Classical electromagnetism and special relativity

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This article is about the contribution of special relativity to the modern theory of classical electromagnetism. For the contribution of classical electromagnetism to the development of special relativity, see History of special relativity.

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. First of all, it gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another. Second of all, it sheds light on the relationship between electricity and magnetism, for example showing that an electric force in one frame of reference may be a magnetic force in another and vice-versa, and likewise that certain laws of magnetism can be "derived" from corresponding laws of electricity and vice-versa. Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

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[edit] Lorentz transformation rules for fields

Consider two inertial frames. As notation, the field variables in one frame are unprimed, and in a frame moving relative to the unprimed frame at velocity v, the fields are denoted with primes. In addition, the fields parallel to the velocity v are denoted by \stackrel{\vec {E}_{\parallel}}{} while the fields perpendicular to v are denoted as \stackrel{\vec {E}_{\bot}}{}. In these two frames moving at relative velocity v, the E-fields and B-fields are related by:[1]

\vec {{E}_{\parallel}}'= \vec {{E}_{\parallel}}            \vec {{B}_{\parallel}}'= \vec {{B}_{\parallel}}
\vec {{E}_{\bot}}'= \gamma \left( \vec {E} + \vec{ v} \times \vec {B} \right)_{\bot}    \vec {{B}_{\bot}}'= \gamma \left( \vec {B} -\frac{1}{c_0^2} \vec{ v} \times \vec {E} \right)_{\bot} \ ,

where

\gamma \overset{\underset{\mathrm{def}}{}}{=}\ \frac{1}{\sqrt{1 - v^2/{c_0}^2}}

is called the Lorentz factor and c0 is the speed of light in free space. The inverse transformations are the same except v → −v.

An equivalent, alternative expression is:

\mathbf{E}' = \gamma \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right ) - \left (\frac{\gamma-1}{v^2} \right ) ( \mathbf{E} \cdot \mathbf{v} ) \mathbf{v}
\mathbf{B}' = \gamma \left( \mathbf{B} - \frac {\mathbf{v} \times \mathbf{E}}{c^2} \right ) - \left (\frac{\gamma-1}{v^2} \right ) ( \mathbf{B} \cdot \mathbf{v} ) \mathbf{v}

Component by component, for relative motion along the x-axis, this works out to be the following:

\displaystyle E'_x = E_x
E'_y = \gamma \left ( E_y - v B_z \right )
E'_z = \gamma \left ( E_z + v B_y \right )
\displaystyle B'_x = B_x
B'_y = \gamma \left ( B_y + \frac{v}{c^2} E_z \right )
B'_z = \gamma \left ( B_z - \frac{v}{c^2} E_y \right )

The claim that the transformation rules for E and B take this particular form is equivalent to the claim that the electromagnetic tensor (defined below) is a covariant tensor.

[edit] Interrelationship between electricity and magnetism

See also: Relativistic electromagnetism

[edit] Fields intermix in different frames

The above transformation rules show that the electric field in one frame contributes to the magnetic field in another frame, and vice versa.[2] This is often described by saying that the electric field and magnetic field are two interrelated aspects of a single object, called the electromagnetic field. Indeed, the entire electromagnetic field can be encoded in a single rank-2 tensor called the electromagnetic tensor; see below.

[edit] Relationships between electric and magnetic laws

Some authors have attempted to derive various laws of magnetism, starting by assuming various laws of electricity, and also assuming that special relativity is true. For example, it has been surmised that the v×B component of the Lorentz force might be derived from Coulomb's law and special relativity if one assumes invariance of electric charge. See Haskell, Landau[3] and Field.[4] For more examples, see relativistic electromagnetism.

[edit] Moving magnet and conductor problem

Main article: Moving magnet and conductor problem

A famous example of the intermixing of electric and magnetic phenomena in different frames of reference is called the "moving magnet and conductor problem", cited by Einstein in his 1905 paper on Special Relativity.

If a conductor moves with a constant velocity through the field of a stationary magnet, eddy currents will be produced due to a magnetic force on the electrons in the conductor. In the rest frame of the conductor, on the other hand, the magnet will be moving and the conductor stationary. Classical electromagnetic theory predicts that precisely the same microscopic eddy currents will be produced, but they will be due to an electric force.[5]

[edit] Covariant formulation

Main article: Covariant formulation of classical electromagnetism

The laws and objects in classical electromagnetism can be written in a form which is "manifestly covariant". In cgs-Gaussian units, the electric and magnetic fields combine into the covariant electromagnetic tensor:

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 charge and current, meanwhile, combine into a four-vector called the four-current:

J^{\alpha} = \, (c \rho, \vec{J} )

where ρ is the charge density, \vec{J} is the current density, and c is the speed of light.

With these definitions, Maxwell's equations take the following manifestly covariant form:

\frac{\partial F^{\alpha\beta}}{\partial x^\alpha}=\frac{4\pi}{c}J^\beta \qquad\hbox{and}\qquad 0=\epsilon^{\alpha\beta\gamma\delta}\frac{\partial F_{\alpha\beta}}{\partial x^\gamma}

where F αβ is the electromagnetic tensor, J α is the 4-current, є αβγδ is the Levi-Civita symbol (a mathematical construct), and the indices behave according to the Einstein summation convention.

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss's Law and Ampere's Law (with Maxwell's correction). The second equation is an expression of the homogenous equations, Faraday's law of induction and Gauss's law for magnetism.

Another covariant electromagnetic object is the electromagnetic stress-energy tensor, a covariant rank-2 tensor which includes the Poynting vector, Maxwell stress tensor, and electromagnetic energy density. Yet another is the four-potential, a four-vector combining the electric potential and magnetic vector potential.

A particularly convenient gauge choice in a relativistic setting is the Lorenz gauge condition, which in terms of the four-potential takes the covariant form:

\partial_\alpha A^\alpha = 0

In the Lorenz gauge, Maxwell's equations can be written as (in cgs):

\Box^2 A^\mu = -\frac{4\pi}{c} J^\mu

where \Box^2 denotes the d'Alembertian.

For a more comprehensive presentation of these topics, see Covariant formulation of classical electromagnetism.

[edit] Footnotes

  1. ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett, Chapter 10.21; p. 402-403 ff. ISBN 0-7637-3827-1. 
  2. ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett, p. 395. ISBN 0-7637-3827-1. 
  3. ^ E M Lifshitz, L D Landau (1980). The classical theory of fields: Vol. 2 (Course of theoretical physics), Fourth Edition, Oxford UK: Butterworth-Heinemann. ISBN 0750627689. 
  4. ^ J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics". Phys. Scr. 74 702-717
  5. ^ David J Griffiths (1999). Introduction to electrodynamics, Third Edition, Prentice Hall, pp. 478-9. ISBN 013805326X.