aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n ENERGY, FORCES, FIELDS AND THE LORENTZ FORCEFORMULA
ARTICE M. DAVISPROFESSOR EMERITUSSAN JOSE STATE UNIVERSITY
Abstract.
We apply a simple decomposition to the energy of a moving par-ticle. Based on this decomposition, we identify the potential and kinetic ener-gies, then use them to give general definitions of momentum and the variouskinds of forces exerted on the particle by fields, followed by the generalizationof Newton’s second law to accomodate these generally defined forces. We showthat our generalization implies the Lorentz force law as well as Lagrange’s equa-tion, along with the usually accepted Lagrangian and the associated velocitydependent potential of a moving charged particle.
Introduction.
The motivation for this paper is to present a rigorous derivationof the Lorentz force law in a nonrelativistic context. The provenance of the law issomewhat obscure. Lorentz’s original paper, , written in French and apparently nottranslated into English, assumed the ether as a medium and contained a number ofunwarranted assumptions and vague definitions. Lorentz apparently then rejectedhis own derivation, choosing in his later monograph “The Theory of Electrons” to simply say the law was “...got by generalizing the results of electromagneticexperiments.” He did not specify which experiments, but he clearly saw fit notto refer to his own earlier paper. Others have derived the formula by assuminga generalized Lagrangian for a moving charged particle, while still others havederived the Lagrangian presuming the Lorentz force formula. This type of analysisresults in a generalized potential, namely ψ = φ − ~A · ~v , for the moving particle,which we will derive. The quantity ~A is the vector potential, which some authors(including Maxwell) referred to as the electromagnetic momentum; but others insistthat the electromagnetic momentum is ǫ h ~E × ~B i . We feel that much of thecontroversy in this matter, as well as in many others, is due to the lack of generaldefinitions. We believe we have supplied such definitions in this work. The work presented here assumes that the energy of a moving particle is a knownquantity. The difficulty of defining energy has been discussed by others, but wewill deem it to be a primitive notion. Otherwise, we will adopt the operationalpoint of view. This means that each fundamental quantity is defined by thedescription of a measuring instrument and a recipe for its use to measure theassociated variable, while a derived quantity is defined by an equation expressingit in terms of previously defined fundamental quantities and/or previously definedderived quantities. We require, of course, that the latter never be self referential;that is, a flow diagram of all the definitions should contain no loops—it must be atree structure.
Particles: Mass and Accelerational Force.
We assume the usual definitionof a particle, namely a vanishingly small region of space having certain propertiesthrough which it interacts with other similarly defined regions of space. We assumethat such interactions are mediated by energy transfer through the empty spacesbetween them, these interactions depending upon the parameters mass and charge.Let us select two particles, remove them from any outside influence, place them ata given location in space with zero velocity, and measure their accelerations due tomutual influence. We will take it as a fact that in any such test the accelerationsare always oppositely directed; furthermore, that the ratio of the magnitudes of theaccelerations is always the same. Selecting one of these particles as a reference anddenoting its acceleration by ~a and that of the other by ~a , we define the mass ofthe other particle by m = a a , (1)where | ~a | = a and | ~a | = a . If the two particles are identical then a = a so m = 1; hence, our reference particle has unit mass. Taking note of the opposingdirections of the accelerations caused by the interaction, we can write ~f a = − ~a = m~a, (2)which we will define to be the accelerational force on the nonreference particle. Thisprocedure for defining mass and force is due to Mach. There exist forces, however,that are not associated with a moving object, for example the force of a spring ona weight it supports. We will offer more general definitions in a subsequent sectionof this paper.
Charge.
