aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n How to get from static to dynamic electromagnetism
J¨urgen K¨onig
Theoretische Physik and CENIDE, Universit¨at Duisburg-Essen, 47057 Duisburg,GermanyE-mail: [email protected]
January 21, 2021
Abstract.
We demonstrate how to derive Maxwell’s equations, including Faraday’slaw and Maxwell’s correction to Amp`ere’s law, by generalizing the description ofstatic electromagnetism to dynamical situations. Thereby, Faraday’s law is introducedas a consequence of the relativity principle rather than an experimental fact, incontrast to the historical course and common textbook presentations. As a by-product, this procedure yields explicit expressions for the infinitesimal Lorentz and,upon integration, the finite Lorentz transformation. The proposed approach helps toelucidate the relation between Galilei and Lorentz transformations and provides analternative derivation of the Lorentz transformation without explicitly referring to thespeed of light.
The theory of electromagnetism, usually taught after a course on mechanics,introduces as a new concept the electromagnetic field. It is characterized by the electricand magnetic field strengths, denoted by E ( r , t ) and B ( r , t ). Maxwell’s famous equationsdescribe how E ( r , t ) and B ( r , t ) arise from static and moving charges, expressed interms of the charge density ρ ( r , t ) and the charge current density j ( r , t ), but also how E ( r , t ) and B ( r , t ) influence each other. The connection to mechanics is provided by theelectromagnetic force on charged bodies, referred to as the Lorentz force.Some textbooks on electrodynamics write down Maxwell’s equations at the verybeginning and, then, specify them when discussing specials limits such as electrostaticsand magnetostatics. Others rather follow the historical course by starting with electro-and magnetostatics, then include Faraday’s law as an experimental fact, and, finally, addMaxwell’s correction to Amp`ere’s law to ensure charge conservation. In both cases, therelativity principle is only discussed after treating electromagnetic waves in vacuum anddemonstrating that their propagation speed c = 1 / √ µ ǫ is independent of the choseninertial frame of reference. The Lorentz transformation as the correct transformationbetween two inertial systems, is, then, derived from combining the relativity principlewith the universality of the vacuum speed of light. It couples time and space coordinatesin some intricate way that leads to astonishing effects such as time dilatation andLorentz contraction, which are at odds with our intuition built on everyday life. Tounify mechanics and electrodynamics within one consistent theory, one has to replace ow to get from static to dynamic electromagnetism Table 1.
Static electromagnetism is described by the static field equations Eqs. (1)and the Lorentz force Eq. (2). Dynamic electromagnetism is achieved in three steps.First, all quantities are made time dependent, leading to the field equations Eqs. (3).Second, charge conservation is used to obtain Maxwell’s correction to Amp`ere’s law.This leads to the field equations Eqs. (6). Finally, Faraday’s law is obtained from therelativity principle. The result is Maxwell’s equations Eqs. (23). static electromagnetism dynamic electromagnetism(1) = ⇒ (3) = ⇒ (6) = ⇒ (22)time dependence charge conservation relativity principleNewtonian mechanics by relativistic mechanics, and one ends up with Einstein’s specialrelativity theory.In this paper, we propose an alternative way to teach electrodynamics. As sketchedin Table 1, Maxwell’s equations are derived by extending the field equations of electro-and magnetostatics in three steps. In the first step, all quantities are simply madetime dependent. The result is, however, in conflict with both charge conservation andthe relativity principle. To guarantee charge conservation, we add, as the second step,Maxwell’s correction to Amp`ere’s law. We show that the resulting field equations areGalilei invariant, while the Lorentz force is not. Therefore, we require, in the third andfinal step, the relativity principle to hold for both the field equations and the Lorentzforce. This leads us to Faraday’s law, which completes Maxwell’s equations. In addition,this procedure provides a recipe for how to correctly transform from one inertial systemto another, first for an infinitesimal relative velocity of inertial systems and then, afterintegration, for a finite relative velocity. The result is the Lorentz transformation.
