electromagnetism.bigb

= Electromagnetism
{wiki}

Electromagnetism describes electric charge, electric and magnetic fields, their forces, and electromagnetic radiation.

= Right-hand rule
{parent=Electromagnetism}
{wiki}

The right-hand rule fixes the orientation of a <cross product> and relates the circulation direction of an <electric current> to the direction of its <magnetic field>.

= Electric charge
{parent=Electromagnetism}
{wiki}

Electric charge is the conserved source of the electric field and the coupling carried by charged matter.

= Charge neutrality
{parent=Electric charge}

A system is charge neutral when its total positive and negative <electric charge> cancel.

= Conservation of electric charge
{parent=Electric charge}
{wiki=Charge_conservation}

Electric charge obeys the <continuity equation>
$$
\partial_t\rho+\nabla\cdot J=0.
$$
For a steady current, this reduces to $\nabla\cdot J=0$.

= Charge density
{parent=Electric charge}
{wiki}

Charge density is electric charge per unit volume, area, or length according to the dimension of its support.

= Electric current
{parent=Electric charge}
{wiki}

Electric current is the rate at which charge crosses a surface.

= Current density
{parent=Electric current}
{wiki}

The current density $J$ is the local charge flux, and its surface integral gives electric current.

= Line current
{parent=Electric current}

A line current idealizes an <electric current> as flowing along a curve; its current-density volume integral reduces to $I\,d\ell$ along that curve.

= Electromotive force
{parent=Electric current}
{wiki}

Electromotive force is work supplied per unit charge around a circuit.

= Electrical resistance
{parent=Electric current}
{wiki=Electrical_resistance_and_conductance}

Electrical resistance relates voltage and current by Ohm's law $V=IR$.

= Ohm's law
{title2=$V=IR$}
{parent=Electrical resistance}
{c}
{wiki}

Ohm's law states that the potential difference across an ohmic component equals its current times its resistance: $V=IR$.

= Electric field
{parent=Electromagnetism}
{wiki}

The electric field is the <force> per unit charge on a stationary test particle.

= Electric potential
{parent=Electric field}
{wiki}

In electrostatics the electric field is $E=-\nabla\Phi$, where $\Phi$ is the electric potential.

= Equipotential
{parent=Electric potential}
{wiki=Equipotential}

An equipotential is a level set of the electric potential. The electric field is normal to every smooth equipotential.

= Magnetic field
{parent=Electromagnetism}
{wiki}

The magnetic field contributes the velocity-dependent term $qv\times B$ to the <Lorentz force>.

= Magnetic flux
{parent=Magnetic field}
{wiki}

The magnetic flux through an oriented surface $S$ is $\Phi_B=\int_S B\mathbin{\cdot}dS$.

= Capacitance
{parent=Electromagnetism}
{wiki}

Capacitance is charge stored per potential difference, $C=Q/V$.

= Capacitance per unit length
{parent=Capacitance}

Capacitance per unit length is the charge per unit length divided by the potential difference. A vacuum coaxial cable with radii $a<b$ has
$$
C'=\frac{2\pi\varepsilon_0}{\log(b/a)}.
$$

= Inductance
{parent=Electromagnetism}
{wiki}

Inductance measures magnetic flux linkage per current and gives the self-induced electromotive force $-L\,dI/dt$.

= Bound charge and bound current
{parent=Electromagnetism}

Polarization and magnetization produce the effective sources
$$
\rho_{\rm b}=-\nabla\cdot P,
\qquad
J_{\rm b}=\partial_tP+\nabla\times M.
$$
Defining $D=\epsilon_0E+P$ and $H=B/\mu_0-M$ moves these sources into the macroscopic fields.

= Electric polarization
{title2=$\mathbf P$}
{parent=Bound charge and bound current}
{wiki=Polarization_density}

Electric polarization is electric dipole moment per unit volume. It produces bound volume and surface charge
$$
\rho_{\rm b}=-\nabla\cdot\mathbf P,
\qquad
\sigma_{\rm b}=\mathbf P\cdot\mathbf n.
$$
Across an interface, the net bound sheet charge is $(\mathbf P_1-\mathbf P_2)\cdot\mathbf n$, where $\mathbf n$ points from medium one to medium two.

