Electromagnetism
Electricity Magnetism Optics History Computational Textbooks Phenomena
Electrostatics Charge density Conductor Coulomb law Electret Electric charge Electric dipole Electric field Electric flux Electric potential Electrostatic discharge Electrostatic induction Gauss's law Insulator Permittivity Polarization Potential energy Static electricity Triboelectricity
Magnetostatics Ampère's law Biot–Savart law Gauss's law for magnetism Magnetic dipole Magnetic field Magnetic flux Magnetic scalar potential Magnetic vector potential Magnetization Permeability Right-hand rule
Electrodynamics Bremsstrahlung Cyclotron radiation Displacement current Eddy current Electromagnetic field Electromagnetic induction Electromagnetic pulse Electromagnetic radiation Faraday's law Jefimenko equations Larmor formula Lenz's law Liénard–Wiechert potential London equations Lorentz force Maxwell's equations Maxwell tensor Poynting vector Synchrotron radiation
Electrical network Alternating current Capacitance Current density Direct current Electric current Electric power Electrolysis Electromotive force Impedance Inductance Joule heating Kirchhoff's laws Network analysis Ohm's law Parallel circuit Resistance Resonant cavities Series circuit Voltage Watt Waveguides
Magnetic circuit AC motor DC motor Electric machine Electric motor Gyrator–capacitor Induction motor Linear motor Magnetomotive force Permeance Reluctance (complex) Reluctance (real) Rotor Stator Transformer
Covariant formulation Electromagnetic tensor Electromagnetism and special relativity Four-current Four-potential Mathematical descriptions Maxwell equations in curved spacetime Relativistic electromagnetism Stress–energy tensor
Scientists Ampère Biot Coulomb Davy Einstein Faraday Fizeau Gauss Heaviside Helmholtz Henry Hertz Hopkinson Jefimenko Joule Kelvin Kirchhoff Larmor Lenz Liénard Lorentz Maxwell Neumann Ohm Ørsted Poisson Poynting Ritchie Savart Singer Steinmetz Tesla Thomson Volta Weber Wiechert
v t e

The theory ofspecial relativity plays an important role in the modern theory ofclassical electromagnetism. It gives formulas for how electromagnetic objects, in particular theelectric andmagnetic fields, are altered under aLorentz transformation from oneinertial frame of reference to another. It sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electric or magnetic laws. It motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity.^[1] Moreover, the apparent coincidences in which the same effect was observed due to different physical phenomena by two different observers would be shown to be not coincidental in the least by special relativity. In fact, half of Einstein's 1905 first paper on special relativity, "On the Electrodynamics of Moving Bodies", explains how to transform Maxwell's equations.

This equation considers twoinertial frames. Theprimed frame is moving relative to the unprimed frame at velocityv. Fields defined in the primed frame are indicated by primes, and fields defined in the unprimed frame lack primes. The field componentsparallel to the velocityv are denoted byE_∥ andB_∥ while the field components perpendicular tov are denoted asE_⟂ andB_⟂. In these two frames moving at relative velocityv, theE-fields andB-fields are related by:^[2]

where

${\displaystyle \gamma \stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\frac{1}{\sqrt{1-v^2/c^2}}}$

is called theLorentz factor andc is thespeed of light infree space. Lorentz factor (γ) is the same in bothsystems. The inverse transformations are the same except for the substitutionv → −v.

An equivalent, alternative expression is:^[3]

where${\displaystyle \hat{\mathbf{v}}=\frac{\mathbf{v}}{\Vert \mathbf{v}\Vert }}$ is the velocityunit vector. With previous notations, one actually has${\displaystyle (\mathbf{E}\cdot \hat{\mathbf{v}})\hat{\mathbf{v}}={\mathbf{E}}_{\parallel }}$ and${\displaystyle (\mathbf{B}\cdot \hat{\mathbf{v}})\hat{\mathbf{v}}={\mathbf{B}}_{\parallel }}$.

Component by component, for relative motion along the x-axisv = (v, 0, 0), this works out to be the following:

If one of the fields is zero in one frame of reference, that doesn't necessarily mean it is zero in all other frames of reference. This can be seen by, for instance, making the unprimed electric field zero in the transformation to the primed electric field. In this case, depending on the orientation of the magnetic field, the primed system could see an electric field, even though there is none in the unprimed system.

This does not mean two completely different sets of events are seen in the two frames, but that the same sequence of events is described in two different ways (see§ Moving magnet and conductor problem below).

If a particle of chargeq moves with velocityu with respect to frameS, then theLorentz force in frameS is:

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

In frameS′, the Lorentz force is:

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

A derivation for the transformation of the Lorentz force for the particular caseu =0 is given here.^[4] A more general one can be seen here.^[5]

The transformations in this form can be made more compact by introducing theelectromagnetic tensor (defined below), which is acovariant tensor.

For theelectric displacementD andmagnetic field strengthH, using theconstitutive relations and the result forc²:

${\displaystyle \mathbf{D}={ϵ}_0\mathbf{E},\mathbf{B}={\mu }_0\mathbf{H},c^2=\frac{1}{{ϵ}_0{\mu }_0},}$

gives

Analogously forE andB, theD andH form theelectromagnetic displacement tensor.

An alternative simpler transformation of the EM field uses theelectromagnetic potentials – theelectric potentialφ andmagnetic potentialA:^[6]

whereA_∥ is the component ofA that is parallel to the direction of relative velocity between framesv, andA_⟂ is the perpendicular component. These transparently resemble the characteristic form of other Lorentz transformations (like time-position and energy-momentum), while the transformations ofE andB above are slightly more complicated. The components can be collected together as:

Analogously for thecharge densityρ andcurrent densityJ,^[6]

Collecting components together:

For speedsv ≪c, the relativistic factorγ ≈ 1, which yields:

so that there is no need to distinguish between the spatial and temporal coordinates in Maxwell's equations.

