for problems submitted late will decrease to 0% after the deadline has passed The wrong answer penalty is 2% per part Multiple choice questions are penalized as described in the online help The unopened hint bonus is 2% per part You are allowed 4 attempts per answer Travelling E-M Waves Learning Goal: To understand the formula representing a traveling electromagnetic wave Traveling Electromagnetic Wave Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves Electromagnetic waves comprise combinations of electric and magnetic fields that are mutually compatible in the sense that the changes in one generate the other The simplest form of a traveling electromagnetic wave is a plane wave For a wave traveling in the x direction whose electric field is in the y direction, the electric and magnetic fields are given by, This wave is linearly polarized in the y direction In these formulas, it is useful to understand which variables are parameters that specify the nature of the wave The variables and are the of the electric and magnetic fields Hint A1 What are parameters? Choose the best answer to fill in the blank maxima amplitudes wavelengths velocities The variable is called the of the wave Choose the best answer to fill in the blank velocity angular frequency wavelength The variable is called the of the wave Choose the best answer to fill in the blank wavenumber wavelength velocity frequency What is the mathematical expression for the electric field at the point at time? Part E For a given wave, what are the physical variables to which the wave responds? Hint E1 What are independent variables? only only only only and and and and This is a plane wave; that is, it extends throughout all space Therefore it exists for any values of the variables and and can be considered a function of,,, and Being an infinite plane wave, however, it is independent of these variables So whether they are considered independent variables is a question of semantics When you appreciate this you will understand the conundrum facing the young Einstein If he traveled along with this wave (ie, at the speed of light ), he would see constant electric and magnetic fields extending over a large region of space with no time variation He would not see any currents or charge, and so he could not see how these fields could satisfy the standard electromagnetic equations for the production of fields Page 1 of 11
Part F What is the wavelength of the wave described in the problem introduction? Hint F1 Finding the wavelength The wave described in the introduction is sinusoidal If we let, then the spatial dependence of the wave is given by The wavelength is defined to be the length in the x direction within which the wave repeats itself Mathematically, we require To find, recall that the sine function repeats itself when its argument changes by : Express the wavelength in terms of the other given variables and constants like Part G What is the period of the wave described in the problem introduction? Express the period of this wave in terms of and any constants Part H What is the velocity Hint H1 of the wave described in the problem introduction? How to find Express the velocity in terms of quantities given in the introduction (such as and ) and any useful constants If this electromagnetic wave were traveling in a vacuum its velocity would be equivalent to, the vacuum speed of light Solving M's Eqns to Find c Triangle Electromagnetic Wave Learning Goal: To show how a propagating triangle electromagnetic wave can satisfy Maxwell's equations if the wave travels at speed c Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves Electromagnetic waves consist of mutually compatible combinations of electric and magnetic fields ("mutually compatible" in the sense that changes in the electric field generate the magnetic field, and vice versa) The simplest form for a traveling electromagnetic wave is a plane wave One particularly simple form for a plane wave is known as a "triangle wave," in which the electric and magnetic fields are linear in position and time (rather than sinusoidal) In this problem we will investigate a triangle wave traveling in the x direction whose electric field is in the y direction This wave is linearly polarized along the y axis; in other words, the electric field is always directed along the y axis Its electric and magnetic fields are given by the following expressions: and, where,, and are constants The constant, which has dimensions of length, is introduced so that the constants and have dimensions of electric and magnetic field respectively This wave is pictured in the figure at time Note that we have only drawn the field vectors along the x axis In fact, this idealized wave fills all space, but the field vectors only vary in the x direction We expect this wave to satisfy Maxwell's equations For it to do so, we will find that the following must be true: 1 The amplitude of the electric field must be directly proportional to the amplitude of the magnetic field 2 The wave must travel at a particular velocity (namely, the speed of light) What is the propagation velocity of the electromagnetic wave whose electric and magnetic fields are given by the expressions in the introduction? 1 Phase velocity Express in terms of and the unit vectors,, and The answer will not involve ; we have not yet shown that this wave travels at the speed of light Page 2 of 11