= Electric displacement field
{title2=$\mathbf D$}
{parent=Electric polarization}
{wiki=Electric_displacement_field}

The electric displacement is $\mathbf D=\epsilon_0\mathbf E+\mathbf P$ and obeys $\nabla\cdot\mathbf D=\rho_{\rm free}$. In a linear isotropic dielectric, $\mathbf D=\epsilon\mathbf E$.

= Magnetization
{title2=$\mathbf M$}
{parent=Bound charge and bound current}
{wiki}

Magnetization is magnetic dipole moment per unit volume. It produces bound volume and surface currents
$$
\mathbf J_{\rm b}=\nabla\times\mathbf M,
\qquad
\mathbf K_{\rm b}=\mathbf M\times\mathbf n.
$$
Across an interface, the net bound sheet current is $(\mathbf M_1-\mathbf M_2)\times\mathbf n$.

= Magnetic field intensity
{title2=$\mathbf H$}
{parent=Magnetization}
{wiki=Magnetic_field#The_H-field}

The magnetic field intensity is $\mathbf H=\mathbf B/\mu_0-\mathbf M$ and obeys $\nabla\times\mathbf H=\mathbf J_{\rm free}$ in magnetostatics. In a linear isotropic medium, $\mathbf B=\mu\mathbf H$.

= Magnetic spherical-shell matching
{parent=Electromagnetism}

Across an interface with no free surface current, the normal component of $B$ and tangential component of $H$ are continuous. For an axisymmetric spherical shell, harmonic scalar potentials have dipole form $(ar+b/r^2)\cos\theta$ in each radial region.

= Maxwell stress tensor
{c}
{parent=Electromagnetism}
{wiki=Maxwell_stress_tensor}

With electromagnetic momentum density $\mathbf g=\epsilon_0\mathbf E\times\mathbf B$, the momentum-flux convention
$$
\sigma_{ij}=-\epsilon_0E_iE_j+\frac{\epsilon_0}{2}E^2\delta_{ij}
-\frac1{\mu_0}B_iB_j+\frac1{2\mu_0}B^2\delta_{ij}
$$
satisfies
$$
\partial_tg_j+\partial_i\sigma_{ij}=-(\rho\mathbf E+\mathbf J\times\mathbf B)_j.
$$

= Proca equation
{parent=Electromagnetism}
{c}
{wiki=Proca_action}

A massive vector field coupled to a conserved current obeys
$$
\partial_\mu F^{\mu\nu}-m^2A^\nu=-\mu_0J^\nu.
$$
The mass term breaks electromagnetic gauge invariance.

= Lorenz constraint in Proca theory
{parent=Proca equation}
{c}

Taking a divergence and using antisymmetry of $F^{\mu\nu}$ and current conservation gives $m^2\partial_\mu A^\mu=0$. For nonzero mass, the Lorenz condition follows from the field equation rather than from a gauge choice.

= Yukawa potential
{parent=Proca equation}
{c}
{wiki}

The static Green function of $\nabla^2-m^2$ in three dimensions is proportional to $e^{-mr}/r$. It describes an interaction screened beyond range $m^{-1}$ and tends to the Coulomb potential as $m\to0$.

= Relativistic Lorentz force
{parent=Electromagnetism}
{wiki=Lorentz_force}

$m\,du^\mu/d\tau=qF^\mu{}_\nu u^\nu$ gives $d\mathbf p/dt=q(\mathbf E+\mathbf v\times\mathbf B)$ and $d(\gamma mc^2)/dt=q\mathbf E\cdot\mathbf v$.

= Relativistic hyperbolic motion in a uniform electric field
{parent=Relativistic Lorentz force}
{wiki=Hyperbolic_motion_(relativity)}

A particle released from rest in a constant electric field parallel to its motion follows a spacetime hyperbola and has rapidity proportional to proper time.

= Null crossed electromagnetic field
{parent=Relativistic Lorentz force}
{wiki=Electromagnetic_tensor\#Classification_of_fields}

Perpendicular fields with $\mathbf E\cdot\mathbf B=0$ and $E^2-c^2B^2=0$ form a null electromagnetic field. The constant-field Lorentz-force generator is nilpotent, so its exponential terminates as a polynomial.