Now let’s select an arbitrary particle and test it against all the otherparticles in the universe, measuring the force between each pair. If no other particlerepels it we will say it is uncharged. If, on the other hand, at least one other particlerepels it, we will say our original particle is positively charged and assign it a chargeof one unit thus making it our reference charge. Now we segregrate all particles inthe universe into two classes: each of those in the first class repels our referencecharge and is said to have a positive charge and each of those in the second classattracts the particle having our reference charge. Next, divide the second classinto two subcategories: those which repel any other particle in the second classand those which attract each other particle in the second class. We will say thatthose in the repelling subcategory are negatively charged and those in the attractingsubcategory uncharged. Thus, each and every particle in the universe is positivelycharged, negatively charged, or uncharged. Finally, we invoke the Coulomb lawto determine the magnitude of a given charge. Thus, we have defined charge, likemass, in an operational manner.
Energy Considerations.
Consider a particle moving freely through space—thatis, a particle with no mechanical constraints—and let U ( ~r, ~v, t ) be its energy. Write U ( ~r, ~v, t ) = U ( ~r, , t ) + [ U ( ~r, ~v, t ) − U ( ~r, , t )] = φ ( ~r, t ) + T ( ~r, ~v, t ) , (3)where φ and T have obvious definitions. The former is called the potential energyand the latter the kinetic energy. We define the generalized momentum by ~p ( ~r, ~v, t ) = ∇ ~v T ( ~r, ~v, t ) = ∇ ~v U ( ~r, ~v, t ) , (4) NERGY, FORCES, FIELDS AND THE LORENTZ FORCE FORMULA 3 where the subscript ~v on the gradient operator refers to differentiation with respectto the components of velocity: ∇ ~v = ˆ e i ∂∂v i . (5)Then we have T ( ~r, ~v, t ) = Z ~v ~p ( ~r, ~α, t ) · d~α, (6)where the integration is that of a line integral in velocity space with both positionand time held fixed. Next, let us apply a similar decomposition to the generalizedmomentum, writing ~p ( ~r, ~v, t ) = ~p ( ~r, , t ) + [ ~p ( ~r, ~v, t ) − ~p ( ~r, , t )] = ~A ( ~r, t ) + ~Q ( ~r, ~v, t ) , (7)where ~A and ~Q have obvious definitions. We will call ~A the potential momentumdue to the field and ~Q the inertial momentum. Letting ~Q = Q j ˆ e j , define ~m j ( ~r, ~v, t ) = ∇ ~v Q j ( ~r, ~v, t ) = ˆ e i ∂Q j ∂v i = m ij ˆ e i . (8)Then Q j ( ~r, ~v, t ) = Z ~v ~m j ( ~r, ~α, t ) · d~α = Z ~v d~α T ~m j = Z ~v m ij dα i . (9)Finally, define the matrix M = [ m ij ] = (cid:20) ∂Q j ∂v i (cid:21) . (10)We will call M = M ( ~r, ~v, t ) the generalized mass tensor. Using it, we have ~Q ( ~r, ~v, t ) = Z ~v d~α T M ( ~r, ~α, t ) , (11)where d~α T is the transpose of the differential of the integration variable and d~α T M denotes the matrix product of the row matrix d~α T and the square matrix M . The Classical Case.
Classically, the mass tensor becomes M = mI = m , (12)where m is a constant which we have already operationally defined. Then theparticle momentum is ~Q ( ~r, ~v, t ) = m~v, (13)the kinetic energy is T ( ~r, ~v, t ) = ~A ( ~r, t ) · ~v + 12 mv , (14)and the total particle energy is U ( ~r, ~v, t ) = φ ( ~r, t ) + T ( ~r, ~v, t ) = φ ( ~r, t ) + ~A ( ~r, t ) · ~v + 12 mv . (15)Following Maxwell, we will call the term ~A · ~v the electrokinetic energy. In whatfollows, we will restrict ourselves to the classical case. ARTICE M. DAVIS PROFESSOR EMERITUS SAN JOSE STATE UNIVERSITY
Generalized Forces.