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

The chosen reference frame determines whether an electromagnetic phenomenon is viewed as an electric or magnetic effect or a combination of the two. Authors usually derive magnetism from electrostatics when special relativity andcharge invariance are taken into account.The Feynman Lectures on Physics (vol. 2, ch. 13–6) uses this method to derive the magnetic force on charge in parallel motion next to a current-carrying wire. See also Haskell^[8] and Landau.^[9]

If the charge instead moves perpendicular to a current-carrying wire, electrostatics cannot be used to derive the magnetic force. In this case, it can instead be derived by considering the relativistic compression of the electric field due to the motion of the charges in the wire.^[10]

The above transformation rules show that the electric field in one frame contributes to the magnetic field in another frame, and vice versa.^[11] This is often described by saying that the electric field and magnetic field are two interrelated aspects of a single object, called theelectromagnetic field. Indeed, the entire electromagnetic field can be represented in a single rank-2 tensor called theelectromagnetic tensor; see below.

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 amagnetic 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 anelectric force.^[12]

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

This section usesEinstein notation, includingEinstein summation convention. See alsoRicci calculus for a summary oftensor index notations, andraising and lowering indices for definition of superscript and subscript indices, and how to switch between them. TheMinkowski metric tensorη here hasmetric signature(+ − − −).

The above relativistic transformations suggest the electric and magnetic fields are coupled together, in a mathematical object with 6 components: anantisymmetric second-ranktensor, or abivector. This is called theelectromagnetic field tensor, usually written asF^μν. In matrix form:^[13]

wherec thespeed of light; innatural unitsc = 1.

There is another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacingE/c →B andB → −E/c, to get itsHodge dualG^μν.

In the context ofspecial relativity, both of these transform according to theLorentz transformation according to

${\displaystyle F^{{\alpha }^{\prime }{\beta }^{\prime }}={\mathrm{\Lambda }}^{{\alpha }^{\prime }}{}_{\mu }{\mathrm{\Lambda }}^{{\beta }^{\prime }}{}_{\nu }{F}^{\mu \nu }}$,

where Λ^α′_ν is the Lorentz transformation tensor for a change from one reference frame to another. The same tensor is used twice in the summation.

The charge and current density, the sources of the fields, also combine into thefour-vector

${\displaystyle J^{\alpha }=\left(c\rho ,J_x,J_y,J_z\right)}$

called thefour-current.

Using these tensors,Maxwell's equations reduce to:^[13]

where the partial derivatives may be written in various ways, see4-gradient. The first equation listed above corresponds to bothGauss's law (forβ = 0) and theAmpère-Maxwell law (forβ = 1, 2, 3). The second equation corresponds to the two remaining equations,Gauss's law for magnetism (forβ = 0) andFaraday's law (forβ = 1, 2, 3).

These tensor equations aremanifestly covariant, meaning they can be seen to be covariant by the index positions. This short form of Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written usingtensors.

Bylowering the indices onF^αβ to obtainF_αβ:

${\displaystyle F_{\alpha \beta }={\eta }_{\alpha \lambda }{\eta }_{\beta \mu }{F}^{\lambda \mu }}$

the second equation can be written in terms ofF_αβ as:

${\displaystyle {\epsilon }^{\delta \alpha \beta \gamma }{\displaystyle \frac{\mathrm{\partial }{F}_{\beta \gamma }}{\mathrm{\partial }{x}^{\alpha }}}={\displaystyle \frac{\mathrm{\partial }{F}_{\alpha \beta }}{\mathrm{\partial }{x}^{\gamma }}}+{\displaystyle \frac{\mathrm{\partial }{F}_{\gamma \alpha }}{\mathrm{\partial }{x}^{\beta }}}+{\displaystyle \frac{\mathrm{\partial }{F}_{\beta \gamma }}{\mathrm{\partial }{x}^{\alpha }}}=0}$

whereε^δαβγ is the contravariantLevi-Civita symbol. Notice thecyclic permutation of indices in this equation:α →β →γ →α from each term to the next.

Another covariant electromagnetic object is theelectromagnetic stress-energy tensor, a covariant rank-2 tensor which includes thePoynting vector,Maxwell stress tensor, and electromagnetic energy density.

The EM field tensor can also be written^[14]

${\displaystyle F^{\alpha \beta }=\frac{\mathrm{\partial }{A}^{\beta }}{\mathrm{\partial }{x}_{\alpha }}-\frac{\mathrm{\partial }{A}^{\alpha }}{\mathrm{\partial }{x}_{\beta }},}$

where

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

is thefour-potential and

${\displaystyle x_{\alpha }=(ct,-x,-y,-z)}$

is thefour-position.

Using the 4-potential in the Lorenz gauge, an alternative manifestly-covariant formulation can be found in a single equation (a generalization of an equation due toBernhard Riemann byArnold Sommerfeld, known as the Riemann–Sommerfeld equation,^[15] or the covariant form of the Maxwell equations^[16]):

where${\displaystyle ◻}$ is thed'Alembertian operator, or four-Laplacian.

• Mathematical descriptions of the electromagnetic field
• Relativistic electromagnetism

• Electromagnetism
• Special relativity

• CS1 maint: archived copy as title
• Articles with short description
• Short description matches Wikidata
• Use American English from March 2019
• All Wikipedia articles written in American English