= Relativistic trajectory in a constant null crossed field
{parent=Null crossed electromagnetic field}

Choose
$$
\mathbf B=B\,\mathbf e_z,\qquad
\mathbf E=cB\,\mathbf e_x,
\qquad
\Omega=\frac{qB}{m}.
$$
A particle released from rest at the origin has the proper-time trajectory
$$
t(\tau)=\tau+\frac{\Omega^2\tau^3}{6},
\qquad
x(\tau)=\frac{c\Omega\tau^2}{2},
\qquad
y(\tau)=-\frac{c\Omega^2\tau^3}{6},
\qquad z(\tau)=0.
$$

= Charged particle in a uniform magnetic field
{parent=Electromagnetism}
{wiki}

A velocity cross magnetic-field force bends perpendicular motion into a circle while doing no work.

= Damped cyclotron motion
{parent=Charged particle in a uniform magnetic field}
{wiki}

Adding linear drag to cyclotron motion produces an exponentially contracting spiral.

= Electrostatics
{parent=Electromagnetism}
{wiki}

Electrostatics studies time-independent electric fields and charge distributions.

= Coulomb's law
{parent=Electrostatics}
{c}
{wiki}

Point charges $q_1,q_2$ separated by a displacement $r$ exert the force
$$
F=\frac{q_1q_2}{4\pi\epsilon_0}\frac{r}{|r|^3}.
$$

= Conductor in electrostatic equilibrium
{parent=Electrostatics}
{wiki=Electrostatics\#Electrostatic_induction}

The electric field vanishes inside a conductor at electrostatic equilibrium, so each connected conductor is equipotential and the exterior field is normal to its surface.

= Electrostatic boundary conditions at a conductor
{parent=Conductor in electrostatic equilibrium}

A Gaussian pillbox gives
$$
(E_{\rm out}-E_{\rm in})\cdot n=\frac\sigma{\epsilon_0}.
$$
Since $E_{\rm in}=0$, the surface charge density is $\sigma=\epsilon_0E_{\rm out}\cdot n$.

= Electrostatic boundary condition
{synonym}

= Surface charge density
{title2=$\sigma$}
{parent=Electrostatic boundary conditions at a conductor}
{wiki}

Surface charge density is electric charge per unit area. At a conductor in electrostatic equilibrium it equals $\epsilon_0\mathbf E_{\rm out}\mathbin{\cdot}\mathbf n$ for the normal pointing out of the conductor.

= Charge sharing between distant connected spheres
{parent=Conductor in electrostatic equilibrium}

Two widely separated conducting spheres of radii $R_1,R_2$ connected by a thin wire have equal potentials, so a total charge $Q$ divides as
$$
Q_i=\frac{R_i}{R_1+R_2}Q.
$$

= Charge on connected concentric spherical shells
{parent=Conductor in electrostatic equilibrium}

Two connected concentric spherical shells form one equipotential conductor. In the absence of charge in the inner cavity, all net charge lies on the exterior surface of the outer shell.

= Induced charge on a conducting sphere in a uniform electric field
{parent=Conductor in electrostatic equilibrium}

A neutral conducting sphere of radius $R$ in the field $E\widehat z$ has exterior potential and induced surface charge
$$
\Phi=-E\left(r-\frac{R^3}{r^2}\right)\cos\theta,
\qquad
\sigma=3\epsilon_0E\cos\theta.
$$
The surface integral of $\sigma$ is zero.

= Electrostatic energy
{parent=Electrostatics}
{wiki}

Electrostatic energy can be written as one half the charge-potential integral or epsilon-zero over two times the electric-field energy integral.

= Coaxial cylindrical capacitor
{parent=Electrostatic energy}

For coaxial cylinders of radii $a<b$ and length $L\gg b$, carrying charges per unit length $\pm\lambda$,
$$
E(r)=\frac{\lambda}{2\pi\epsilon_0r}\widehat r,
\qquad
V=\frac{\lambda}{2\pi\epsilon_0}\log\frac ba,
\qquad
C=\frac{2\pi\epsilon_0L}{\log(b/a)}.
$$
The field energy is $U=\lambda^2L\log(b/a)/(4\pi\epsilon_0)=QV/2$.

