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| [[File:060618 conductor magnet.svg|thumb|300px|right|Conductor moving in a magnetic field.]]
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| The '''moving magnet and conductor problem''' is a famous [[thought experiment]], originating in the 19th century, concerning the intersection of [[classical electromagnetism and special relativity]]. In it, the current in a [[Electrical conductor|conductor]] moving with constant velocity, ''v'', with respect to a [[magnet]] is calculated in the [[inertial frame|frame of reference]] of the magnet and in the frame of reference of the conductor. The observable quantity in the experiment, the current, is the same in either case, in accordance with the basic ''principle of relativity'', which states: "Only ''relative'' motion is observable; there is no absolute standard of rest".<ref>The ''Laws of Physics'' are the same in all [[inertial frames]].</ref> However, according to Maxwell's equations, the charges in the conductor experience a '''magnetic force''' in the frame of the magnet and an '''electric force''' in the frame of the conductor. The same phenomenon would seem to have two different descriptions depending on the frame of reference of the observer.
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| This problem, along with the [[Fizeau experiment]], the [[aberration of light]], and more indirectly the [[Tests of special relativity|negative aether drift tests]] such as the [[Michelson–Morley experiment]], formed the basis of Einstein's development of the theory of relativity.<ref name="norton">{{Citation|last=Norton, John D.|year=2004|first1=John D.|journal=Archive for History of Exact Sciences|title= Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905|pages= 45–105|volume=59|url=http://philsci-archive.pitt.edu/archive/00001743/|doi=10.1007/s00407-004-0085-6|bibcode=2004AHES...59...45N}}</ref>
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| ==Introduction==
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| [[Albert Einstein|Einstein's]] 1905 paper that introduced the world to relativity opens with a description of the magnet/conductor problem.[http://www.fourmilab.ch/etexts/einstein/specrel/www/]
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| {{Quotation
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| |It is known that Maxwell's electrodynamics – as usually understood at the present time – when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighborhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighborhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise – assuming equality of relative motion in the two cases discussed – to electric currents of the same path and intensity as those produced by the electric forces in the former case.
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| |A. Einstein|On the electrodynamics of moving bodies (1905)
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| }}
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| An overriding requirement on the descriptions in different frameworks is that they be [[consistency|consistent]]. Consistency is an issue because [[Newton's laws of motion|Newtonian mechanics]] predicts one transformation (so-called [[Galilean invariance]]) for the ''forces'' that drive the charges and cause the current, while electrodynamics as expressed by [[Maxwell's equations]] predicts that the ''fields'' that give rise to these forces transform differently (according to [[Lorentz invariance]]). Observations of the aberration of light, culminating in the [[Michelson–Morley experiment]], established the validity of Lorentz invariance, and the development of [[special relativity]] resolved the resulting disagreement with Newtonian mechanics. Special relativity revised the transformation of forces in moving reference frames to be consistent with Lorentz invariance. The details of these transformations are discussed below.
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| In addition to consistency, it would be nice to consolidate the descriptions so they appear to be frame-independent. A clue to a framework-independent description is the observation that magnetic fields in one reference frame become electric fields in another frame. Likewise, the [[solenoidal field|solenoidal]] portion of electric fields (the portion that is not originated by electric charges) becomes a magnetic field in another frame: that is, the solenoidal electric fields and magnetic fields are aspects of the same thing.<ref>There are ''two'' constituents of electric field: a [[solenoidal field]] (or ''incompressible field'') and a [[conservative field]] (or ''irrotational field''). The first is transformable to a magnetic field by changing the frame of reference, the second originates in electric charge, and transforms always into an electric field, albeit of different magnitude.</ref> That means the paradox of different descriptions may be only [[semantic gap|semantic]]. A description that uses scalar and vector potentials φ and '''''A''''' instead of '''''B''''' and '''''E''''' avoids the semantical trap. A Lorentz-invariant [[four vector]] ''A''<sup>α</sup> = (φ / ''c''<sub>0</sub>, '''''A''''' ) replaces '''E''' and '''B'''<ref>The symbol ''c''<sub>0</sub> represents the [[speed of light]] in [[free space]].</ref> and provides a frame-independent description (albeit less visceral than the '''E'''– '''B'''–description).<ref>However, φ and '''''A''''' are not completely disentangled, so the two types of ''E''-field are not separated completely. See Jackson [http://arxiv.org/abs/physics/0204034 ''From Lorenz to Coulomb and other explicit gauge transformations''] The author stresses that ''Lorenz'' is ''not'' a typo.</ref> An alternative unification of descriptions is to think of the physical entity as the [[electromagnetic field tensor]], as described later on. This tensor contains both '''E''' and '''B''' fields as components, and has the same form in all frames of reference.