We will now define three generalized forces. The first willbe “positional force,” the force on the particle caused by change of position. It willbe defined by ~f P = −∇ ~r h φ ( ~r, t ) − ~A ( ~r, t ) · ~v i , (16)where ∇ ~r = ˆ e i ∂/∂x i . Here are the reasons for our choice of signs. If the field exertsforce on the particle, the particle moves from a region of higher potential energy toa region of lower potential energy. On the other hand, a force exerted by the fieldon a particle tends to increase its kinetic energy—and ~A · ~v is part of the kineticenergy.The second generalized force we will define is the “inertial force,” given by thetime rate of change of the generalized momentum: ~f I ( ~r, ~v, t ) = ddt ~p ( ~r, ~v, t ) = ddt ~A ( ~r, t ) + ddt Q ( ~r, ~v, t ) = ddt ~A ( ~r, t ) + m~a. (17)We now generalize Newton’s second law by postulating that ~f P = ~f I . (18)Supressing arguments for simplicity of notation and applying standard vector iden-tities to equations (16), (17), and (18) , we obtain − ∇ ~r h φ − ~A · ~v i = ∇ ~r h ~A · ~v i − ~v × h ∇ ~r × ~A i + ∂ t ~A + m~a. (19)Our third and last generalized force is that part of the inertial force which we havealready defined to be the ’‘accelerational force,” ~f a = m~a . Using it in equation (19)gives ~f a = −∇ ~r φ − ∂ t ~A + ~v × h ∇ ~r × ~A i . (20)If we define the ‘positional field” by ~E = −∇ ~r φ − ∂ t ~A (21)and the “motional field” by ~B = ∇ ~r × ~A, (22)we can rewrite equation (20) in the extremely simple form ~f a = ~E + ~v × ~B. (23)We see at once that ~E is the accelerational force on a particle at rest and ~v × ~B theadded accelerational force due to its motion. These interpretations clearly serve asoperational definitions of these two fields. The Lorentz Force Law.
We now recognize that the energy of a particle mightdepend upon both mass and charge. At first suppressing the position, velocity, andtime arguments and then reintroducing them, we write U = U ( m, q ) = U ( m,
0) + [ U ( m, q ) − U ( m, V m ( ~r, ~v, t ) + W q ( ~r, ~v, t ) , (24)with obvious definitions of V m and W q . Next, we perform, for each of V m and W q the decomposition in equation (15), using obvious notation and assuming that eachcomponent is normalized to unit mass or charge as appropriate: V m ( ~r, ~v, t ) = mφ m ( ~r, t ) + 12 mv . (25) NERGY, FORCES, FIELDS AND THE LORENTZ FORCE FORMULA 5 and W q ( ~r, ~v, t ) = qφ q ( ~r, t ) + q ~A ( ~r, t ) · ~v. (26)We have made two key assumptions here, namely(1) V m has no component due to field momentum.(2) W q is independent of particle mass.These assumptions can, of course, be removed at the expense of a more complexresulting theory. It is also convenient in working strictly with electrodynamics toassume that mφ m << qφ q ≈ qφ . These assumptions, taken together, permit us toremove all subscripts and rewrite the total particle energy in equation (15) as U ( ~r, ~v, t ) = qφ ( ~r, t ) + q ~A ( ~r, t ) · ~v + 12 mv (27)and our field force equation in (23) as ~f a = q ~E + q~v × ~B, (28)where all terms except ~f a are purely electrical in nature. We now see that ~E isclearly the electrical field intensity and ~B the magnetic field as these quantitiesare normally defined in electromagnetic field theory, the former being the per-unitforce on a stationary charged particle and the second term in equation (28) beingthe incremental force added by the magnetic field. Lagrange’s Equation.