= Electric dipole moment
{parent=Electrostatics}
{wiki}

The electric dipole moment is the integral of position weighted by charge density.

= Time derivative of the electric dipole moment
{parent=Electric dipole moment}

For a localized charge and current distribution satisfying <charge conservation from Maxwell equations>,
$$
\dot{\mathbf p}=\int\mathbf J(x,t)\,d^3x.
$$

= Electric dipole
{parent=Electric dipole moment}
{wiki}

An electric dipole is a pair of equal and opposite charges in a small separation limit with their charge-times-separation vector held fixed.

= Electric multipole expansion
{parent=Electrostatics}
{wiki=Multipole_expansion}

For a localized charge distribution, the far <electric potential> is an inverse-radius expansion whose first terms are the total charge monopole, the electric dipole moment, and the electric quadrupole moment.

= Multipole expansion
{synonym}

= Electric quadrupole
{parent=Electric multipole expansion}
{wiki}

An electric quadrupole is a charge distribution whose total charge and <electric dipole moment> vanish while its second charge moment does not. Its potential decays as $r^{-3}$.

= Multipole expansion of three collinear charges
{parent=Electric multipole expansion}

For charges $-Q,NQ,-MQ$ at $z=d,0,-d$, respectively,
$$
\Phi(r,\theta)=\frac{Q}{4\pi\epsilon_0}\left[
\frac{N-M-1}{r}+\frac{(M-1)d\cos\theta}{r^2}
-\frac{(M+1)d^2P_2(\cos\theta)}{r^3}+O(r^{-4})\right].
$$
The monopole vanishes when $N=M+1$, the dipole vanishes when $M=1$, and both vanish when $(N,M)=(2,1)$.

= Dipole and quadrupole scaling limits of three collinear charges
{parent=Multipole expansion of three collinear charges}

Subject to $N=M+1$, keeping $Qd$ fixed as $d\to0$ gives a finite point-dipole limit when $M\ne1$. Keeping $Qd^2$ fixed gives a finite pure-quadrupole limit only when $M=1$; otherwise the dipole diverges.

= Transverse force from a collinear electric quadrupole
{parent=Multipole expansion of three collinear charges}

For charges $-Q,2Q,-Q$ at $z=d,0,-d$, the exact force on a charge $-Q$ at $(x,0,0)$ is
$$
F=-\frac{Q^2x}{2\pi\epsilon_0}
\left(\frac1{|x|^3}-\frac1{(x^2+d^2)^{3/2}}\right)\widehat x.
$$
It points toward the origin for every nonzero $x$.

= Maxwell equations
{parent=Electromagnetism}
{c}
{wiki}

Maxwell’s equations relate electric and magnetic fields to charge and current:
$$
\nabla\mathbin{\cdot}E=\rho/\epsilon_0,
\qquad
\nabla\mathbin{\cdot}B=0,
\qquad
\nabla\times E=-\partial_tB,
\qquad
\nabla\times B=\mu_0J+\mu_0\epsilon_0\partial_tE.
$$

= Maxwell equations in matter
{c}
{parent=Maxwell equations}
{wiki=Maxwell%27s_equations#Macroscopic_formulation}

In macroscopic matter with free charge density $\rho$ and free current density $J$,
$$
\nabla\cdot D=\rho,
\quad
\nabla\cdot B=0,
\quad
\nabla\times E=-\partial_tB,
\quad
\nabla\times H=J+\partial_tD.
$$
The electric displacement $D$ and magnetic field strength $H$ encode the material response.

= Linear anisotropic dielectric
{parent=Maxwell equations in matter}
{wiki=Permittivity#Anisotropic_medium}

A linear anisotropic medium has constitutive relations $D_i=\varepsilon_{ij}E_j$ and $B_i=\mu_{ij}H_j$. Symmetric, time-independent tensors make the field-energy density a quadratic form.

