Classical electromagnetism and special relativity

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. For a fully covariant discussion, see Covariant formulation of classical electromagnetism.

Electromagnetism
Electricity · Magnetism
Electrostatics Electric charge · Coulomb's law · Electric field · Electric flux · Gauss's law · Electric potential energy · Electric potential · Electrostatic induction · Electric dipole moment · Polarization density
Magnetostatics Ampère's law · Electric current · Magnetic field · Magnetization · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss's law for magnetism
Electrodynamics Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy current
Electrical Network Electrical conduction · Electrical resistance · Capacitance · Inductance · Impedance · Resonant cavities · Waveguides
Covariant formulation Electromagnetic tensor · EM Stress-energy tensor · Four-current · Electromagnetic four-potential
Scientists Ampère · Coulomb · Faraday · Gauss · Heaviside · Henry · Hertz · Lorentz · Maxwell · Tesla · Volta · Weber · Ørsted
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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. Secondly, it sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

Joules-Bernoulli equation for fields and forces

This equation considers 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]

$\vec {{E}_{\parallel}}'= \vec {{E}_{\parallel}}$            $\vec {{B}_{\parallel}}'= \vec {{B}_{\parallel}}$

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

where

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

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

An equivalent, alternative expression is:

$\mathbf{E}' = \gamma \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right ) - \left ({\gamma-1} \right ) ( \mathbf{E} \cdot \mathbf{\hat{v}} ) \mathbf{\hat{v}}$

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

where :$\mathbf{\hat{v}}$ is velocity's unit vector.

If a particle of charge q moves with velocity $\mathbf{u}$ with respect to frame S, then the Lorentz force in frame S is:

$\mathbf{F}=q\mathbf{E}+q \mathbf{u} \times \mathbf{B}$

In frame S', the Lorentz force is:

$\mathbf{F'}=q\mathbf{E'}+q \mathbf{u'} \times \mathbf{B'}$

If S and S' have aligned axes then^[2]:

$u_x'=\frac{u_x+v}{1 + (v \ u_x)/c^2}$

$u_y'=\frac{u_y/\gamma}{1 + (v \ u_x)/c^2}$

$u_z'=\frac{u_z/\gamma}{1 + (v \ u_x)/c^2}$

A derivation for the transformation of the Lorentz force for the particular case $\mathbf{u=0}$ is given here.^[3] A more general one can be seen here.^[4]

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

$\displaystyle E'_x = E_x$

$E'_y = \gamma \left ( E_y - v B_z \right )$

$E'_z = \gamma \left ( E_z + v B_y \right )$

$\displaystyle B'_x = B_x$

$B'_y = \gamma \left ( B_y + \frac{v}{c^2} E_z \right )$

$B'_z = \gamma \left ( B_z - \frac{v}{c^2} E_y \right )$

and in Gaussian-cgs units, the transformation is given by:^[5]

$\displaystyle E'_x = E_x$

$E'_y = \gamma \left ( E_y - \beta B_z \right )$

$E'_z = \gamma \left ( E_z + \beta B_y \right )$

$\displaystyle B'_x = B_x$

$B'_y = \gamma \left ( B_y + \beta E_z \right )$

$B'_z = \gamma \left ( B_z - \beta E_y \right )$

where $\beta \ \overset{\underset{\mathrm{def}}{}}{=} \ v/c$.

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.

Relationship between electricity and magnetism

"One part of the force between moving charges we call the magnetic force. It is really one aspect of an electrical effect."

- Feynman Lectures vol. 2, ch. 1-1

Deriving magnetism from electrostatics

Main article: relativistic electromagnetism

The chosen reference frame determines if an electromagnetic phenomenon is viewed as an effect of electrostatics or magnetism. Authors usually derive magnetism from electrostatics when special relativity and charge invariance are taken into account. The Feynman Lectures on Physics (vol. 2, ch. 13-6) uses this method to derive the "magnetic" force on a moving charge next to a current-carrying wire. See also Haskell,^[6] Landau,^[7] and Field.^[8]

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.^[9] 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.

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.^[10]

Covariant formulation in vacuum

Main article: Covariant formulation of classical electromagnetism

The laws and objects in classical electromagnetism can be written in a form which is manifestly covariant. Here, this is only done so for vacuum (or for the microscopic Maxwell equations, not using macroscopic descriptions of materials such as electric permittivity). 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.

Footnotes

1. ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. p. Chapter 10.21; p. 402–403 ff. ISBN 0-7637-3827-1. http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn:0763738271#PPA368,M1. 
2. ^ R.C.Tolman "Relativity Thermodynamics and Cosmology" pp25
3. ^ Force Laws and Maxwell's Equations http://www.mathpages.com/rr/s2-02/2-02.htm at MathPages
4. ^ http://www.hep.princeton.edu/~mcdonald/examples/EM/ganley_ajp_31_510_62.pdf
5. ^ Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X
6. ^ http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm
7. ^ E M Lifshitz, L D Landau (1980). The classical theory of fields. Course of Theoretical Physics. Vol. 2 (Fourth Edition ed.). Oxford UK: Butterworth-Heinemann. ISBN 0750627689. http://worldcat.org/isbn/0750627689. 
8. ^ 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
9. ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. p. 395. ISBN 0-7637-3827-1. http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn:0763738271#PPR6,M1. 
10. ^ David J Griffiths (1999). Introduction to electrodynamics (Third Edition ed.). Prentice Hall. pp. 478–9. ISBN 013805326X. http://worldcat.org/isbn/013805326X.