Moving magnet and conductor problem

Conductor moving in a magnetic field.

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 conductor moving with constant velocity, 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. 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".^[1] 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.

This problem, along with Michelson-Morley experiment, formed the basis of Einstein's theory of relativity.

Introduction

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 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.

— On the electrodynamics of moving bodies, A. Einstein, 1905

An overriding requirement on the descriptions in different frameworks is that they be consistent. Consistency is an issue because 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.

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 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.^[2] That means the paradox of different descriptions may be only 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^α = (φ / c₀, A ) replaces E and B^[3] and provides a frame-independent description (albeit less visceral than the E– B–description).^[4] 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.

Background

Electromagnetic fields are not directly observable. The existence of 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.

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.

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.^[5]

Transformation of fields, assuming Galilean transformations

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.^[6]

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.

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.

Magnet frame

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.

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):

$\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),$

where $q \$ is the charge on the particle, $\mathbf{v} \$ is the particle velocity and F is the Lorentz force. Here, however, the electric field is zero, so the force on the particle is

$\mathbf{F} = q \mathbf{v} \times \mathbf{B} .$

Conductor frame

In the conductor frame, the magnetic field B' will be related to the magnetic field B in the magnet frame according to:^[7]

$\mathbf{B}'(\mathbf{x}',t) = \mathbf{B}(\mathbf{x}'+\mathbf{v}t) .$

In this frame, there is an electric field, generated by the Maxwell-Faraday equation:

$\mathbf{ \nabla \times E}' =- \frac { \partial \mathbf{B}'}{\partial t} \ .$

Using the above expression for B',

$\mathbf{ \nabla \times E}' = - (\mathbf{v} \cdot \nabla) \mathbf{B} = -\nabla\times(\mathbf{B} \times \mathbf{v}) - \mathbf{v}(\nabla\cdot \mathbf{B}) = -\nabla\times(\mathbf{B} \times \mathbf{v})$

(using the chain rule and Gauss's law for magnetism). This has the solution:

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

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

$\mathbf{F}' = q\mathbf{E}' = q\mathbf{v}\times \mathbf{B} \ .$

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.

Galilean transformation formula for fields

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,

$\mathbf{E}' = \mathbf{E} + \mathbf{v}\times \mathbf{B}$

$\mathbf{B}' = \mathbf{B} - \frac{1}{{c_0}^2}\ \mathbf{v} \times \mathbf{E}\ ,$

with c₀ the speed of light in free space.

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.

Transformation of fields as predicted by Maxwell's equations

See also: Classical electromagnetism and special relativity

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 Maxwell's equations transform as:^[8]

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

where

$\gamma = \frac{1}{\sqrt{1 - {(v/c_0)}^2}}$

is called the Lorentz factor and c₀ 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 c₀. 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.

Consequently, the force on the charge is

$\mathbf{F}' = q \mathbf{E}' = q \gamma \mathbf{v} \times \mathbf{B}\ .$

This expression differs from the expression obtained from the nonrelativistic Newton's law of motion by a factor of γ. Special relativity modifies space and time in a manner such that the forces and fields transform consistently.

Modification of dynamics for consistency with Maxwell's equations

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

The Lorentz force has the same form in both frames, though the fields differ, namely:

$\mathbf{F} = q \left[\mathbf{E} + \mathbf{v} \times \mathbf{B} \right] \ .$

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

$F_y = -q\ v\ B \ ,$

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:

${F_y}' = q\ E' = -q\ \gamma v \ B .$

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.

Relativity takes the Lorentz transformation of space-time suggested by invariance of Maxwell's equations and imposes it upon 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 ) :^[9]

$x'=\gamma (x-vt)\$  $x=\gamma(x'+vt')\ ,$

$t'=\gamma (t-\frac{vx}{c_0^2})$  $t=\gamma(t'+\frac{vx'}{c_0^2})\ .$

These transformations lead to a change in the y-component of a force:

${F_y}' = \gamma {F_y} \ .$

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:

$\gamma F_y \ = -q\ \gamma v \ B \ ,$    ${F_y}' \ = -q\ \gamma v \ B \ ,$

which agrees completely. So the force on the charge is not the same in both frames, but it transforms as expected according to relativity.

Newton's law of motion in modern notation

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

The modern approach to obtaining the relativistic version of Newton's law of motion can be obtained by writing Maxwell's equations in covariant form and identifying a covariant form that is a generalization of Newton's law of motion.

Newton's law of motion 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:

$F^{\alpha \beta} = \left( \begin{matrix} 0 & {E_x} & {E_y} & {E_z} \\ -{E_x} & 0 & cB_z & -cB_y \\ -{E_y} & -cB_z & 0 & cB_x \\ -{E_z} & cB_y & -cB_x & 0 \end{matrix} \right) .$

Alternatively, using the four vector:

$A^{\alpha} = \left( \phi / c,\ A_x,\ A_y,\ A_z \right) \ ,$

related to the electric and magnetic fields by:

$\mathbf{E = -\nabla} \phi - \partial_t \mathbf{A}$    $\mathbf{B = \nabla \times A } \ ,$

the field tensor becomes:^[10]

$F^{\alpha \beta} = \frac {\partial A^{\beta}}{\partial x_{\alpha}} - \frac {\partial A^{\alpha}}{\partial x_{\beta}} \ ,$

where:

$x_{\alpha} = \left( -ct,\ x,\ y,\ z \right) \ .$

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, 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.

References and notes

1. ^ The Laws of Physics are the same in all inertial frames.
2. ^ 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.
3. ^ The symbol c₀ represents the speed of light in free space.
4. ^ However, φ and A are not completely disentangled, so the two types of E-field are not separated completely. See Jackson From Lorenz to Coulomb and other explicit gauge transformations The author stresses that Lorenz is not a typo..
5. ^ Roger Penrose (Martin Gardner: foreword) (1999). The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press. p. 248. ISBN 0192861980. http://books.google.com/books?id=oI0grArWHUMC&pg=PA248&dq=reference+%22laws+of+physics%22. 
6. ^ See Jackson, Classical Electrodynamics, Section 5.15.
7. ^ 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.
8. ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. Chapter 10.21; p. 402–403 ff. ISBN 0-7637-3827-1. http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271&sig=PgEEBA6TQEZ5fD_AhJQ8dd7MGHo#PPA368,M1. 
9. ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. Chapter 10.5; p. 368 ff. ISBN 0-7637-3827-1. http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271#PPA368,M1. 
10. ^ DJ Griffiths (1999). Introduction to electrodynamics. Saddle River NJ: Pearson/Addison-Wesley. p. 541. ISBN 0-13-805326-X. 

External links

• Magnets and conductors in special relativity

Further reading

[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. 

[5] C Møller (1976). The Theory of Relativity (Second Edition ed.). Oxford UK: Oxford University Press. ISBN 019560539X. http://worldcat.org/oclc/220221617&referer=brief_results. 

See also

Principle of relativity Galilean invariance Lorentz transformation Theory of special relativity

Faraday's law Lenz's law Inertial frame Annus Mirabilis Papers

Electric motor Eddy current Faraday paradox Darwin Lagrangian