= Ampère-Maxwell equation
{parent=Maxwell equations}
{c}
{wiki=Amp%C3%A8re%27s_circuital_law}

The Ampère-Maxwell equation
$$
\nabla\times B=\mu_0J+\mu_0\epsilon_0\frac{\partial E}{\partial t}
$$
relates magnetic circulation to conduction and displacement current.

= Faraday's law
{parent=Maxwell equations}
{c}
{wiki=Faraday%27s_law_of_induction}

Faraday's law is
$$
\nabla\times E=-\frac{\partial B}{\partial t},
\qquad
\mathcal E=-\frac{d\Phi_B}{dt}.
$$

= Lenz's law
{parent=Faraday's law}
{c}
{wiki=Lenz%27s_law}

The induced current flows so that its magnetic effect opposes the change of flux that produced it.

= Gauss's law
{parent=Maxwell equations}
{c}
{wiki=Gauss%27s_law}

Gauss's law states
$$
\nabla\mathbin{\cdot}E=\rho/\epsilon_0,
\qquad
\int_{\partial V}E\mathbin{\cdot}dS=Q_{\rm enclosed}/\epsilon_0.
$$

= Charge conservation from Maxwell equations
{parent=Maxwell equations}

Taking the divergence of the Ampère-Maxwell equation and using Gauss's law gives
$$
\partial_t\rho+\nabla\cdot J=0.
$$
Hence the charge in a fixed volume changes only through current crossing its boundary.

= Covariant Maxwell equation with the minus-plus-plus-plus metric
{parent=Maxwell equations}
{c}

With metric signature $(-,+,+,+)$, $A_0=-\Phi/c$, and $j^\mu=(c\rho,J)$, the field tensor satisfies
$$
\partial_\nu F^{\mu\nu}=\mu_0j^\mu,
\qquad
\partial_{[\lambda}F_{\mu\nu]}=0.
$$
Its components are $F_{0i}=-E_i/c$ and $F_{ij}=\epsilon_{ijk}B_k$, so these tensor equations reproduce all four Maxwell equations.

= Source-free Maxwell equations in a linear medium
{parent=Maxwell equations}

For $D=\epsilon E$ and $B=\mu H$ with no free charge or current,
$$
\nabla\cdot D=0,\qquad
\nabla\cdot B=0,\qquad
\nabla\times E=-\partial_tB,\qquad
\nabla\times H=\partial_tD.
$$

= Plane electromagnetic wave in a linear medium
{parent=Source-free Maxwell equations in a linear medium}

A plane wave in a homogeneous linear medium satisfies
$$
\omega^2=\frac{|k|^2}{\epsilon\mu},
\qquad
v=\frac1{\sqrt{\epsilon\mu}},
\qquad
B_0=\frac1\omega k\times E_0.
$$

= Magnetic vector potential
{parent=Electromagnetism}
{wiki}

A magnetic vector potential satisfies B equal to its curl and is defined up to a gradient.

= Neumann's mutual-inductance formula
{parent=Magnetic vector potential}
{c}
{wiki=Inductance#Mutual_inductance}

For thin closed wire loops $C_1,C_2$, the mutual inductance is
$$
L_{12}=L_{21}
=\frac{\mu_0}{4\pi}
\oint_{C_1}\oint_{C_2}
\frac{dx_1\cdot dx_2}{|x_1-x_2|}.
$$

= Coulomb gauge
{parent=Magnetic vector potential}
{c}
{wiki}

The Coulomb gauge imposes
$$
\nabla\cdot A=0
$$
on the <magnetic vector potential>. A <gauge transformation> $A\mapsto A+\nabla\chi$ reaches this gauge when $\chi$ solves
$$
\nabla^2\chi=-\nabla\cdot A.
$$

= Gauge transformation
{parent=Magnetic vector potential}
{wiki}

The transformation $A\mapsto A+\nabla\chi$ leaves $B=\nabla\times A$ unchanged because the <curl> of a <gradient> vanishes.

= Magnetic dipole moment
{parent=Magnetic vector potential}
{wiki}

For a localized steady current density,
$$
m=\frac12\int x\times J(x)\,d^3x.
$$
It has dimensions $\mathrm{A\,m^2}$.

