MISN-0-211 z ENERGY AND MOMENTUM IN ELECTROMAGNETIC WAVES y È B` x ENERGY AND MOMENTUM IN ELECTROMAGNETIC WAVES by J. S. Kovas and P. Signell Mihigan State University 1. Desription................................................ 1 2. Stati Fields a. Energy in Stati Fields.................................. 1 b. Energy Density in Stati Fields..........................2 3. Eletromagneti Waves a. Energy Density..........................................2 b. Energy Current Density................................. 3. Momentum Current Density............................. 3 d. Time-Average Quantities................................ 3 e. Pressure From Absorption and Refletion................ 4 Aknowledgments............................................4 Glossary...................................................... 4 Projet PHYSNET Physis Bldg. Mihigan State University East Lansing, MI 1
ID Sheet: MISN-0-211 Title: Energy and Momentum in Eletromagneti Waves Author: J. S. Kovas and P. Signell, Physis, Mihigan State Version: 2/1/2000 Evaluation: Stage 0 Length: 1 hr; 12 pages Input Skills: 1. Voabulary: eletromagneti wave (MISN-0-210), harge density (MISN-0-147), urrent density (MISN-0-144). 2. Desribe the storage of energy in a apaitor (MISN-0-135). 3. Desribe the storage of energy in an indutor (MISN-0-144). 4. Relate the eletromagneti wave s eletri and magneti fields to eah other and to the diretion of propagation of the wave. (MISN- 0-210). 5. Relate fore to rate of hange of momentum (MISN-0-15). Output Skills (Knowledge): K1. Voabulary: eletri energy urrent density, magneti energy urrent density, energy urrent density, momentum urrent density. K2. Given the energy density in an eletri fields, and the knowledge that the eletri field energy is given by E e = (1/ )E 2 where E is the eletri field, derive the expression for the time-average energy urrent density in the wave. Output Skills (Rule Appliation): R1. Relate the eletri and magneti field amplitudes to the intensity, energy density, and momentum density assoiated with an eletromagneti wave of a given frequeny. Post-Options: 1. Brewster s Law and Polarization (MISN-0-225). THIS IS A DEVELOPMENTAL-STAGE PUBLICATION OF PROJECT PHYSNET The goal of our projet is to assist a network of eduators and sientists in transferring physis from one person to another. We support manusript proessing and distribution, along with ommuniation and information systems. We also work with employers to identify basi sientifi skills as well as physis topis that are needed in siene and tehnology. A number of our publiations are aimed at assisting users in aquiring suh skills. Our publiations are designed: (i) to be updated quikly in response to field tests and new sientifi developments; (ii) to be used in both lassroom and professional settings; (iii) to show the prerequisite dependenies existing among the various hunks of physis knowledge and skill, as a guide both to mental organization and to use of the materials; and (iv) to be adapted quikly to speifi user needs ranging from single-skill instrution to omplete ustom textbooks. New authors, reviewers and field testers are welome. PROJECT STAFF Andrew Shnepp Eugene Kales Peter Signell Webmaster Graphis Projet Diretor ADVISORY COMMITTEE D. Alan Bromley Yale University E. Leonard Jossem The Ohio State University A. A. Strassenburg S. U. N. Y., Stony Brook Views expressed in a module are those of the module author(s) and are not neessarily those of other projet partiipants. 2001, Peter Signell for Projet PHYSNET, Physis-Astronomy Bldg., Mih. State Univ., E. Lansing, MI 48824; (517) 355-3784. For our liberal use poliies see: http://www.physnet.org/home/modules/liense.html. 3 4
MISN-0-211 1 ENERGY AND MOMENTUM IN ELECTROMAGNETIC WAVES by J. S. Kovas and P. Signell Mihigan State University 1. Desription Eletromagneti waves travel at the speed of light, and they transport energy and momentum from the plaes where they originate to the plaes they travel to. The energy and momentum they arry an be easily omputed from a knowledge of their amplitudes. Knowing the energy arried by a wave, one an alulate the heating produed when an eletromagneti wave is absorbed by a material objet. Similarly, knowing the momentum being arried by a wave allows us to alulate the pressure exerted on an objet that absorbs, deflets, or reflets the wave. 2. Stati Fields 2a. Energy in Stati Fields. Elsewhere we have shown that the energy stored in the eletri field of a apaitor is: 1 E e = 1 2 CV 2. (1) This amount of energy an be reovered if the apaitor is disharged so its voltage and hene its internal eletri field drops to zero. Similarly, the energy stored in the magneti field of an indutor is: 2 E m = 1 2 LI2, (2) and this amount of energy an be reovered if the urrent through the indutor is shut off so its magneti field drops to zero. A ruial point is that one an prove in general that the energy stored in eletri and magneti fields, inluding the examples shown above, an 1 See Capaitane and Capaitors, (MISN-0-135). 2 See Indutane and Indutors, (MISN-0-144). MISN-0-211 2 be alulated from: 3 E e = all spae where E is the eletri field, and E m = where B is the magneti field. all spae ( 1 ( 1 8πk m ) E 2 dv, (3) ) B 2 dv, (4) 2b. Energy Density in Stati Fields. The integrands in the right hand sides of Eqns. (3) and (4) are over all spae, so their integrands an be interpreted as energy densities, energy per unit volume: and de e dv = 1 E 2, (5) de m dv = 1 8πk m B 2. (6) 3. Eletromagneti Waves 3a. Energy Density. For the ase of an eletromagneti wave the eletri and magneti amplitudes are related: 4 E = B, so we an use that and the relation between the eletri and magneti fore onstants: 5 k m = k e / 2 to rewrite Eq. (6) as the magneti energy density in an eletromagneti wave: de m dv = 1 E 2. (7) Note that this is exatly the same as the eletri energy density, Eq. (5), so the total energy density for the eletromagneti wave is: de dv = 1 E 2. (8) 3 See any advaned Eletriity and Magnetism textbook. 4 See Eletromagneti Waves From Maxwell s Equations, MISN-0-110. 5 See Magneti Interations, (MISN-0-124). 5 6