Let’s return to Newton’s generalized second law, ~f p = ~f I ,and write it in terms of the basic definitions: − ∇ ~r h qφ − q ~A · ~v i = ddt ∇ ~v (cid:20) q ~A · ~v + 12 mv (cid:21) . (29)Note that T = 12 mv + q ~A · ~v. (30)In component form (29) becomes ddt ∂T∂v i + q ∂φ∂x i = ∂∂x i h q ~A · ~v i . (31)Next, we define L = T − qφ = 12 mv − q h φ − ~A · ~v i (32)and note that φ is not a function of ~v to obtain ddt ∂L∂v i − ∂L∂x i = 0 . (33)We now see that ψ ( r, v, t ) = φ ( r, t ) − A ( r, t ) · ~v (34)has the character of a velocity dependent potential. We also see that L is the com-monly accepted Lagrangian for a particle in the electromagnetic field, usually eitherassumed on an ad hoc basis or derived from the Lorentz force law as an assuption. Equation (33) can, of course, be immediately extended by expressing the cartesianposition variables in terms of generalized coordinates. Standard procedures canthen be applied to show that Lagrange’s equation is invariant under this transfor-mation. Thus, the theory just outlined produces both the Lorentz force equationand Lagrange’s equation as results, rather than as assumptions.
ARTICE M. DAVIS PROFESSOR EMERITUS SAN JOSE STATE UNIVERSITY
Finally, we note that the steps leading from the generalized Newton’s secondlaw expresssd by equation (18) with its associated force definitions to Lagrange’sequation in (33) are all reversible; in other words the two equations are equivalentmathematical assertions. Hence, if one accepts the latter, one must accept theformer. We feel, therefore, that the theory outlined in this paper offers a solid,theoretically sound approach to introducing both topics in introductory courses infields and classical mechanics.
Acknowledgements.
I would like to express my appreciation to Vladimir Onoochinfor a number of productive discussions about the topics treated her.
Notes H. A. Lorentz,
La Th´eorie El´ectromagn´etique de Maxwell et son Application aux Corps Mou-vants (Leiden, E. J. Brill, 1892). H. A. Lorentz,
The Theory of Electrons and its Applications to the Phenomena of Light andRadiant Heat , 2 nd ed, (Teubner, Leipzig, 1916). See the sentence immediately after equation (23), page 14 in reference [2] See, for example, J. R. Taylor,
Classical Mechanics , (University Science Books, 2005), page273; Goldstein, Poole, and Safco,
Classical Mechanics , (Addison-Wesley, San Francisco,2002), sec.1.5, or M. G. Calkin,
Lagrangian and Hamiltonian Mechanics ,(World Scientific, Singapore, 1996),p. 46. O. D. Johns,
Analytical Mechanics for Relativity and Quantum Mechanics , (Oxford UniversityPress, 2005), sec. 2.17. See also E. J. Konopinski, “What the Electromagnetic Vector PotentialDescribes,” Am. J. Phys., , 499-502 (1978). See D. Griffiths, “Resource Letter EM-1: Electromagnetic Momentum,” Am. J. Phys, ,7-18, (2012) for a thorough discussion of this issue as well as for an extensive list of references. Our definitions do not generally resolve the aforementioned bone of contention, though itdoes imply that ~A is the appropriate quantity insofar as the motion of a single charged particle isconcerned. See, for example, Feynman, Leighton, and Sands,
The Feynman Lectures on Physics , (Addison-Wesley, Reading, 1964), v1, sec. 4.1. But see Falk, Hermann, and Schmid, “Energy Forms orEnergy Carriers?”, Am. J. Phys., , 1074-1076, (1983). P. Bridgman,
The Logic of Modern Physics , (McMillan, New York, 1958), chapter 1. Because their velocities are initially zero we are defining rest mass and ignoring relativisticeffects. We will also assume “near” interactions so that retardation effects may be neglected We will assume throughout the following work that a vector can be thought of as a 3 × W. Mach,
The Science of Mechanics , (The Open Court Publishing Company, 1983) 4 th ed.,chapter 2, section 5. There are additional consistency requirements which were pointed out byothers after Mach advanced this definition. See L. Eisenbud, “On the Classical Laws of Motion,”Am. J. Phys., , 144-159, (2005). Note that the uncharged particles attract every other particle (weakly!) because of gravita-tional forces. We are assuming the particle has no energy of rotation (so no explicit dependence uponrotational angles or velocities). We are mixing matrix and vector notation here, but as all our vectors are column matricesthis should not create confusion.16