1. From Static to Dynamic Electromagnetism in Three Steps
The starting point for deriving Maxwell’s equation is the description of staticelectromagnetism, summarized by the static field equations (in SI units) ∇ · E ( r ) = ρ ( r ) ǫ (1 a ) ∇ × E ( r ) = (1 b ) ∇ · B ( r ) = 0 (1 c ) ∇ × B ( r ) = µ j ( r ) (1 d )as well as the Lorentz force F = Q ( E + v × B ) (2) ow to get from static to dynamic electromagnetism Q that moves with velocity v .In static electromagnetism, an electromagnetic field can be considered as thecoexistence of two independent physical entities, namely an electric field generatedby a static charge density ρ ( r ) and a magnetic field generated by a stationary anddivergence-free charge current density j ( r ) with ∇ · j ( r ) = 0. The respective sourceterms ρ ( r ) and j ( r ) can be understood as being independent of each other (althoughphysical currents are, of course, composed of moving charges). On the one hand,pure electrostatics appears for a static charge distribution, such that j = . Puremagnetostatics, on the other hand, can be realized by a stationary current distributioncomposed of moving charges of one type (e.g., negatively charged electrons) while staticcharges of another type (e.g., positively charged ions) guarantee charge neutrality, suchthat ρ = 0. In that respect, static electromagnetism is nothing more than a collectiveterm for the two independent theories of electrostatics, described by Eqs. (1 a ) and (1 b ),and magnetostatics, described by Eqs. (1 c ) and (1 d ). The Lorentz force Eq. (2) on atest charge is, then, nothing but the superposition of the electric and magnetic forcegenerated by the electric and magnetic field, respectively.Since static electromagnetism is a special case of dynamic electromagnetism, it isimpossible to deduce Maxwell’s equations from the static field equations Eqs. (1 a )-(1 d ).Therefore, we use inductive reasoning to derive Maxwell’s equations by enlarging ad-hocthe applicability range of the field equations and, then, introducing those modificationsenforced by the compatibility with fundamental principles, namely charge conservationand the relativity principle. ρ , j , E , and B time dependent In a first step, we simply make the charge density ρ , the current density j , the electricfield strength E , and the magnetic field strength B time dependent. This leads to thefield equations ∇ · E ( r , t ) = ρ ( r , t ) ǫ (3 a ) ∇ × E ( r , t ) = (3 b ) ∇ · B ( r , t ) = 0 (3 c ) ∇ × B ( r , t ) = µ j ( r , t ) . (3 d )The Lorentz force is still given by Eq. (2), but now with time-dependent electric andmagnetic field strengths.While the expression for the Lorentz force is already the correct one, the fieldsequation are not yet complete. We nowadays know that the time derivative of boththe electric and the magnetic field strength have to enter. The incompleteness of theabove field equations may be demonstrated experimentally. A much stronger argumentto include them comes, however, from checking the consistency with fundamentalprinciples. ow to get from static to dynamic electromagnetism One of these fundamental principles is charge conservation, expressed through thecontinuity equation ∂ρ ( r , t ) ∂t + ∇ · j ( r , t ) = 0 . (4)To demonstrate that the field equations violate charge conservation, we take thedivergence of Eq. (3 d ) and use that div curl ≡
0, so that0 = ∇ · ( ∇ × B ) = ∇ · ( µ j ) = − µ ∂ρ∂t = 0 . (5) As noticed by Maxwell, the inconsistency with charge conservation can be cured byadding to the current density the so-called displacement current density, j j + ǫ ∂ E ∂t .This leads to the field equations ∇ · E = ρǫ (6 a ) ∇ × E = (6 b ) ∇ · B = 0 (6 c ) ∇ × B = µ j + µ ǫ ∂ E ∂t (6 d )where the last term in Eq. (6 d ) is called Maxwell’s correction to Amp`ere’s law. It fixescharge conservation since now0 = ∇ · ( ∇ × B ) = ∇ · (cid:18) µ j + µ ǫ ∂ E ∂t (cid:19) = µ (cid:18) ∇ · j + ∂ρ∂t (cid:19) = 0 X . (7)In the static case, the field equations for the electric and the magnetic fieldstrengths were independent of each other. They could be split into two equationsfor electrostatics and two for magnetostatics, with ρ ( r ) and j ( r ) being considered asindependent quantities. The generalization to the dynamic case makes all four fieldequations coupled to each other, since now the source terms ρ ( r , t ) for the electric and j ( r , t ) for the magnetic field strength are connected to each other via the continuityequation. In addition, Maxwell’s correction to Amp`ere’s law introduces an explicitcoupling between E ( r , t ) and B ( r , t ) for time-dependent electric field strengths. Whatdoes it mean for the interpretation of the electromagnetic field? At the present stage, wecould still consider the electric and the magnetic field as two separate physical entities(a notion that we have to revise after the discussion of the relativity principle below),but now they are at least coupled to each other.In Eq. (6 d ), the combination µ ǫ appears. It is convenient to introduce theabbreviation µ ǫ = 1 c (8)where the constant c has the dimension of a velocity. This constant is identical tothe vacuum speed of light derived from the full electrodynamic theory (which includes ow to get from static to dynamic electromagnetism a )-(6 d ) do notsupport propagating solutions for the electromagnetic field in vacuum, i.e., the conceptof light is so far meaningless. Nevertheless, c provides a scale for expressing velocitiessuch as the relative velocity between two inertial systems.To arrive at Maxwell’s equations, we need now to include Faraday’s law.Historically, the motivation for this came from experiment. Here, we suggest, as analternative, to derive Faraday’s law from the relativity principle. The relativity principle statesThe laws of nature have the same form in all inertial systems.The relativity principle does not make a statement of how to transform from oneinertial system to another. If a candidate for such a transformation is given, onecan check whether the relativity principle is respected for some theory with the giventransformation. But one can also turn the argument the other way around and construct the transformation by postulating the relativity principle to hold for the consideredtheory. We will see both ways of reasoning in the following.Within Newtonian mechanics, which is usually taught prior to electrodynamics,the situation is clear. It respects the relativity principle when using the Galileitransformation to change between different inertial systems. Newtonian mechanics is,therefore, called Galilei-invariant.
