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Encyclopedia > Moving magnet and conductor problem
Conductor moving in a magnetic field.
Conductor moving in a magnetic field.

In the Moving magnet and conductor problem the force on a conductor moving with constant speed v with respect to a magnet is calculated in the frame of reference of the magnet and in the frame of reference of the conductor. It is found that the conductor experiences a magnetic force in the frame of the magnet and an electric force in the frame of the conductor. The same phenomena would seem to have two different descriptions depending on the frame of reference of the observer. If more than one description of a physical phenomenon exists, then, to avoid a paradox, one expects the descriptions to be consistent with each other. The paradox is resolved by noting that magnetic fields in one reference frame are transformed into electric fields in another frame. Electric fields can also be transformed into magnetic fields. The paradox is complicated by the observation that Newtonian classical mechanics predicts one form for the transformation of fields, while electrodynamics as expressed by Maxwell's equations predicts another transformation. This paradox, along with the Michelson Morley experiment and the aberration of light, forms the experimental impetus for the development of special relativity in which it is concluded that classical mechanics must be revised such that transformation of fields and forces in moving reference frames is consistent with electrodynamics and Maxwell's equations. Einstein's 1905 paper that introduced the world to relativity opens with a description of the magnet/conductor problem.[1] Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... The Michelson-Morley experiment, one of the most important and famous experiments in the history of physics, was performed in 1887 at what is now Case Western Reserve University, and is considered to be the first strong evidence against the theory of a luminiferous aether. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. ... Albert Einstein, photographed in 1947 by Oren J. Turner. ...

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 neighbourhood 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 neighbourhood 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. A. Einstein, 1905


Transformation of fields as predicted by Newtonian mechanics

The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation (SI units): In physics, an electric field or E-field is an effect produced by an electric charge (or a time-varying magnetic field) that exerts a force on charged objects in the field. ... Current flowing through a wire produces a magnetic field (B, labeled M here) around the wire. ... In physics, the Lorentz force is the force exerted on a charged particle in an electromagnetic field. ...

where is the charge on the particle and is the particle velocity.

For a conductor moving in the frame of the magnet, the force on the conductor is

since there is no electric field in the magnet frame.

The force on the conductor in the conductor frame is due to an electric field (an electromotive force) generated by the magnetic field changing in time as the magnet approaches the conductor. This force can be written

since the conductor is at rest in that frame.

The force on the conductor should be independent of the frame of reference. Therefore there must be an electric field in the conductor frame that is related to the magnetic field in the magnet frame by the expression

This expression, however, is not consistent with the transformation of fields as predicted by Maxwell's equations.

Transformation of fields as predicted by Maxwell's equations

In the moving magnet and conductor problem, the fields transform according to Maxwell's equations as


is called the Lorentz factor and c is the speed of light in a vacuum. It has been suggested that Lorentz term be merged into this article or section. ... The speed of light in a vacuum is denoted by the letter c. ...

This expression differs from the expression obtained from the Lorentz force by a factor of γ. Special relativity modifies the Lorentz force in a manner such that the fields transform consistently with Maxwell's equations.

Modification of the Lorentz force to be consistent with Maxwell's equations

To be consistent with Maxwell's equations the Lorentz force equation must be modified to

The change of energy due to the fields is

Lorentz force equation in modern notation

The modern approach to obtaining the relativistic version of the Lorentz force can be obtained by writing Maxwell's equations in covariant form and identifying a covariant form that is a generalization of the Lorentz equation. In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... It has been suggested that this article or section be merged with Covariant. ...

The Lorentz force equation can be written in modern covariant notation in terms of the field strength tensor as (cgs units):

where m is the particle mass, q is the charge, and Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ...

is the 4-velocity of the particle. Here, τ is c times the proper time of the particle and η is the Minkowski metric tensor. In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector (vector in four-dimensional spacetime) that replaces classical velocity (a three-dimensional vector). ... Proper time is time as measured by the clock for an observer who is traveling through spacetime. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

The field strength tensor is written in terms of fields as:

The fields are transformed to a frame moving with constant relative velocity by:

where is a Lorentz transformation. A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed (=rotation of Minkowski space). ...

In the magnet/conductor problem this gives

which agrees with the traditional transformation when one takes into account the difference between SI and cgs units.

See also

Principle of relativity
Galilean invariance
Annus Mirabilis Papers
Faraday's law
Lenz's law
Electromagnetic induction
Electric motor
Eddy current

Note: The principle of relativity should not be confused with the Theory of relativity. ... Galilean invariance is a principle which states that the fundamental laws of physics are the same in all inertial (uniform-velocity) frames of reference. ... Einstein, in 1905, when he wrote the Annus Mirabilis Papers The Annus Mirabilis Papers (from Annus mirabilis, Latin for year of wonders) are the papers of Albert Einstein submitted to the Annalen der Physik journal in 1905. ... Faradays law can mean: Faradays law of induction (electromagnetic fields) Faradays law of electrolysis This is a disambiguation page — a navigational aid which lists pages that might otherwise share the same title. ... Lenzs law (pronounced (IPA) ) was formulated by German physicist Heinrich Lenz and gives the direction of the induced emf resulting from electromagnetic induction, thus: Definition The induced current produced in the conductor always flows in a direction such that it opposes the change that is producing it. ... Electromagnetic induction is the production of an electrical potential difference (or voltage) across a conductor situated in a changing magnetic flux. ... Rotating magnetic field as a sum of magnetic vectors from 3 phase coils. ... As the circular plate moves down through a small region of constant magnetic field directed into the page, eddy currents are induced in the plate. ...

External links

Magnets and conductors in special relativity


[1] Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-029618.
[2] Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.
[3] Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.
[4] Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 047130932X.

edit General subfields within physics

Atomic, molecular, and optical physics | Classical mechanics | Condensed matter physics | Continuum mechanics | Electromagnetism | Special relativity | General relativity | Particle physics | Quantum field theory | Quantum mechanics | Statistical mechanics | Thermodynamics The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Atomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess the property of electric charge, and is in turn affected by the presence and motion of such particles. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. ... For a non-technical introduction to the topic, please see Introduction to General relativity. ... Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... ‹ The template below has been proposed for deletion. ...



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