In the next few parts, we will use Faraday's law of induction to find a relationship between and Faraday's law relates the line integral of the electric field around a closed loop to the rate of change in magnetic flux through this loop: To use Faraday's law for this problem, you will need to constuct a suitable loop, around which you will integrate the electric field In which plane should the loop lie to get a nonzero electric field line integral and a nonzero magnetic flux? the xy plane the yz plane the zx plane Consider the loop shown in the figure It is a square loop with sides of length, with one corner at the origin and the opposite corner at the coordinates, Recall that What is the value of the line integral of the electric field around loop at arbitrary time? 1 Integrating along segments 1 and 2 2 Integrating along segments 3 and 4 Hint C3 Integrating around the entire loop Express the line integral in terms of,,,, and/or = Recall that Find the value of the magnetic flux through the surface in the xy plane that is bounded by the loop, at arbitrary time Hint D1 2 Simplifying the integrand Evaluating the integral Express the magnetic flux in terms of,,,, and/or = Part E Now use Faraday's law to establish a relationship between and Part E1 Using Faraday's law Express in terms of and other quantities given in the introduction Page 3 of 11
If the electric and magnetic fields given in the introduction are to be self-consistent, they must obey all of Maxwell's equations, including the Ampère-Maxwell law In these last few parts (again, most of which are hidden) we will use the Ampère-Maxwell law to show that self-consistency requires the electromagnetic wave described in the introduction to propagate at the speed of light The Ampère-Maxwell law relates the line integral of the magnetic field around a closed loop to the rate of change in electric flux through this loop: In this problem, the current is zero (For to be nonzero, we would need charged particles moving around In this problem, there are no charged particles present We assume that the electromagnetic wave is propagating through a vacuum) Part F To use the Ampère-Maxwell law you will once again need to construct a suitable loop, but this time you will integrate the magnetic field around the loop In which plane should the loop lie to get a nonzero magnetic field line integral and hence nonzero electric flux? the xy plane the yz plane the zx plane Part G Use the Ampère-Maxwell law to find a new relationship between and Hint G1 Part G2 Part G3 Part G4 How to approach the problem Find an expression for the left-hand side of the equation Find an expression for the right-hand side of the equation Use the Ampère-Maxwell law Express in terms of,,, and other quantities given in the introduction Part H Finally we are ready to show that the electric and magnetic fields given in the introduction describe an electromagnetic wave propagating at the speed of light If the electric and magnetic fields are to be self-consistent, they must obey all of Maxwell's equations Using one of Maxwell's equations, Faraday's law, we found a certain relationship between and You derived this in Part E Using another of Maxwell's equations, the Ampère-Maxwell law, we found what appears to be a different relationship between and You derived this in Part I If the results of Parts E and I are to agree, what does this imply that the speed of propagation Express in terms of only and must be? You have just worked through the details of one of the great triumphs of physics: Maxwell's equations predict a form of traveling wave consisting of a matched pair of electric and magnetic fields moving at a very high velocity We can measure and independently in the laboratory, and these experimentally determined values lead to a speed of, the speed of light After thousands of years of speculation about the nature of light, Maxwell had developed a plausible and quantitatively testable theory about it Faraday had a hunch that light and magnetism were related, as demonstrated by the Faraday effect (Glass, put in a large magnetic field, will rotate the plane of polarization of light that passes through it) Now Maxwell had predicted an electromagnetic wave with the following properties: 1 It was transverse, with two possible polarizations (which agreed with an already known characteristic of light) 2 It had an extraordinarily high velocity (relative to waves in air or on strings) that agreed with the experimentally determined value for the speed of light Any doubt that light waves were in fact electromagnetic waves vanished as various optical phenomena (such as the behavior of electromagnetic waves at glass surfaces) were predicted and found to agree with the behavior of light This theory showed that lower frequency waves could be created and detected by their interactions