In the following, we denote by S ′ an inertial system thatmoves with constant velocity u relative to the reference system S . In both systems,any event is characterized by the respective time and space coordinates. The Galileitransformation describes how the coordinates in S ′ are connected to those in S . It reads t ′ = t (9 a ) r ′ = r − u t . (9 b )By writing t ( r ′ , t ′ ) = t ′ and r ( r ′ , t ′ ) = r ′ + u t ′ as functions of r ′ and t ′ and using thechain rule, we find ∂∂t ′ = ∂t∂t ′ ∂∂t + ∂ r ∂t ′ · ∇ = ∂∂t + u · ∇ as well as ∂∂x ′ = ∂t∂x ′ ∂∂t + ∂ r ∂x ′ · ∇ = ∂∂x and similarly for y and z . This implies the transformation rules ∂∂t ′ = ∂∂t + u · ∇ (10 a ) ∇ ′ = ∇ (10 b )for the derivatives with respect to time and space. For Newtonian mechanics to beGalilei invariant, the equation of motion F = m ¨ r has to keep its form under a Galileitransformation. The Galilei transformation implies ¨ r ′ = ¨ r for the acceleration of a ow to get from static to dynamic electromagnetism m ′ = m , then Galilei invariance of the equation of motion implies F ′ = F , (11)i.e., the force on a body has the same value in all inertial systems. We note thatEq. (11) is a stronger statement than mere form invariance of the equations of motions.Form invariance of an equation is, per definition, guaranteed by the relativity principle,irrespective of which transformation between inertial systems is considered. In contrast,the equal values of the force in all inertial systems is a special property of the Galileitransformation. The relativity principle holds formechanics, and in favor of a unified description of nature we demand that it also holdsfor electromagnetism. Since Newtonian mechanics is Galilei invariant, we check whetheror not electromagnetism can be formulated in a Galilei-invariant way. This includes notonly the field equations Eqs. (6 a )-(6 d ) but also the Lorentz force Eqs. (2), which connectselectromagnetism to mechanics.The source terms for the electric and magnetic field strengths are the charge densityand the current density. Under Galilei transformation they transform according to ρ ′ ( r ′ , t ′ ) = ρ ( r , t ) (12 a ) j ′ ( r ′ , t ′ ) = j ( r , t ) − u ρ ( r , t ) . (12 b )As a consequence, the continuity equation is Galilei invariant, which is proven by ∂ρ ′ ∂t ′ + ∇ ′ · j ′ = (cid:18) ∂∂t + u · ∇ (cid:19) ρ + ∇ · ( j − u ρ ) = ∂ρ∂t + ∇ · j = 0 X . (13)We now turn to the field equations Eqs. (6 a )-(6 d ). By postulating their Galileiinvariance, we can deduce how the electric and magnetic field strengths transform.Since ∇ ′ = ∇ and ρ ′ = ρ , it is obvious that Eqs. (6 a ) and (6 b ) are Galilei invariant ifthe electric field strength has the same value in all inertial systems, E ′ = E . To findthe transformation behavior of the magnetic field strength, we rewrite Amp`ere’s lawEq. (6 c ) as ∇ × B = µ j + µ ǫ ∂ E ∂t = µ ( j ′ + u ρ ) + µ ǫ (cid:18) ∂∂t ′ − u · ∇ (cid:19) E = µ j ′ + µ ǫ ∂ E ′ ∂t ′ + µ ǫ [ u ( ∇ · E ) − ( u · ∇ ) E ] | {z } ∇× ( u × E ) . (14)From ∇ ′ × (cid:16) B − u c × E (cid:17)| {z } B ′ = µ j ′ + µ ǫ ∂ E ′ ∂t ′ (15) ow to get from static to dynamic electromagnetism B ′ . Finally, we check that also the last field equationEq. (6 d ) is Galilei invariant, ∇ ′ · B ′ = ∇ · (cid:16) B − u c × E (cid:17) = ∇ · B | {z } − c ∇ · ( u × E )= 1 c u · ( ∇ × E ) | {z } = 0 X (16)To summarize, we find that the field equations Eqs. (6 a )-(6 d ) are, indeed, Galileiinvariant if the field strengths transform according to E ′ ( r ′ , t ′ ) = E ( r , t ) (17 a ) B ′ ( r ′ , t ′ ) = B ( r , t ) − u c × E ( r , t ) . (17 b )An immediate consequence of Eq. (17 b ) is that the value of the magnetic fieldstrength depends on the chosen inertial system. It is, therefore, no longer possible toconsider the electric and the magnetic field as two separate physical entities. The electricand the magnetic field strength E and B should rather be viewed as two facets of thevery same physical entity, namely the electromagnetic field .We have seen that not only Newtonian mechanics but also the field equationsEqs. (6 a )-(6 d ) of the electromagnetic field (which do not include Faraday’s law) areGalilei invariant. Does it mean we have found a consistent theory that unifies mechanicsand electromagnetism in a Galilei-invariant way? No, it doesn’t. What remains to bechecked is the connection between mechanics and electromagnetism, provided by theLorentz force. And here comes the trouble: the Lorentz force is not Galilei invariant.To transform the Lorentz force Eq. (2) on a test charge Q that moves with velocity v in S into another inertial system S ′ that moves with u relative to S , we make use of Q ′ = Q , v ′ = v − u , F ′ = F , E ′ = E , and B ′ = B − u c × E to find F ′ = Q ′ ( E ′ + v ′ × B ′ ) + Q ′ h u × B ′ + ( v ′ + u ) × (cid:16) u c × E ′ (cid:17)i (18)which clearly differs in its form from Eq. (2). The terms containing u obviously breakGalilei invariance.Another way to drastically demonstrate the breakdown of Galilei invariance is toconsider a point charge Q that is moving with velocity v in a static magnetic field B that is generated by a magnet at rest. In the rest frame S of the magnet, there is a finiteLorentz force F = Q v × B . If we switch to the (momentary) rest frame S ′ of the pointcharge with the help of the Galilei transformation with u = v , we find that neither theelectric field (due to E ′ = E = ) nor the magnetic field (due to v ′ = ) contributes tothe Lorentz force, hence F ′ = , in contradiction to Eq. (11). Not only are the valuesfor the force different in different inertial systems. The force is even vanishing in onesystem while being finite in another one. So, depending on the chosen inertial