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| ==Background==
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| Electromagnetic fields are not directly observable. The existence of [[Classical physics|classical]] electromagnetic fields can be inferred from the motion of charged particles, whose trajectories are observable. Electromagnetic fields do explain the observed motions of classical charged particles.
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| A strong requirement in [[physics]] is that all observers of the motion of a particle agree on the trajectory of the particle. For instance, if one observer notes that a particle collides with the center of a bullseye, then all observers must reach the same conclusion. This requirement places constraints on the nature of electromagnetic fields and on their transformation from one reference frame to another. It also places constraints on the manner in which fields affect the acceleration and, hence, the trajectories of charged particles.
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| Perhaps the simplest example, and one that Einstein referenced in his 1905 paper introducing [[special relativity]], is the problem of a conductor moving in the field of a magnet. In the frame of the magnet, a conductor experiences a ''magnetic'' force. In the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an ''electric'' field. The magnetic field in the magnet frame and the electric field in the conductor frame must generate consistent results in the conductor. At the time of Einstein in 1905, the field equations as represented by [[Maxwell's equations]] were properly consistent. Newton's law of motion, however, had to be modified to provide consistent particle trajectories.<ref name=Penrose>{{cite book |title=The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics |author=Roger Penrose (Martin Gardner: foreword) |page= 248 |url=http://books.google.com/books?id=oI0grArWHUMC&pg=PA248&dq=reference+%22laws+of+physics%22
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| |isbn=0-19-286198-0 |year=1999 |publisher=Oxford University Press }}</ref>
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| ==Transformation of fields, assuming Galilean transformations==
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| Assuming that the magnet frame and the conductor frame are related by a [[Galilean transformation]], it is straightforward to compute the fields and forces in both frames. This will demonstrate that the induced current is indeed the same in both frames. As a byproduct, this argument will ''also'' yield a general formula for the electric and magnetic fields in one frame in terms of the fields in another frame.<ref>See Jackson, ''Classical Electrodynamics'', Section 5.15.</ref>
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| In reality, the frames are ''not'' related by a Galilean transformation, but by a [[Lorentz transformation]]. Nevertheless, it will be a Galilean transformation ''to a very good approximation'', at velocities much less than the speed of light.
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| Unprimed quantities correspond to the rest frame of the magnet, while primed quantities correspond to the rest frame of the conductor. Let '''v''' be the velocity of the conductor, as seen from the magnet frame.
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| ===Magnet frame===
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| In the rest frame of the magnet, the magnetic field is some fixed field '''B'''('''r'''), determined by the structure and shape of the magnet. The electric field is zero.