with currents in wires (later called antennas) and paved the way to the creation and detection of radio waves Poynting Vector and Power in E-M Waves Poynting Flux and Power Dissipation in a Resistor When a steady current flows through a resistor, the resistor heats up We say that "electrical energy is dissipated" by the resistor, that is, converted into heat But if energy is dissipated, where did it come from? Did it come from the voltage source through the wires? This problem will show you an alternative way to think about the flow of energy and will introduce a picture in which the energy flows in many unexpected places--but not through the wires! We will calculate the Poynting flux, the flow of electromagnetic energy, across the surface of the resistor The Poynting flux, or Poynting vector, has units of energy per unit area per unit time and is related to the electric field vector and the magnetic field vector by the equation Page 4 of 11
, where is the permeability of free space Consider a cylindrical resistor of radius, length, and resistance with a steady current flowing along the axis of the cylinder Which of the following is the most accurate qualitative description of the the magnetic field vector inside the cylindrical resistor? Answer not displayed What can you say about the electric field vector inside the resistor? Answer not displayed Part E In what direction does the Poynting vector point? Hint E1 Cross products in cylindrical coordinates Answer not displayed Part F An electromagnetic wave is traveling through vacuum Its electric field vector is given by Poynting Flux, where is the unit vector in the y direction If is the amplitude of the magnetic field vector, find the complete expression for the magnetic field vector of the wave Hint A1 Relative orientation of and for a wave in vacuum Hint A2 Orientation of and relative to the direction of propagation 3 Determine the direction of propagation of the wave Hint A4 Phase relationship between and Page 5 of 11
What is the Poynting vector, that is, the power per unit area associated with the electromagnetic wave described in the problem introduction? Hint B1 Definition of the Poynting vector Give your answer in terms of some or all of the variables,,,,,, and Specify the direction of the Poynting vector using the unit vectors,, and as appropriate Energy in Electromagnetic Waves Electromagnetic waves transport energy This problem shows you which parts of the energy are stored in the electric and magnetic fields, respectively, and also makes a useful connection between the energy density of a plane electromagnetic wave and the Poynting vector In this problem, we explore the properties of a plane electromagnetic wave traveling at the speed of light along the x axis through vacuum Its electric and magnetic field vectors are as follows: Throughout, use these variables (,,,,,, and ) in your answers You will also need the permittivity of free space and the permeability of free space Note: To indicate the square of a trigonometric function in your answer, use the notation sin(x)^2 NOT sin^2(x) What is the instantaneous energy density in the electric field of the wave? Hint A1 Energy density in an electric field Give your answer in terms of some or all of the variables in What is the instantaneous energy density in the magnetic field of the wave? Hint B1 Energy density in a magnetic field Give your answer in terms of some or all of the variables in What is the average energy density in the electric field of the wave? Hint C1 Average value of Give your answer in terms of and What is the average energy density in the magnetic field of the wave? Page 6 of 11
Hint D1 Average value of Give your answer in terms of and Part E From the previous results, derive an expression for, the average energy density in the whole wave Hint E1 Relationship among,, and Hint E2 Relationship between and for electromagnetic waves in vacuum Hint E3 Relationship among, and for electromagnetic waves in vacuum Express the average energy density in terms of and only Part F The Poynting vector gives the energy flux per unit area of electromagnetic waves It is defined by the relation Calculate the time-averaged Poynting vector of the wave considered in this problem Hint F1 Relationship between and for electromagnetic waves in vacuum Hint F2 Relationship among, and for electromagnetic waves in vacuum Give your answer in terms of, and and unit vectors,, and/or Do not use or If you compare this expression for the time-averaged Poynting flux to the one obtained for the overall energy density, you find the simple relation Thus, the energy density of the electromagnetic field times the speed at which it moves gives the energy flux, which is a logical result Radiation Pressure Radiation Pressure A communications satellite orbiting the earth has solar panels that completely absorb all sunlight incident upon them The total area of the panels is The intensity of the sun's radiation incident upon the earth is about total solar power absorbed by the panels? Suppose this is the value for the intensity of sunlight incident upon the satellite's solar panels What is the Hint A1 Definition of intensity Express your answer numerically in kilowatts to two significant figures Answer not displayed kw What is the total force on the panels exerted by radiation pressure from the sunlight? Page 7 of 11