systemwe come to different conclusions of whether the point charge remains in uniform motionor not. This clearly contradicts the relativity principle. ow to get from static to dynamic electromagnetism Q v × B . The advantageof achieving a consistent, Galilei-invariant, unified description of both mechanics andelectrodmagnetism would, thus, come with the drawback of totally removing magnetism:the magnetic field strength B would still be present in the field equations but it wouldhave no effect whatsoever, neither on charged bodies nor on the electric field strength.A world without magnetic phenomena, however, is not the world we are living in.If the Lorentz force is not changed, then the relativity principle can only be saved bymodifying the transformation that connects different inertial systems, i.e., the Galileitransformation has to be replaced by some other transformation, which we call, bydefinition, the Lorentz transformation. The task is now to derive the explicit expressionsof the Lorentz transformation by imposing the requirement that electromagnetism isLorentz invariant. As we will see below, following this route yields Faraday’s law as anecessary consequence of the relativity principle. At the same time, it leads in a naturaland transparent way to explicit expressions for the Lorentz transformation. The Lorentztransformation involves a so-far unknown u -dependence of all the quantities t ′ , r ′ , ∂∂t ′ , ∇ ′ , ρ ′ , j ′ , E ′ , and B ′ . For an infinitesimally small u , an expansion up to linear order issufficient. We refer to this linear expansion as the infinitesimal Lorentz transformation.While for half of these quantities, namely r ′ , ∂∂t ′ , j ′ , and B ′ , the correct linear term isalready provided by the Galilei transformation, we need to determine the correspondinglinear terms for the other half, namely t ′ , ∇ ′ , ρ ′ , and E ′ .We start to derive the infinitesimal Lorentz transformation of the electric andmagnetic field strength. For this, we consider an inertial system S which hosts a staticelectromagnetic field with field strengths E and B and a point charge Q that moveswith infinitesimal velocity u . The force on the point charge is F = Q ( E + u × B ). Inthe (momentary) rest frame S ′ of the point charge, the force is F ′ = Q E ′ . By requiring F ′ = F + O ( u ) (19)and understanding Eq. (17 b ) as the linear approximation of the finite Lorentztransformation of the magnetic field, we obtain for the infinitesimal Lorentz ow to get from static to dynamic electromagnetism E ′ = E + u × B + O ( u ) (20 a ) B ′ = B − u c × E + O ( u ) (20 b )By inductive reasoning, we assume that this transformation behavior is not restrictedto the case of static E and B , but valid in general.We remark that, in contrast to the Galilei transformation, not only the magneticbut also the electric field strength depends on the chosen inertial system. This confirmsonce more that the electromagnetic field should not be considered as a mere coexistenceof two separate physical entities.Since we have changed, as compared to the Galilei transformation, thetransformation behavior of the electromagnetic field, we cannot expect the fieldequations Eqs. (6 a )-(6 d ) to be form invariant under the new transformation. In fact,invariance under the new transformation enforces a modification of Eqs. (6 b ) that leadsto Faraday’s law. Let us consider a static magnetic field, i.e., ∂ B ∂t = , in theabsence of an electric field, E = . Applying the infinitesimal Lorentz transformationyields E ′ = u × B + O ( u ) and B ′ = B + O ( u ). Using ∇ ′ = ∇ + O ( u ) as well as ∂∂t ′ = ∂∂t + u · ∇ + O ( u ), we find ∇ ′ × E ′ = ∇ ′ × ( u × B ) + O ( u ) = ∇ × ( u × B ) + O ( u )= u ( ∇ · B ) | {z } − ( u · ∇ ) | {z } ∂∂t ′ − ∂∂t B + O ( u )= ∂ B ∂t |{z} − ∂ B ∂t ′ + O ( u ) = − ∂ B ′ ∂t ′ + O ( u ) . (21)This is nothing but Faraday’s law, derived for an inertial system that movesinfinitesimally slow with respect to a system in which only a static magnetic field exists.Again, we use inductive reasoning to assume that the relation ∇ × E = − ∂ B ∂t is generallyvalid also beyond this specific scenario.In conclusion, Faraday’s law appears as a by-product in trying to save the relativityprinciple for electromagnetism. There is no need to rely on experimental input at thisstage. By including Faraday’s law, we arrive at the final form of the field equations, thecelebrated Maxwell equations ∇ · E = ρǫ (22 a ) ∇ × E = − ∂ B ∂t (22 b ) ow to get from static to dynamic electromagnetism ∇ · B = 0 (22 c ) ∇ × B = µ j + µ ǫ ∂ E ∂t . (22 d )Faraday’s law makes the field equations more symmetric with respect to the electricand the magnetic field strengths. Not only does a time-dependent electric field strength E serve as a source of a magnetic field strength B , but also a time-dependent magneticfield strength B generates an electric field strength E . This mutual interdependence of E and B is an import prerequisite for propagating solutions of the electromagnetic field.In that respect, we have arrived at a truly dynamic theory of electromagnetism, usuallyreferred to as electrodynamcis .What still needs to be checked, of course, is that all four of Maxwell’s equationsare Lorentz invariant. We will do that in the next section by applying an infinitesimalLorentz transformation. Thereby, we make use of the freedom to choose the so-farundetermined linear terms (in u ) of the transformed charge density ρ ′ , the gradient ∇ ′ ,and the time t ′ such that the relativity principle is fulfilled for electrodynamics. Sincetime and space define the stage not only for electrodynamics but also for mechanics, wefinally end up with a consistent theory for both of them.