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| In general, the force exerted upon a particle of charge ''q'' in the conductor by the [[electric field]] and [[magnetic field]] is given by (SI units):
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| :<math>\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),</math>
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| where <math>q</math> is the charge on the particle, <math>\mathbf{v}</math> is the particle velocity and '''F''' is the [[Lorentz force]]. Here, however, the electric field is zero, so the force on the particle is
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| :<math>\mathbf{F} = q \mathbf{v} \times \mathbf{B}.</math>
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| ===Conductor frame===
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| In the conductor frame, the magnetic field '''B'''' will be related to the magnetic field '''B''' in the magnet frame according to:<ref>This expression can be thought of as an assumption based on our experience with magnets, that their fields are independent of their velocity. At relativistic velocities, or in the presence of an electric field in the magnet frame, this equation would not be correct.</ref>
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| :<math>\mathbf{B}'(\mathbf{x}',t) = \mathbf{B}(\mathbf{x}'+\mathbf{v}t).</math>
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| In this frame, there ''is'' an electric field, generated by the [[Faraday's law of induction#The Maxwell-Faraday equation|Maxwell-Faraday equation]]:
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| :<math>\mathbf{\nabla \times E}' = -\frac{\partial \mathbf{B}'}{\partial t}.</math>
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| Using the above expression for '''B'''',
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| :<math>\mathbf{\nabla \times E}' = -(\mathbf{v} \cdot \nabla) \mathbf{B} = -\nabla\times(\mathbf{B} \times \mathbf{v}) - \mathbf{v}(\nabla\cdot \mathbf{B}) = -\nabla\times(\mathbf{B} \times \mathbf{v})</math>
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| (using the [[chain rule]] and [[Gauss's law for magnetism]]). This has the solution:
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| :<math>\mathbf{E}' = -\mathbf{B} \times \mathbf{v} = \mathbf{v}\times \mathbf{B}.</math>
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| A charge ''q'' in the conductor will be at rest in the conductor frame. Therefore, the magnetic force term of the [[Lorentz force]] has no effect, and the force on the charge is given by
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| :<math>\mathbf{F}' = q\mathbf{E}' = q\mathbf{v} \times \mathbf{B}.</math>
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| This demonstrates that ''the force is the same in both frames'' (as would be expected), and therefore any observable consequences of this force, such as the induced current, would also be the same in both frames. This is despite the fact that the force is seen to be an electric force in the conductor frame, but a magnetic force in the magnet's frame.
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| ===Galilean transformation formula for fields===
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| A similar sort of argument can be made if the magnet's frame also contains electric fields. (The [[Ampere-Maxwell equation]] also comes into play, explaining how, in the conductor's frame, this moving electric field will contribute to the magnetic field.) The end result is that, in general,
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| :<math>\mathbf{E}' = \mathbf{E} + \mathbf{v}\times \mathbf{B}</math>
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| :<math>\mathbf{B}' = \mathbf{B} - \frac{1}{{c_0}^2} \mathbf{v} \times \mathbf{E},</math>
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| with ''c''<sub>0</sub> the [[speed of light]] in [[free space]].
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| By plugging these transformation rules into the full [[Maxwell's equations]], it can be seen that if Maxwell's equations are true in one frame, then they are ''almost'' true in the other, but contain incorrect terms pro by the [[Lorentz transformation]], and the field transformation equations also must be changed, according to the expressions given below.
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| ==Transformation of fields as predicted by Maxwell's equations==
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| {{see also|Classical electromagnetism and special relativity}}
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| In a frame moving at velocity '''v''', the '''E'''-field in the moving frame when there is no '''E'''-field in the stationary magnet frame [[Relativistic_electromagnetism#More_rigorous_analysis|Maxwell's equations]] transform as:<ref name=Chow>
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| {{cite book
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| |author=Tai L. Chow
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| |title=Electromagnetic theory
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| |year= 2006
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| |publisher=Jones and Bartlett
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| |location=Sudbury MA
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| |isbn=0-7637-3827-1
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| |url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271&sig=PgEEBA6TQEZ5fD_AhJQ8dd7MGHo#PPA368,M1
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| |nopp=true
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| |pages =Chapter 10.21; p. 402–403 ff}}
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| </ref>
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| :<math>\mathbf{E}' = \gamma \mathbf{v} \times \mathbf{B}</math>
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| where
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| :<math>\gamma = \frac{1}{\sqrt{1 - {(v/c_0)}^2}}</math>
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| is called the [[Lorentz factor]] and ''c''<sub>0</sub> is the [[speed of light]] in [[free space]]. This result is a consequence of requiring that observers in all [[inertial frames]] arrive at the same form for Maxwell's equations. In particular, all observers must see the same speed of light ''c''<sub>0</sub>. That requirement leads to the [[Lorentz transformation]] for space and time. Assuming a Lorentz transformation, invariance of Maxwell's equations then leads to the above transformation of the fields for this example.
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| Consequently, the force on the charge is
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| :<math>\mathbf{F}' = q \mathbf{E}' = q \gamma \mathbf{v} \times \mathbf{B}.</math> | |
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| This expression differs from the expression obtained from the nonrelativistic Newton's law of motion by a factor of [[Lorentz factor|<math>\gamma </math>]]. Special relativity modifies space and time in a manner such that the forces and fields transform consistently.