Hint B1 2 Hint B3 Time derivative of a kinetic energy in relation to momentum Working out the power incident upon the panels Getting the units right Express the total force numerically, to two significant figures, in units of newtons Answer not displayed N Solar Sail A solar sail allows a spacecraft to use radiation pressure, instead of rockets, for propulsion (similar to the way wind propels a sailboat) The sails of such spacecraft are usually made out of a large reflecting panel The size of each panel is maximized to allow the largest possible flux of incident photons, leading to the largest possible total momentum transfer from the incident radiation Because the surface is reflective, the momentum transferred by the photons is twice what they carry For such spacecraft to work, the force from the radiation pressure exerted by the photons must be greater than the gravitational attraction from the star providing the photons The critical parameter turns out to be the mass per unit area of the sail To solve this problem you will need to know the following: mass of the sun:, intensity of sunlight as a function of the distance from the sun:, and gravitational constant: Suppose that a perfectly reflecting circular mirror is initially at rest a distance away from the sun and is oriented so that the solar radiation is incident upon, and perpendicular to, the plane of the mirror What is the critical value of mass/area for which the radiation pressure exactly cancels out the force due to gravity? 1 2 Hint A3 Find the force due to radiation Find the force due to gravity Solving for mass/area Express your answer numerically, to two significant figures, in units of kilograms per meter squared mass/area = 160 10 3 When choosing the material for a solar sail, density, strength, and reflectivity are the principal concerns Given a representative thickness of the sail of 1 sufficiently low density and high strength are carbon fibers These have a density of 160, roughly one fifth that of iron, the only current material with a The Electromagnetic Spectrum Electromagnetic radiation is more common than you think Radio and TV stations emit radio waves when they broadcast their programs; microwaves cook your food in a microwave oven; dentists use X rays to check your teeth Even though they have different names and different applications, these types of radiation are really all the same thing: electromagnetic (EM) waves, that is, energy that travels in the form of oscillating electric and magnetic fields Consider the following: radio waves emitted by a weather radar system to detect raindrops and ice crystals in the atmosphere to study weather patterns; microwaves used in communication satellite transmissions; infrared waves that are perceived as heat when you turn on a burner on an electric stove; the multicolor light in a rainbow; the ultraviolet solar radiation that reaches the surface of the earth and causes unprotected skin to burn; and X rays used in medicine for diagnostic imaging Which of the following statements correctly describe the various forms of EM radiation listed above? A They have different wavelengths B They have different frequencies C They propagate at different speeds through a vacuum depending on their frequency D They propagate at different speeds through nonvacuum media depending on both their frequency and the material in which they travel E They require different media to propagate Hint A1 The electromagnetic spectrum Page 8 of 11
Hint A2 Frequency and wavelength of an EM wave Enter the letters of all the correct options in alphabetical order For instance, if you thought that A, B, and D were correct, then you would enter ABD ABD The frequency and wavelength of EM waves can vary over a wide range of values Scientists refer to the full range of frequencies that EM radiation can have as the electromagnetic spectrum Electromagnetic waves are used extensively in modern technology Many devices are built to emit and/or receive EM waves at a very specific frequency, or within a narrow band of frequencies Here are some examples followed by their frequencies of operation: garage door openers: 400, standard cordless phones: 400 to 500, baby monitors: 490, FM radio stations: 880 to 108, cell phones: 800 to 900, Global Positioning System: 1227 to 1575, microwave ovens: 2450, wireless Internet technology: 24 to 26 Which of the following statements correctly describe the various applications listed above? A All these technologies use radio waves, including low-frequency microwaves B All these technologies use radio waves, including high-frequency microwaves C All these technologies use a combination of infrared waves and high-frequency microwaves D Microwave ovens emit in the same frequency band as some wireless Internet devices E The radiation emitted by wireless Internet devices has the shortest wavelength of