= Bohr magneton
{title2=$\mu_B$}
{c}
{parent=Magnetic dipole moment}
{wiki}

The Bohr magneton,
$$
\mu_B=\frac{e\hbar}{2m_e},
$$
is the natural unit of an electron's orbital and spin magnetic moment.

= Coulomb-gauge vector potential of a localized steady current
{c}
{parent=Magnetic dipole moment}

If $\nabla\cdot J=0$ and the current decays enough for boundary terms to vanish, then
$$
A(x)=\frac{\mu_0}{4\pi}\int\frac{J(x')}{|x-x'|}\,d^3x'
$$
satisfies $\nabla\cdot A=0$. Its far field is
$$
A(x)=\frac{\mu_0}{4\pi}\frac{m\times x}{|x|^3}+\cdots.
$$

= Magnetic dipole moment of a rigidly rotating charge distribution
{parent=Magnetic dipole moment}

A circular charge distribution rotating with angular speed $\omega$ can be decomposed into current loops. A hoop of line density $\eta$ and radius $R$ has $m=\pi\eta\omega R^3\hat n$; a disc of surface density $\sigma$ has $m=\pi\sigma\omega R^4\hat n/4$.

= Biot-Savart law
{parent=Electromagnetism}
{c}
{wiki}

For a steady <current density> $J$, the <magnetic field> is
$$
B(r)=\frac{\mu_0}{4\pi}\int
\frac{J(r')\times(r-r')}{|r-r'|^3}\,d^3r'.
$$

= On-axis magnetic field of a circular current loop
{parent=Biot-Savart law}

A circular loop of radius $R$ carrying <electric current> $I$ has, on its axis,
$$
B(z)=\frac{\mu_0IR^2}{2(R^2+z^2)^{3/2}}\,\hat z,
$$
where the sign of $\hat z$ is fixed by the <right-hand rule>. At its centre, $B(0)=\mu_0I/(2R)$.

= Magnetic-field cancellation between oppositely driven coaxial loops
{parent=Biot-Savart law}

If coaxial loops of radii $R$ and $2R$, separated by $D$, carry currents $I$ and $2I$ in opposite senses, their on-axis fields cancel between the loops at distance $D/3$ from the smaller loop.

= Beltrami field
{c}
{parent=Electromagnetism}
{wiki}

A Beltrami field is a vector field parallel to its curl:
$$
\nabla\times J=\lambda J.
$$
If it is divergence-free and used as a magnetostatic current with constant nonzero $\lambda$, then $B=\mu_0J/\lambda$ solves $\nabla\times B=\mu_0J$ and $\nabla\cdot B=0$.

= Relativistic electromagnetism
{parent=Electromagnetism}
{wiki}

Relativistic electromagnetism combines electric and magnetic fields into Lorentz tensors.

= Relativistic charged-particle action
{parent=Relativistic electromagnetism}

For metric signature $(-,+,+,+)$, a particle of mass $m$ and charge $q$ coupled to a four-potential has the reparametrization-invariant action
$$
S=-mc\int\sqrt{-\dot x^2}\,d\lambda
+q\int A_\mu(x)\dot x^\mu\,d\lambda.
$$

= Gauge invariance of the charged-particle worldline action
{parent=Relativistic charged-particle action}

Under $A_\mu\mapsto A_\mu+\partial_\mu\chi$, the interaction changes by
$$
q\int\frac{d\chi(x(\lambda))}{d\lambda}\,d\lambda
=q\,[\chi]_{\rm endpoints}.
$$
Fixed-endpoint equations of motion are therefore gauge invariant.

= Lorentz-force equation from the worldline action
{parent=Relativistic charged-particle action}
{c}

Varying the <relativistic charged-particle action> with fixed endpoints gives
$$
m\frac{du_\mu}{d\tau}=qF_{\mu\nu}u^\nu,
$$
or the equivalent raised-index equation.

= Electromagnetic four-potential
{parent=Relativistic electromagnetism}

The four-potential combines the <electric potential> and <magnetic vector potential> as $A^\mu=(\phi/c,\mathbf A)$. The electromagnetic four-potential $A_\mu$ transforms under
$$
A_\mu\mapsto A_\mu+\partial_\mu\chi,
$$
while the antisymmetric field tensor $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ is unchanged.

