Moving magnet and conductor problem

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Conductor moving in a magnetic field.
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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]

   
“
 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

Contents

[edit] Background

Electromagnetic fields are not directly observable. The existence of classical electromagnetic fields is inferred from the motion of charged particles, whose trajectories are observable. It can be claimed that electromagnetic fields were invented in order to explain the observed motions of classical charged particles.

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 bulls eye, 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.

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 at constant speed with respect 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 consistently generate the same motion in the conductor and all observers must be able to agree on that motion. The motion of a charged particle is calculated from a force equation. Not only must the field equations be consistent in order to generate a commonly agreed upon motion, but the force equation must also be consistent. At the time of Einstein in 1905, the field equations as represented by Maxwell's equations, were properly consistent. The Lorentz force equation, however, had to be modified in order to provide unique particle trajectories upon which all observers could agree.

[edit] Transformation of fields as predicted by Newtonian mechanics

In nonrelativistic 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):

{d \left ( m \mathbf{v} \right ) \over dt } = \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),

where q \ is the charge on the particle and \mathbf{v} \ is the particle velocity.

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

\mathbf{F} = q \mathbf{v} \times \mathbf{B} ,

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

\mathbf{F}' = q \mathbf{E}',

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

\ \mathbf{E}' = \mathbf{v} \times \mathbf{B}.

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

[edit] 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

\ \mathbf{E}' = \gamma \mathbf{v} \times \mathbf{B}

where

\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}

is called the Lorentz factor and c is the speed of light in a vacuum.

This expression differs from the expression obtained from the nonrelativistic 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.

[edit] 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

{d \left ( \gamma m \mathbf{v} \right ) \over dt } = \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}).

The change of energy due to the fields is

{d \left ( \gamma m c^2 \right ) \over dt } = q \mathbf{E} \cdot \mathbf{v} .

[edit] Lorentz force equation in modern notation

Main article: Formulation of Maxwell's equations in special relativity

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.

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

m c { d u^{\alpha} \over { d \tau } } = { {} \over {} }F^{\alpha \beta} q u_{\beta}

where m is the particle mass, q is the charge, and

u_{\beta} = \eta_{\beta \alpha } u^{\alpha } = \eta_{\beta \alpha } { d x^{\alpha } \over {d \tau} }

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.

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

F^{\alpha \beta} = \left( \begin{matrix} 0 & {E_x} & {E_y} & {E_z} \\ -{E_x} & 0 & B_z & -B_y \\ -{E_y} & -B_z & 0 & B_x \\ -{E_z} & B_y & -B_x & 0 \end{matrix} \right) .

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

\acute{F}^{\mu \nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta} ,

where {\Lambda^{\mu}}_{\alpha} is a Lorentz transformation.

In the magnet/conductor problem this gives

\ \mathbf{E}' = \gamma { \mathbf{v} \over c } \times \mathbf{B},

which agrees with the traditional transformation when one takes into account the difference between SI and cgs units. Thus, with the relativistic modification to the Lorentz force, both the Lorentz force and Maxwell's equations yield predictions for the motion of particles that are coonsistent in all frames of reference.

[edit] See also

[edit] External links

[edit] References

[1] Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8.
[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 0-471-30932-X.


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