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The Lorentz Force and the Radiation Pressure of Light Haverford College Haverford Scholarship Faculty Publications Astronomy 2009 The Lorentz force and the radiation pressure of light Tony Rothman Stephen P. Boughn Haverford College, [email protected] Follow this and additional works at: https://scholarship.haverford.edu/astronomy_facpubs Repository Citation "The Lorentz Force and the Radiation Pressure of Light" (with Tony Rothman), Am. J. Phys, 77, 122 (2009). This Journal Article is brought to you for free and open access by the Astronomy at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact [email protected]. The Lorentz force and the radiation pressure of light Tony Rothman and Stephen Boughn Citation: Am. J. Phys. 77, 122 (2009); doi: 10.1119/1.3027432 View online: http://dx.doi.org/10.1119/1.3027432 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v77/i2 Published by the American Association of Physics Teachers Related Articles A missing magnetic energy paradox Am. J. Phys. 81, 298 (2013) Bound charges and currents Am. J. Phys. 81, 202 (2013) Noether's theorem and the work-energy theorem for a charged particle in an electromagnetic field Am. J. Phys. 81, 186 (2013) Obtaining Maxwell's equations heuristically Am. J. Phys. 81, 120 (2013) A low voltage “railgun” Am. J. Phys. 81, 38 (2013) Additional information on Am. J. Phys. Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html Downloaded 22 Mar 2013 to 165.82.168.47. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission The Lorentz force and the radiation pressure of light ͒ Tony Rothmana Department of Physics, Princeton University, Princeton, New Jersey 08544 ͒ Stephen Boughnb Department of Astronomy, Haverford College, Haverford, Pennsylvania 19041 ͑Received 8 July 2008; accepted 27 October 2008͒ To make plausible the idea that light exerts a pressure on matter, some introductory physics texts consider the force exerted by an electromagnetic wave on an electron. The argument as presented is mathematically incorrect and has several serious conceptual difficulties without obvious resolution at the classical, yet alone introductory, level. We discuss these difficulties and propose an alternative argument. © 2009 American Association of Physics Teachers. ͓DOI: 10.1119/1.3027432͔ I. THE FRESHMAN ARGUMENT ated by the E-field in the +x-direction and acquires a velocity Ͼ vx 0. The magnetic field then exerts a force on the charge The interaction of light and matter plays a central role, not equal to qvϫB, which points in the +z-direction, the direc- only in physics itself, but in any introductory electricity and tion of propagation of the wave. The electromagnetic wave magnetism course. To develop this topic, most courses intro- therefore carries a momentum in this direction. Reference 2 duce the Lorentz force law, which gives the electromagnetic states that “…the motion of the charge is mainly due to E. force acting on a charged particle, and later discuss Max- Thus v is along E and reverses direction at the same rate that well’s equations. Students are then persuaded that Maxwell’s E reverses direction. But B reverses whenever E reverses. equations admit wave solutions that travel at the speed of Thus vϫB always has the same sign.”4 light, thus establishing the connection between light and A moment’s reflection shows that the last assertion is electromagnetic waves. At this point instructors generally false. After one-half cycle, both E and B change sign. But state that electromagnetic waves carry momentum in the di- because during this time the E-field has accelerated the rection of propagation via the Poynting flux vector and that charge entirely in the +x-direction, the electron at that point light therefore exerts a radiation pressure on matter. The as- ͑ 1 still has a positive x-velocity. In other words, the velocity sertion is not controversial: Maxwell recognized that light and acceleration are essentially 90° out of phase, as in a should cause a radiation pressure, but his demonstration is harmonic oscillator.