= Electromagnetic multipole expansion
{parent=Electromagnetic four-potential}
{wiki=Multipole_expansion#Applications}

An electromagnetic multipole expansion expresses the potentials or fields of a localized source as successively smaller monopole, dipole, quadrupole, and higher moments in the ratio of source size to observation distance.

= Retarded electromagnetic potential
{parent=Electromagnetic four-potential}
{wiki=Retarded_potential}

A retarded potential evaluates each source at the event on the source's past light cone. This accounts for propagation at the speed of light and enforces <causality>.

= Retarded time
{title2=$t_{\rm ret}$}
{parent=Retarded electromagnetic potential}
{wiki}

For a source point $y(s)$ and field event $(x,t)$, the retarded time solves
$$
t_{\rm ret}+\frac{|x-y(t_{\rm ret})|}{c}=t.
$$

= Uniqueness of retarded time for a subluminal source
{parent=Retarded time}

For $R(s)=|x-y(s)|$ and $|v|<c$, the function $F(s)=s+R(s)/c$ has
$$
F'(s)=1-\widehat R\cdot v/c>0.
$$
It is strictly increasing, so the retarded-time equation has at most one solution.

= Lienard-Wiechert potentials
{c}
{parent=Retarded electromagnetic potential}
{wiki=Li%C3%A9nard%E2%80%93Wiechert_potential}

The potentials of a point charge $q$ are
$$
\phi=\frac{q}{4\pi\epsilon_0}
\frac1{R-\mathbf v\cdot\mathbf R/c},
\qquad
\mathbf A=\frac{\mathbf v}{c^2}\phi,
$$
with every source quantity evaluated at the <retarded time>.

= Lorenz gauge
{c}
{parent=Electromagnetic four-potential}
{wiki=Lorenz_gauge_condition}

The Lorenz gauge condition is
$$
\frac1{c^2}\partial_t\phi+\nabla\cdot\mathbf A=0,
$$
or $\partial_\mu A^\mu=0$ in covariant notation.

= Electromagnetic field tensor
{title2=$F^{\mu\nu}$}
{parent=Electromagnetic four-potential}
{wiki}

The electromagnetic field tensor is
$$
F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu.
$$
It packages the electric and magnetic fields into an antisymmetric rank-two Lorentz tensor and transforms as
$$
F'^{\mu\nu}(x')
=\Lambda^\mu{}_\rho\Lambda^\nu{}_\sigma F^{\rho\sigma}(x).
$$

= Lorentz force
{parent=Electromagnetic field tensor}
{c}
{wiki}

For <four-momentum> $p^\mu$ and <four-velocity> $u^\mu$, the covariant Lorentz-force law is
$$
\frac{dp^\mu}{d\tau}=qF^{\mu\nu}u_\nu.
$$
Its spatial and temporal components are
$$
\frac{d\mathbf p}{dt}=q(\mathbf E+\mathbf v\times\mathbf B),
\qquad
\frac{d\mathcal E}{dt}=q\mathbf E\cdot\mathbf v.
$$

= E-cross-B drift
{title2=$\mathbf E\times\mathbf B/B^2$}
{c}
{parent=Lorentz force}
{wiki=Guiding_center#Electric_field}

In uniform perpendicular electric and magnetic fields, a charged particle's cyclotron orbit has guiding-centre velocity
$$
\mathbf v_D=\frac{\mathbf E\times\mathbf B}{B^2},
$$
independent of its charge and mass.

= Relativistic electromagnetic work-energy theorem
{parent=Lorentz force}

For $\mathcal E=\gamma mc^2$, the temporal component of the covariant Lorentz-force equation gives
$$
\frac{d\mathcal E}{dt}=q\mathbf E\cdot\mathbf v.
$$
The magnetic field does no work because $\mathbf v\cdot(\mathbf v\times\mathbf B)=0$.