2. Infinitesimal Lorentz transformation
First, we postulate the Lorentz invariance of ∇ · B = 0 to derive how the gradienttransforms. Combining0 = ∇ · B = ∇ · (cid:16) B ′ + u c × E (cid:17) + O ( u )= ∇ · B ′ − u c · ( ∇ × E ) | {z } − ∂ B ∂t + O ( u ) = (cid:18) ∇ + u c ∂∂t (cid:19)| {z } ∇ ′ · B ′ + O ( u ) (23)with the postulate ∇ ′ · B ′ = 0 fixes the expression for ∇ ′ . Therefore, the derivativeswith respect to space and time transform according to ∂∂t ′ = ∂∂t + u · ∇ + O ( u ) (24 a ) ∇ ′ = ∇ + u c ∂∂t + O ( u ) . (24 b )Next, we derive the transformation of the coordinates themselves. From ∂t∂t ′ =( ∂∂t + u · ∇ ) t + O ( u ) = 1 + O ( u ) as well as ∇ ′ t = ( ∇ + u c ∂∂t ) t + O ( u ) = u c + O ( u ),we get t = t ′ + u · r ′ c + O ( u ) and, therefore, t ′ = t − u · r c + O ( u ) (25 a ) r ′ = r − u t + O ( u ) . (25 b )In contrast to the Galilei transformation, time remains no longer invariant under achange of the inertial system. ow to get from static to dynamic electromagnetism Next, we postulate Lorentz invariance of Gauß’s law. This leads to ρǫ = ∇ · E = (cid:18) ∇ ′ − u c ∂∂t (cid:19) · ( E ′ − u × B ) + O ( u )= ∇ ′ · E ′ − u c · ∂ E ′ ∂t − ∇ ′ · ( u × B ) + O ( u )= ρ ′ ǫ − u c · ∂ E ∂t − ∇ · ( u × B ) | {z } − u · ( ∇× B ) + O ( u )= ρ ′ ǫ + u · (cid:18) ∇ × B − c ∂ E ∂t (cid:19)| {z } µ j + O ( u ) (26)from which we deduce ρ ′ = ρ − u · j c + O ( u ) (27 a ) j ′ = j − u ρ + O ( u ) . (27 b )As a consequence, the value of the charge density depends on the chosen inertial system.It is not constant, as the Galilei transformation would suggest. Finally, we check the Lorentz invariance of Faraday’s and Amp`ere’s law. We get ∇ ′ × E ′ + ∂ B ′ ∂t ′ = (cid:18) ∇ + u c ∂∂t (cid:19) × ( E + u × B )+ (cid:18) ∂∂t + u · ∇ (cid:19) (cid:16) B − u c × E (cid:17) + O ( u )= ∇ × E + ∂ B ∂t | {z } + [ ∇ × ( u × B ) + ( u · ∇ ) B ] | {z } u ( ∇· B )= + O ( u )= O ( u ) X (28)as well as ∇ ′ × B ′ − µ j ′ − c ∂ E ′ ∂t ′ = (cid:18) ∇ + u c ∂∂t (cid:19) × (cid:16) B − u c × E (cid:17) − µ j + µ u ρ − c (cid:18) ∂∂t + u · ∇ (cid:19) ( E + u × B ) + O ( u )= ∇ × B − µ j − c ∂ E ∂t | {z } + µ u ρ − c [ ∇ × ( u × E ) + ( u · ∇ ) E ] | {z } u ( ∇· E )= u ρ/ǫ + O ( u )= O ( u ) X . (29) ow to get from static to dynamic electromagnetism Table 2.
Comparison of the Galilei transformation with the infinitesimal Lorentztransformation. Here, u denotes the velocity of inertial system S ′ with respect toinertial system S . Galilei transformation infinitesimal Lorentz transformation t ′ = t t ′ = t − u · r c + O ( u ) r ′ = r − u t r ′ = r − u t + O ( u ) ∂∂t ′ = ∂∂t + u · ∇ ∂∂t ′ = ∂∂t + u · ∇ + O ( u ) ∇ ′ = ∇ ∇ ′ = ∇ + u c ∂∂t + O ( u ) ρ ′ = ρ ρ ′ = ρ − u · j c + O ( u ) j ′ = j − u ρ j ′ = j − u ρ + O ( u ) E ′ = E E ′ = E + u × B + O ( u ) B ′ = B − u c × E B ′ = B − u c × E + O ( u )In summary, we have shown that Maxwell’s equations as well as the Lorentzforce are Lorentz invariant. Furthermore, we have derived the infinitesimal Lorentztransformation of t ′ , r ′ , ∇ ′ , ∂∂t ′ , ρ ′ , j ′ , E ′ , and B ′ . The results are summarized inTable 2.