MISN-0-211 3 3b. Energy Current Density. As an eletromagneti wave travels at the speed of light, the energy it arries an be expressed as an energy urrent in analogy with eletri urrent. Reall that eletri urrent density j is defined as the amount of harge rossing unit area perpendiular to the urrent diretion, per seond. This urrent density an be expressed in terms of its harge density ρ and the drift veloity v of the harges: j = ρ v. (9) Similarly, we an define the energy urrent density E as the energy per unit area perpendiular to the urrent diretion, per seond. Then it is given by the analog of Eq. (9): E = 1 E 2 v. (10) Making use of the diretions of the fields, this is usually written: E = 2 E B. (11) 3. Momentum Current Density. For partiles that are traveling at high speeds, speeds not negligible ompared to the speed of light, momentum and energy are related by: 6 p = E v. (12) 2 Applying this equation to our eletromagneti wave, where v =, we get, from Eq. (11), the momentum urrent density: P = E B. (13) 3d. Time-Average Quantities. The eletri and magneti amplitudes in the energy and momentum urrent densities, Eqns. (11) and (13), are osillating funtions of time while in most appliations we are interested in time-averaged quantities. The onversion is easy beause the time averages that our in those equations are simply osine-squared funtions of time, and any osine-squared funtion has an average value of one-half (0.500). Thus: E = 6 See Relativisti Momentum, MISN-0-24. 2 E0 B 0. (14) MISN-0-211 4 P = E0 B 0. (15) 3e. Pressure From Absorption and Refletion. Sine fore is hange of momentum per unit time, the time-average pressure exerted on a surfae by an eletromagneti wave is the hange of the wave s timeaverage momentum urrent density due to its interation with the surfae: P = P. For the ase of absorption of the wave, P = P = E0 B 8πk 0. (16) e For the ase of perfet refletion where the wave s momentum is simply reversed in diretion: P = 2P = E0 B 4πk 0. (17) e Aknowledgments Preparation of this module was supported in part by the National Siene Foundation, Division of Siene Eduation Development and Researh, through Grant #SED 74-20088 to Mihigan State University. Glossary eletri energy urrent density: the eletri-field energy being transported, normal to a plane, per seond per unit area in the plane. The SI unit is watts per square meter. magneti energy urrent density: the magneti-field energy being transported, normal to a plane, per seond per unit area in the plane. The SI unit is watts per square meter. energy urrent density: the energy being transported, normal to a plane, per seond per unit area in the plane. The SI unit is watts per square meter. momentum urrent density: the normal omponent of the momentum at a plane, per unit area in the plane. The SI unit is kilogram meters per seond per square meter. 7 8
MISN-0-211 PS-1 MISN-0-211 PS-2 Brief Answers: PROBLEM SUPPLEMENT 1. 3.32 10 4 watts/m 2 Note: Problem 3 also appears in this module s Model Exam. 2. 6.64 10 4 W/m 2 1. The eletri field of a plane eletromagneti wave in vauum is represented by: E x = 0, E y = 0.50 (N/C) os [ 2.09 m 1 (x t) ], E z = 0. Determine the wave s intensity, its average energy per unit area per unit time. 2. Solve Problem 1 for the wave represented by: A. ŷ. B. 6.75 10 9 N. C. 0.750 J/(m 2 s). D. 1.75 10 5 joules falls on area. E. ŷ. F. 0.250 10 8 J/m 3. G. Zero. E x = 0, E y = 0.50 (N/C) os [ 0.419 m 1 (x t) ], E z = 0.50 (N/C) os [ 0.419 m 1 (x t) ]. 3. The amplitude of the eletri field of an eletromagneti wave is 180π volts per meter. The eletri field is in the ˆx-diretion, the magneti field in the ẑ-diretion. Determine: a. The average energy urrent density assoiated with this wave. [C] b. The average momentum urrent density arried by this wave. [F]. The diretion of this momentum. [A] d. The energy inident in one day upon a surfae oriented in the x-z plane that has an area of 2.7 m 2. [D] e. The energy inident upon this surfae if the surfae is oriented in the x-y plane. [G] f. The fore exerted on this surfae by the radiation if it is ompletely absorbed by the surfae. [B] g. The diretion of this fore. [E] 9 10
MISN-0-211 ME-1 MODEL EXAM E = 2 E B. 1. See Output Skills K1-K2 in this module s ID Sheet. 2. The amplitude of the eletri field of an eletromagneti wave is 180π volts per meter. The eletri field is in the ˆx-diretion, the magneti field in the ẑ-diretion. Determine: a. The average energy urrent density assoiated with this wave. b. The average momentum urrent density arried by this wave.. The diretion of this momentum. d. The energy inident in one day upon a surfae oriented in the x-z plane that has an area of 2.7 m 2. e. The energy inident upon this surfae if the surfae is oriented in the x-y plane. f. The fore exerted on this surfae by the radiation if it is ompletely absorbed by the surfae. g. The diretion of this fore. Brief Answers: 1. See this module s text. 2. See Problem 3 in this module s Problem Supplement, 11 12