all the technologies listed above F All these technologies emit waves with a wavelength in the range 010 to 100 G All the technologies emit waves with a wavelength in the range 001 to 100 Hint B1 Hint B2 Hint B3 Frequency and wavelength of an EM wave Hertz, megahertz, and gigahertz Meters and kilometers Enter the letters of all the correct options in alphabetical order For instance, if you thought that A, B, and D were correct, then you would enter ABD ADEF The frequency band used in wireless technology is strictly regulated by government agencies to avoid undesired interference effects In the United States, the Federal Communications Commission (FCC) is responsible for assigning specific radio frequency bands to different wireless communication systems Despite their extensive applications in communication systems, radio waves are not the only form of EM waves present in our atmosphere Another form of EM radiation plays an even more important role in our life (and the life of our planet): sunlight The sun emits over a wide range of frequencies; however, the fraction of its radiation that reaches the earth's surface is mostly in the visible spectrum (Note that about 35% of the radiation coming from the sun is absorbed directly by the atmosphere before even reaching the earth's surface) The earth, then, absorbs this radiation and reemits it as infrared waves Based on this information, which of the following statements is correct? A The earth absorbs visible light and emits radiation with a shorter wavelength B The earth absorbs visible light and emits radiation with a longer wavelength C The earth absorbs visible light and emits radiation with a lower frequency D The earth absorbs visible light and emits radiation with a higher frequency Hint C1 Relation between frequency and wavelength Enter the letters of the correct options in alphabetical order For instance, if you thought that A, B, and D were correct, then you would enter ABD BC Even though our atmosphere absorbs a very small amount of visible light, it strongly reflects and absorbs infrared waves Therefore the radiation emitted by the earth does not leave the atmosphere Instead, it is reflected back into it, contributing to a warming effect known as the greenhouse effect A large fraction of the ultraviolet (UV) radiation coming from the sun is absorbed by the atmosphere The main UV absorber in our atmosphere is ozone, radiation with frequencies around 938 10 14 What is the wavelength of the radiation absorbed by ozone? In particular, ozone absorbs Page 9 of 11
Hint D1 Frequency and wavelength of an EM wave Hint D2 Meters and nanometers Express your answer in nanometers 320 Electromagnetic Waves and Human Vision The photoreceptors in the human eye, called rods and cones, have different sensitivities to different wavelengths of electromagnetic waves (Notice that the y axis in the figure is a logarithmic scale) The rods, which number over 100 million, are not senstive to color In other words, they note differences in shades of grey (from black to white) and are responsible for a person's ability to see in dim light Cones, which number around 6 million, are responsible for color vision Cones come in three different kinds: 64 of cones are sensitive to long wavelengths of visible light (toward the red end of the spectrum), 32 are sensitive to medium wavelengths, and the remaining 2 are sensitive to short wavelengths (toward the blue end of the spectrum) Colors are differentiated on the basis of the extent to which visible light stimulates each kind of cone Do rods have their peak sensitivity at a higher or lower frequency than cones? Hint A1 Relationship between wavelength and frequency higher lower Do rods and cones have similar sensitivities near the red or near the violet edge of the visible spectrum? Hint B1 Visible light red violet Is it easier to detect a dim red source or a dim violet source of light? Hint C1 Which curve to use red violet At 500, which of the following statements is true? Hint D1 Logarithmic scales Rods are about 1000 times more sensitive than cones Rods are about 3 times more sensitive than cones Rods and cones are about equally sensitive Cones are about 3 times more sensitive than rods Cones are about 1000 times more sensitive than rods Part E Since rods are about 1000 times more sensitive than cones (at 500 ), they should be able to detect smaller values of the electric field Assuming rods and cones are sensitive to the average energy density of an electromagnetic wave, which of the following statements is correct? Hint E1 Average energy density of an electromagnetic wave Hint E2 Relating energy density to electric field Page 10 of 11
Rods are able to detect electric fields 1000 times smaller than the fields detectable by cones Rods are able to detect electric fields times smaller than the fields detectable by cones Rods are able to detect electric fields times smaller than the fields detectable by cones Rods are able to detect electric fields 3 times smaller than the fields detectable by cones Summary 7 of 9 problems complete (7989% avg score) 3595 of 35 points Page 11 of 11