͒ A similar argument holds for the not immediately transparent to present-day students. z-velocity. Thus the cross product vϫB reverses sign and At least two texts, the Berkeley Physics Course2 and Tipler 3 now points in the negative z-direction, opposite to the direc- and Mosca’s Physics for Scientists and Engineers, attempt tion of propagation. Furthermore, because there is an to make the suggestion that light carries momentum more x-component to the force, one needs to argue that on average plausible by calculating the Lorentz force exerted by an elec- it is zero. tromagnetic wave on an electron. In their discussion the au- Reference 2 claims that the first two terms in Eq. ͑2͒ av- thors claim—with differing degrees of rigor—to show that a erage to zero, the first because E varies sinusoidally, the light wave exerts an average force on the electron in the second because B varies sinusoidally as well and because we direction of propagation. Tipler and Mosca, for example, “…can assume that the increment of velocity along z during then derive an expression for the radiation pressure produced 3 one cycle is negligible, that is, we can take the slowly in- by a light wave. A cursory look at their argument shows that 4 creasing velocity vz to be constant during one cycle.” With it is incorrect in several obvious ways and in other respects these assumptions Ref. 2 concludes that the average force on leads rapidly into deep waters. ͗ ͘ ͗ ͘ ˆ The purpose of this note is threefold: to demonstrate why the charge is F =q vxBy k. Although the result might seem the “freshman” argument is incorrect, to show that nontrivial plausible, it is also incorrect because the velocity and mag- physics must be introduced to correct it, and to present a netic field are out of phase and consequently the time aver- more plausible derivation of radiation pressure that is acces- age of their product vanishes. That this is so, as well as the sible to first-year students. Consider, then, the situation previous claim, can be seen by an integration of the equation shown in Fig. 1. We assume that a light wave propagates in of motion. the +z-direction, its E-field oscillates in the x-direction, and its B-field oscillates in the y-direction. The wave impinges on a stationary particle with charge q, exerting on it a force according to the Lorentz force law. In units with c=1 the II. EQUATION OF MOTION Lorentz force is To determine the momentum of the charge, which we take F = q͑E + v ϫ B͒, ͑1͒ to be an electron, assume the electric and magnetic fields ͑␻ ␾͒ˆ which reduces to of the light wave are given by E=E0 sin t+ i and ͑␻ ␾͒ˆ ␾ ͑ ͒ˆ ˆ ͑ ͒ B=B0 sin t+ j, where is an arbitrary phase angle. In F = q Ex − vzBy i + qvxByk. 2 / ͑ ͒ our units E0 =B0. We set FLorentz=mdv dt in Eq. 2 to obtain The freshman argument goes like this: Assume that a pair of coupled ordinary linear first-order equations for the Eϳsin͑␻t͒ and Bϳsin͑␻t͒. The particle is initially acceler- electron velocity: 122 Am. J. Phys. 77 ͑2͒, February 2009 http://aapt.org/ajp © 2009 American Association of Physics Teachers 122 Downloaded 22 Mar 2013 to 165.82.168.47. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission x 0.2 E 0.15 z vx,vz q 0.1 y B 0.05 Fig. 1. An electromagnetic wave traveling in the z-direction strikes a point particle with charge q.TheE-field is taken in the x-direction and the B-field is taken in the y-direction. 0 10 20 30 40 ωt ␻ ␻ /␻ ␾ Fig. 2. The x- and z-velocities versus t for c =0.1 and =0. Note that ␻ /␻ ͑ ͒ ӷ / from the small c approximation see text vx vz; in this case vz vx dv =0.1. z = ␻ sin͑␻t + ␾͒v , ͑3a͒ dt c x dv x = ␻ sin͑␻t + ␾͓͒1−v ͔, ͑3b͒ Additional insight into the solutions can be obtained by c z ␻ /␻Ӷ dt examining the limit c 1. For ordinary light sources at ␻ϳ 16 / ␻ ϵ / optical frequencies 10 rad s, consideration of the where we have let c qB0 m, the cyclotron frequency. ͑ ͒ ␻ /␻ϳ −11 ͑ ͒ Poynting flux in the following section gives c 10 , Equations 3 have the somewhat surprising analytic solu- and so the limit is well satisfied. For high-powered lasers, tions such as those at the National Ignition Facility with pulse ϳ ␻ ␻ ␻ Ӷ␻ ␻ energy 2 MJ, it is possible that c exceeds . For c , v ͑t͒ = c cosͫ c cos͑␻t + ␾͒ͬ ͑ ͒ ␻ /␻ z 1 ␻ expansion of the solutions 6 to lowest order in c for ␾=0 yields ␻ + c sinͫ c cos͑␻t + ␾͒ͬ +1, ͑4a͒ 1 ␻ 2 2 ␻ v Х ͩ c ͪ ͓cos͑␻t͒ −1͔2, ͑7a͒ z 2 ␻ ␻ ͑ ͒ ͫ c ͑␻ ␾͒ͬ ␻ v t = c sin cos t + c x 1 ␻ Х ͩ ͓ͪ1 − cos͑␻t͔͒. ͑7b͒ vx ␻ ␻ ͫ c ͑␻ ␾͒ͬ ͑ ͒ Both v and v are positive definite, as shown in Fig. 2. − c2 cos cos t + , 4b z x ␻ Therefore their averages must be as well. This in itself con- ͗ ͘ ͗ ͘ where c and c are the integration constants.
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