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| ==Modification of dynamics for consistency with Maxwell's equations==
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| [[File:Moving magnet.PNG|thumb|400px| Figure 1: Conducting bar seen from two inertial frames; in one frame the bar moves with velocity '''v'''; in the ''primed'' frame the bar is stationary because the primed frame moves at the same velocity as the bar. The '''B'''-field varies with position in the ''x''-direction]]
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| The Lorentz force has the same ''form'' in both frames, though the fields differ, namely:
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| :<math>\mathbf{F} = q \left[\mathbf{E} + \mathbf{v} \times \mathbf{B} \right].</math>
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|
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| See Figure 1. To simplify, let the magnetic field point in the ''z''-direction and vary with location ''x'', and let the conductor translate in the positive ''x''-direction with velocity ''v''. Consequently, in the magnet frame where the conductor is moving, the Lorentz force points in the negative ''y''-direction, perpendicular to both the velocity, and the ''B''-field. The force on a charge, here due only to the ''B''-field, is
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| :<math>F_y = -qvB,</math>
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| while in the conductor frame where the magnet is moving, the force is also in the negative ''y''-direction, and now due only to the '''E'''-field with a value:
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| :<math>{F_y}' = qE' = -q\gamma vB.</math>
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| The two forces differ by the Lorentz factor γ. This difference is expected in a relativistic theory, however, due to the change in space-time between frames, as discussed next.
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| Relativity takes the Lorentz transformation of space-time suggested by invariance of Maxwell's equations and imposes it upon [[Dynamics (physics)|dynamics]] as well (a revision of [[Newton's laws of motion]]). In this example, the Lorentz transformation affects the ''x''-direction only (the relative motion of the two frames is along the ''x''-direction). The relations connecting time and space are ( ''primes'' denote the moving conductor frame ) :<ref name=Chow2>
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| {{cite book
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| |author=Tai L. Chow
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| |title=Electromagnetic theory
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| |year= 2006
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| |publisher=Jones and Bartlett
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| |location=Sudbury MA
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| |isbn=0-7637-3827-1
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| |url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271#PPA368,M1
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| |nopp=true
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| |pages =Chapter 10.5; p. 368 ff}}
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| </ref>
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| :<math>x' = \gamma (x - vt), \quad x = \gamma(x' + vt'),</math>
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| :<math>t' = \gamma (t - \frac{vx}{c_0^2}), \quad t = \gamma(t' + \frac{vx'}{c_0^2}).</math>
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| These transformations lead to a change in the ''y''-component of a [[Special relativity#Force|force]]:
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| :<math>{F_y}' = \gamma F_y.</math>
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| That is, within [[Lorentz invariance]], force is '''''not''''' the same in all frames of reference, unlike Galilean invariance. But, from the earlier analysis based upon the Lorentz force law:
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| :<math>\gamma F_y = -q\gamma vB, \quad {F_y}' = -q\gamma v B,</math>
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| which agrees completely. So the force on the charge is '''''not''''' the same in both frames, but it transforms as expected according to relativity.
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| ==Newton's law of motion in modern notation==
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| {{main| Formulation of Maxwell's equations in special relativity }}
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| The modern approach to obtaining the relativistic version of Newton's law of motion can be obtained by writing Maxwell's equations in [[Covariant transformation|covariant form]] and identifying a covariant form that is a generalization of Newton's law of motion.
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| Newton's law of motion can be written in modern covariant notation in terms of the field strength tensor as (cgs units):
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| :<math>mc\frac{du^\alpha}{d\tau} = F^{\alpha\beta} q u_\beta,</math>
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| where ''m'' is the particle [[mass]], ''q'' is the [[Electric charge|charge]], and
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| :<math>u_\beta = \eta_{\beta\alpha } u^\alpha = \eta_{\beta\alpha} \frac{dx^\alpha}{d\tau}</math>
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| is the [[four-velocity|4-velocity]] of the particle. Here, <math>\tau</math> is ''c'' times the [[proper time]] of the particle and <math>\eta</math> is the [[Minkowski metric]] tensor.