3. Finite Lorentz transformation
Based on the results for the infinitesimal Lorentz transformation, we derive the finiteLorentz transformation by repeatedly applying infinitesimal ones. In order to keep thenotation transparent, we choose the boost direction (i.e., the direction of the relativevelocity between two inertial systems) from now on always along the x -axis. Since the y -and z -components (i.e., those perpendicular to the boost direction) of r , ∇ , and j remainunchanged under an infinitesimal Lorentz transformation, they also remain unchangedunder a finite one. The same holds true to the x -components (i.e., the one parallel tothe boost direction) of the electric and magnetic field strength E and B . The othercomponents ( x , ∂∂x , j x as well as E y , E z , B y , B z ), on the other hand, do change underLorentz transformation, and we need to determine how.We start with the Lorentz transformation of time and space coordinates and theirderivatives. Afterwards, we determine the transformation of the charge and currentdensity and, finally, of the electric and the magnetic field strength. ow to get from static to dynamic electromagnetism Let S , S ′ , and S ′′ be three inertial systems, where S ′ moves with finite velocity u ˆ e x relative to S , S ′′ moves with infinitesimal velocity du ˆ e x relative to S ′ , and the relativevelocity of S ′′ with respect to S is denoted by ˜ u ˆ e x . The finite Lorentz transformationfrom S to S ′ can be written in a vector-matrix notation ct ′ x ′ ! = Λ( u ) ctx ! (30)by introducing the Lorentz-transformation matrix Λ( u ). Since ct and x have the sameunits, the matrix elements of Λ( u ) are all dimensionless. Similarly, Λ(˜ u ) connects thecoordinates specified in inertial systems S ′′ and S , respectively, and Λ( du ) + O ( du )those in inertial systems S ′′ and S ′ . The relationΛ(˜ u ) = Λ( du )Λ( u ) + O (cid:0) ( du ) (cid:1) . (31)describes that the finite Lorentz transformation from S to S ′′ can be decomposed intoa finite one from S to S ′ and an infinitesimal one from S ′ to S ′′ .From the previous section, we know how to perform the infinitesimal Lorentztransformation. This is expressed by the symmetric matrixΛ( du ) = − duc − duc ! + O (cid:0) ( du ) (cid:1) (32)which we plug into Eq. (31) to getΛ(˜ u ) = Λ( u ) + duc − − ! Λ( u ) + O (cid:0) ( du ) (cid:1) . (33)To proceed, we need to know how ˜ u depends on u and du . For this, we consider aparticle that is at rest in S . In S ′ , it has velocity v ′ = dx ′ dt ′ = − u and in S ′′ , the velocityis given by v ′′ = dx ′′ dt ′′ = − ˜ u . This yields˜ u = − dx ′′ dt ′′ = − d ( x ′ − t ′ du ) d ( t ′ − x ′ duc ) + O (cid:0) ( du ) (cid:1) = − dx ′ + dt ′ dudt ′ − dx ′ duc + O (cid:0) ( du ) (cid:1) = u + du uduc + O (cid:0) ( du ) (cid:1) = ( u + du ) (cid:18) − uduc (cid:19) + O (cid:0) ( du ) (cid:1) = u + (cid:18) − u c (cid:19) du + O (cid:0) ( du ) (cid:1) . (34)We note that ˜ u = u + du for finite u , i.e., the Galilean rule of adding velocities is no longervalid. Furthermore, at this stage, the role of c as a limiting velocity becomes apparent:for u = c , the extra Lorentz boost by du does not increase the velocity anymore. It is,thus, impossible to achieve ˜ u > c , i.e., velocities larger than the speed of light.Plugging Eq. (34) into Λ(˜ u ) and expanding up to linear order in du yieldsΛ(˜ u ) = Λ( u ) + (cid:18) − u c (cid:19) d Λ du du + O (cid:0) ( du ) (cid:1) . (35) ow to get from static to dynamic electromagnetism d Λ d (cid:2) arctanh (cid:0) uc (cid:1)(cid:3) = − − ! Λ , (36)for the Lorentz-transformation matrix Λ, where we have made use of the derivative ddx [arctanh x ] = − x .It is quite intriguing that in Eq. (36) the velocity u appears only in the combination θ = arctanh (cid:16) uc (cid:17) . (37)This suggest a change of variable from the velocity u to the rapidity θ . In terms of therapidity, the differential equation for Λ simplifies to d Λ dθ = − − ! Λ (38)Instead of solving this first-order differential equation, it is convenient to calculate thesecond derivative. We make use of − − ! − − ! = and end up withthe second-order differential equation d Λ dθ = Λ (39)that is easy to solve. Each matrix element of Λ is a linear combination ofcosh θ = 1 p − tanh θ = 1 q − u c = γ (40)and sinh θ = tanh θ p − tanh θ = uc q − u c = γ uc (41)where γ = 1 / q − u c denotes the famous Lorentz factor.The coefficients of the linear combinations of cosh θ and sinh θ for the matrixelements of Λ have to be chosen such that the expansion up to linear order in θ isgiven by the infinitesimal Lorentz transform. Because of cosh θ = 1 + O ( u ) andsinh θ = uc + O ( u ), this results inΛ = cosh θ − sinh θ − sinh θ cosh θ ! = γ − uc − uc ! . (42)In comparison to the infinitesimal Lorentz transformation, the finite one is obtainedsimply multiplying with the Lorentz factor γ in the time coordinate and the spatialcoordinate along the boost direction. In conclusion, the final result for the finite Lorentztransformation is t ′ = γ (cid:16) t − uxc (cid:17) (43 a ) x ′ = γ ( x − ut ) . (43 b ) ow to get from static to dynamic electromagnetism t ( x ′ , t ′ ) = γ (cid:0) t ′ + ux ′ c (cid:1) and x ( x ′ , t ) = γ ( x ′ + ut ′ ) as functions of x ′ and t ′ and use the chain rule for derivatives. This leads to ∂∂t ′ = γ (cid:18) ∂∂t + u ∂∂x (cid:19) (44 a ) ∂∂x ′ = γ (cid:18) ∂∂x + uc ∂∂t (cid:19) . (44 b )So, again, the finite Lorentz transformation differs from the infinitesimal one just bythe Lorentz factor γ in the derivatives of the time coordinate and the spatial coordinatealong the boost direction. The infinitesimal Lorentz transformation of ρ and j is identical to the one of t and r .Therefore, also the finite Lorentz transformations have to be identical. This leads tothe result ρ ′ = γ (cid:18) ρ − uj x c (cid:19) (45 a ) j ′ x = γ ( j x − uρ ) (45 b )which, again, differs from the infinitesimal transformation only by the Lorentz factor γ in the charge density and the current density along the boost direction. Finally, we consider the electric and magnetic field strength. The field strengths alongthe boost direction remain unchanged. For the perpendicular directions, we define theLorentz-transformation matrix M ( u ) by E ′ y E ′ z cB ′ y cB ′ z = M ( u ) E y E z cB y cB z (46)Since E and cB have the same units, the matrix elements of M ( u ) are all dimensionless.Analogous to the procedure for the time and space coordinates, we derive a differentialequation for M ( u ) from M (˜ u ) = M ( du ) M ( u ) + O (cid:0) ( du ) (cid:1) . (47)by using the infinitesimal Lorentz transformation M ( du ) = − duc duc duc − duc + O ( du ) (48) ow to get from static to dynamic electromagnetism u in terms of u and du through Eq. (34), and finally performing thevariable change from velocity to rapidity. This results in the first-order differentialequation dMdθ = −
10 0 1 00 1 0 0 − M (49)Again, it is convenient to calculate the second derivative. This leads to the second-orderdifferential equation d Mdθ = M (50)which is easily solved by linear combinations of cosh θ and sinh θ in each matrix element.The coefficients of these linear combination are, again, fixed by comparison with theinfinitesimal Lorentz transformation. This yields the final result E ′⊥ = γ ( E ⊥ + u × B ⊥ ) (51 a ) B ′⊥ = γ (cid:16) B ⊥ − u c × E ⊥ (cid:17) (51 b )where the y - and z -components of the electric and magnetic field strength are combinedinto E ⊥ and B ⊥ . Once more, the finite Lorentz transformation differs from theinfinitesimal one only by the Lorentz factor γ in the perpendicular field-strengthcomponents.The comparison between the infinitesimal and the finite Lorentz transformation forthe boost direction along the x -axis is summarized in Table 3.
4. Discussion
In the presented approach, the Lorentz transformation has been constructed out of theGalilei transformation, via the infinitesimal Lorentz transformation as an intermediatestep. To clarify the relation between the Galilei and the Lorentz transformation, wenow go the opposite way (as it is usually done in the literature) and ask how to obtainthe Galilei transformation as a limiting case of the Lorentz transformation.In contrast to what is sometimes claimed in textbooks, the Galilei transformationis not just the low-velocity limit of the Lorentz transformation. It is true that in thelow-velocity limit, u ≪ c , the Lorentz factor γ drops out because of γ = 1 + O ( u ).To linearize the Lorentz transformation in u/c does, however, not lead to the Galileitransformation but rather to the infinitesimal Lorentz transformation, see Table 3.The Galilei transformation is only achieved after dropping the term linear in u in thetransformations of time, spatial derivative, charge density, and electric field strength, seeTable 2. This means that, in addition to the low-velocity limit, another approximationis involved. Neglecting the linear term in the transformation of the time coordinate ow to get from static to dynamic electromagnetism Table 3.
Comparison of infinitesimal with finite Lorentz transformation. Here, u denotes the velocity of inertial system S ′ with respect to inertial system S along the x -direction. infinitesimal Lorentz transformation finite Lorentz transformation t ′ = t − uxc + O ( u ) t ′ = γ (cid:16) t − uxc (cid:17) x ′ = x − ut + O ( u ) x ′ = γ ( x − ut ) y ′ = y y ′ = yz ′ = z z ′ = z∂∂t ′ = ∂∂t + u ∂∂x + O ( u ) ∂∂t ′ = γ (cid:18) ∂∂t + u ∂∂x (cid:19) ∂∂x ′ = ∂∂x + uc ∂∂t + O ( u ) ∂∂x ′ = γ (cid:18) ∂∂x + uc ∂∂t (cid:19) ∂∂y ′ = ∂∂y ∂∂y ′ = ∂∂y∂∂z ′ = ∂∂z ∂∂z ′ = ∂∂zρ ′ = ρ − uj x c + O ( u ) ρ ′ = γ (cid:18) ρ − uj x c (cid:19) j ′ x = j x − uρ + O ( u ) j ′ x = γ ( j x − uρ ) j ′ y = j y j ′ y = j y j ′ z = j z j ′ z = j z E ′ x = E x E ′ x = E x E ′⊥ = E ⊥ + u × B ⊥ + O ( u ) E ′⊥ = γ ( E ⊥ + u × B ⊥ ) B ′ x = B x B ′ x = B x B ′⊥ = B ⊥ − u c × E ⊥ + O ( u ) B ′⊥ = γ (cid:16) B ⊥ − u c × E ⊥ (cid:17) means that the time difference of two events considered within this approximation mustbe taken much larger than the spatial difference, c ∆ t ≫ | ∆ r | , a condition referred to as largely timelike [1, 2]. To make this approximation consistent, also the linear term in thetransformation of the electric field strength has to be neglected, i.e., the magnetic fieldstrength must be much smaller than the electric one. For this reason, the description ofthe electromagnetic field within this approximation, given by Eqs. (6 a )-(6 d ), has beencalled the electric limit [2, 3]. As discussed in the first part of the paper, the problemwith this Galilean description of electromagnetism is that, in order to be consistent, alsothe term linear in u in the Lorentz force, i.e., the magnetic force needs to be neglected.As a consequence, the magnetic field strength does not have any impact.Dropping the linear terms in the transformations of time, spatial derivative, charge ow to get from static to dynamic electromagnetism c ∆ t ≪ | ∆ r | , a condition referred to as largely spacelike [1, 2]. Furthermore, the magnetic field strength must be much largerthan the electric one, which motivates the term magnetic limit [2, 3] for the resultingdescription of electromagnetism. In the magnetic limit, there is no problem with theLorentz force but now the continuity equation is not satisfied, as discussed in Ref. [2]. Science aims at describing how nature is and not why it is the way it is. It is, therefore,useless to ask why electrodynamics is a relativistic theory. Nevertheless, it is interestingfrom the pedagogical point of view trying to identify the origin of relativistic behaviorof electrodynamics, which may be considered as an answer to the why question.Before doing so, however, we need to clarify what is meant by relativistic .For historical reasons, electrodynamics is usually called relativistic , while Newtonianmechanics is referred to as nonrelativistic . This nomenclature is, however, veryunfortunate because, as a matter of fact, both theories are in accord with the relativityprinciple. The distinction between the two theories consists in how to transform fromone inertial system to another. We, therefore, prefer to call electrodynamics
Lorentz-relativistic and Newtonian mechanics