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| The field strength tensor is written in terms of fields as:
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| :<math>F^{\alpha\beta} = \left(\begin{matrix}
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| 0 & {E_x} & {E_y} & {E_z} \\
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| -{E_x} & 0 & cB_z & -cB_y \\
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| -{E_y} & -cB_z & 0 & cB_x \\
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| -{E_z} & cB_y & -cB_x & 0
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| \end{matrix}\right).</math>
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| Alternatively, using the four vector:
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| :<math>A^\alpha = \left(\phi/c, A_x, A_y, A_z \right), </math>
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| related to the electric and magnetic fields by:
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| :<math>\mathbf{E} = -\nabla\phi - \partial_t \mathbf{A}, \quad \mathbf{B} = \nabla \times \mathbf{A},</math>
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| the field tensor becomes:<ref name =DJ_Griffiths>{{cite book |author=DJ Griffiths |title=Introduction to electrodynamics |publisher=Pearson/Addison-Wesley |year=1999 |location=Saddle River NJ |page=541 |isbn =0-13-805326-X}}</ref>
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| :<math>F^{\alpha\beta} = \frac{\partial A^\beta}{\partial x_\alpha} - \frac{\partial A^\alpha}{\partial x_\beta},</math>
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| where:
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| :<math>x_\alpha = \left(-ct, x, y, z \right).</math>
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| The fields are transformed to a frame moving with constant relative velocity by:
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| :<math>\acute{F}^{\mu\nu} = {\Lambda^\mu}_\alpha {\Lambda^\nu}_\beta F^{\alpha\beta},</math>
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| where <math>{\Lambda^\mu}_\alpha</math> is a [[Lorentz transformation]].
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| In the magnet/conductor problem this gives
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| :<math>\mathbf{E}' = \gamma \frac{\mathbf{v}}{c} \times \mathbf{B},</math>
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| which agrees with the traditional transformation when one takes into account the difference between SI and cgs units. Thus, the relativistic modification to Newton's law of motion using the traditional Lorentz force yields predictions for the motion of particles that are consistent in all frames of reference with Maxwell's equations.
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| ==References and notes==
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| <references/>
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| ==External links==
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| * [http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/special_relativity/special_relativity.html Magnets and conductors in special relativity]
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| ==Further reading==
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| * {{cite book | author=Einstein, A. | author-link = Albert Einstein | title=Relativity: The Special and General Theory | location= New York | publisher=Crown| year=1961 | isbn=0-517-02961-8}}
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| * {{Cite book| last1 = Feynman| first1 = Richard P. |author-link = Richard Feynman |first2 = Robert B. |last2 = Leighton |author2-link = Robert B. Leighton |first3 = Matthew |last3 = Sands |author3-link = Matthew Sands |title = [[The Feynman Lectures on Physics]] |volume = Vol 2 |year = 2006 |isbn = 0-8053-9045-6
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| |page= 13-6 Chapter 13}} (The relativity of magnetic and electric fields)
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| * {{cite book | author=Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | isbn=0-7167-0344-0}}
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| * {{cite book | author=Landau, L. D. and Lifshitz, E. M.| title=Classical Theory of Fields (Fourth Revised English Edition) | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7}}
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| * {{cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|isbn=0-471-30932-X}}
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| * {{cite book |author=C Møller |title=The Theory of Relativity |publisher=Oxford University Press |location=Oxford UK |isbn=0-19-560539-X |year=1976 |url=http://worldcat.org/oclc/220221617&referer=brief_results |edition=Second Edition}}
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| ==See also==
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| {{multicol}}
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| * [[Principle of relativity]]
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| * [[Galilean invariance]]
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| * [[Lorentz transformation]]
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| * [[Theory of special relativity]]
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| {{multicol-break}}
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| * [[Faraday's law of induction|Faraday's law]]
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| * [[Lenz's law]]
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| * [[Inertial frame]]
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| * [[Annus Mirabilis Papers]]
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| {{multicol-break}}
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| * [[Electric motor]]
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| * [[Eddy current]]
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| * [[Faraday paradox]]
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| * [[Darwin Lagrangian]]
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| {{multicol-end}}
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| [[Category:Electromagnetism]]
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| [[Category:Special relativity]]
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