Galilei-relativistic . So, the question we pose inhere is what makes electrodynamics Lorentz-relativistic instead of Galilei-relativistic.The crucial feature of the Lorentz transformation is the existence of a velocity scale,set by a special velocity c . Lorentz transformations with different values of u/c have verydifferent appearances. The Galilei transformation, on the other hand, is scale free, i.e.,there is no special velocity. In this sense, we identify the origin of Lorentzian relativityof a theory through the existence of a fixed velocity scale.In most textbook presentations, the Lorentz transformation is introduced afterdiscussing propagating electromagnetic waves in vacuum, i.e., light. According toMaxwell’s equations, the vacuum speed of light is universal and given by the constant c = 1 / √ µ ǫ . The Lorentz-relativistic nature of electrodynamics is, thus, based on theproperties of a dynamic phenomenon, namely the propagation of light. While there isnothing wrong with this interpretation, the approach presented in this paper suggestsan alternative way to explain why electrodynamics must be Lorentz-relativistic, but nowbased on static electromagnetism.In our derivation of Maxwell’s equation, Galilei invariance had to be abandonedduring the third step only because the Lorentz force is not Galilei invariant. The Lorentzforce on a charged body consists of an electric and a magnetic part. The electric forceis independent of, and the magnetic force is proportional to the velocity of the body.In order to compare electric with magnetic forces or to add them to one resulting force, ow to get from static to dynamic electromagnetism c = 1 / √ µ ǫ in static electromagnetismis the situation studied in Problem 5.13 of Ref. [4]. There, two positively charged,infinitely long, infinitely thin, parallel wires are considered, which move together withconstant velocity v along the wires’ direction. In this scenario, both the charge densityand the current density are time independent, leading to time-independent electric andmagnetic field strengths. Because of the charges, there is an electrostatic repulsive forcebetween the wires. In addition, since the moving charges constitute a current, there isan attractive magnetostatic force. It is a simply exercise of electro- and magnetostaticsto calculate the ratio of the magnetic over the electric force. The ratio turns out tobe ( v/c ) . As a consequence, it vanishes for v = 0 and approaches unity for v = c .Therefore, c is the velocity limit at which the magnetic force between the two wirescompensates the electric one.In conclusion, the very existence of a magnetic force in addition to an electricone can, therefore, be viewed as the origin of the Lorentz-relativistic nature ofelectrodynamics. Most textbook presentations of electrodynamics follow the historical course: aftertreating electro- and magnetostatics, Faraday’s law is introduced as an experimental factand, then, the magnetostatic version of Amp`ere’s law is modified by adding Maxwell’scorrection. We suggest to change this modus operandi in two respects.First, the order of introducing Faraday’s law and Maxwell’s correction should beinterchanged. The advantage of doing so has already been emphasized in Ref. [5].Second, we suggest to derive Faraday’s law from the relativity principle rather thanintroducing it as an experimental fact. This has, to the best of our knowledge, not yetbeen discussed in the literature so far.There are several advantages of the proposed approach. To begin with, Faraday’slaw is not introduced as an experimental fact but rather as a necessity to complywith fundamental principles. This gives students a stronger justification for whyFaraday’s law looks like the way it does. Furthermore, the proposed order of stepsallows us to discuss the principle of relativity at an earlier stage than it is usuallydone. The pedagogical virtue of this discussion is that it better connects Newtonianmechanics to electrodynamics, it highlights the principle of relativity as a fundamentalconcept, and it employs the Galilei transformation which is more intuitive and betteraccessible to students than the Lorentz transformation. The practical exercise ofderiving the transformation behavior of the electric and magnetic field strength undera Galilei transformation provides an opportunity to become acquainted with therelativity principle before dealing with the complications associated with the Lorentztransformation. In addition, the infinitesimal Lorentz transformation obtained during ow to get from static to dynamic electromagnetism
5. Acknowledgements.
We thank Eric Kleinherbers for drawing out attention to Ref. [5] and for helpfuldiscussions. [1] J.-M. L´evy-Leblond, Une nouvelle limite non-relativiste du groupe de Poincar´e, Annales del’I.H.P., section A , 1 (1965).[2] M. Le Bellac and J.-M. L´evy-Leblond, Galilean Electromagnetism, Il Nuovo Cimento ,217 (1973).[3] J.A. Heras, The Galilean limits of Maxwell’s equations, Am. J. Phys. , 1048 (2010).[4] D.J. Griffiths, Introduction to Electrodynamics, 4th Edition, (Cambridge University Press,Cambridge, 2017).[5] M. Jammer and J. Stachel, If Maxwell had worked between Amp`ere and Faraday: Anhistorical fable with a pedagogical moral, Am. J. Phys. , 5 (1980).[6] I. Galili and D. Kaplan, Changing approach to teaching electromagnetism in a conceptuallyoriented introductory physics course, Am. J. Phys.65