= Coordinate acceleration of a relativistic charged particle
{parent=Lorentz force}

Combining $d(\gamma m\mathbf v)/dt=q(\mathbf E+\mathbf v\times\mathbf B)$ with $mc^2\,d\gamma/dt=q\mathbf E\cdot\mathbf v$ gives
$$
\frac{d\mathbf v}{dt}
=\frac q{m\gamma}
\left[
\mathbf E+\mathbf v\times\mathbf B
-\frac{\mathbf v(\mathbf v\cdot\mathbf E)}{c^2}
\right].
$$

= Electromagnetic field invariants
{title2=$F_{\mu\nu}F^{\mu\nu},\,F_{\mu\nu}\widetilde F^{\mu\nu}$}
{parent=Electromagnetic field tensor}
{wiki}

For signature $(-,+,+,+)$ and a consistent orientation,
$$
F_{\mu\nu}F^{\mu\nu}
=2\left(B^2-\frac{E^2}{c^2}\right),
\qquad
F_{\mu\nu}\widetilde F^{\mu\nu}
=-\frac4c\mathbf E\cdot\mathbf B.
$$
Overall signs depend on tensor and orientation conventions, while their vanishing does not.

= Null electromagnetic field
{parent=Electromagnetic field invariants}
{wiki}

A null field has both electromagnetic invariants zero:
$$
\mathbf E\cdot\mathbf B=0,
\qquad
E^2=c^2B^2.
$$
For a nonzero constant null field, the mixed tensor is nilpotent of index three,
$$
F^\mu{}_\rho F^\rho{}_\sigma F^\sigma{}_\nu=0.
$$

= Electromagnetic stress-energy tensor
{parent=Electromagnetic field tensor}
{wiki=Electromagnetic_stress%E2%80%93energy_tensor}

For metric signature $(-,+,+,+)$,
$$
T^{\mu\nu}=\frac1{\mu_0}
\left(F^{\mu\rho}F^\nu{}_{\rho}
-\frac14\eta^{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}\right).
$$
In particular, $T^{00}=\tfrac12(\epsilon_0E^2+B^2/\mu_0)$.

= Null energy condition for the electromagnetic field
{parent=Electromagnetic stress-energy tensor}

For every null vector $k$, put $q^\rho=k_\mu F^{\mu\rho}$. Antisymmetry gives $q\cdot k=0$, so $q$ has nonnegative norm and
$$
T^{\mu\nu}k_\mu k_\nu=\frac1{\mu_0}q^\rho q_\rho\geq0.
$$

= Lorentz transformation of electromagnetic fields
{parent=Relativistic electromagnetism}
{c}
{wiki}

Lorentz boosts mix transverse electric and magnetic field components.

= Electromagnetic wave
{parent=Electromagnetism}
{wiki}

An electromagnetic wave is a propagating coupled oscillation of electric and magnetic fields governed by Maxwell's equations.

= Electromagnetic wave equation
{parent=Electromagnetic wave}
{wiki}

In vacuum, each field obeys $\nabla^2E-c^{-2}E_{tt}=0$ and $\nabla^2B-c^{-2}B_{tt}=0$.

= Plane electromagnetic wave
{parent=Electromagnetic wave}
{wiki}

For a vacuum plane wave, $k\cdot E_0=0$, $\omega=c|k|$, and $B_0=k\times E_0/\omega$.

= Perfect conductor
{parent=Plane electromagnetic wave}
{wiki}

At the surface of a perfect conductor, the tangential <electric field> and normal <magnetic field> vanish. An incident <plane electromagnetic wave> is therefore accompanied by a reflected wave whose tangential electric field cancels the incident field at the surface.

= Normal reflection of an electromagnetic wave from a perfect conductor
{parent=Perfect conductor}

At normal incidence, the reflected electric amplitude is the negative of the incident amplitude at the surface, while the reflected magnetic amplitude has the same sign. Their superposition forms a <standing wave> with an electric node and a magnetic antinode at the conductor.

= Dielectric-interface boundary conditions
{parent=Plane electromagnetic wave}
{wiki=Interface_conditions_for_electromagnetic_fields}

Without free surface charge or current, the tangential components of $E$ and $H$ and the normal components of $D$ and $B$ are continuous across an interface.

= Phase matching at a planar wave interface
{parent=Dielectric-interface boundary conditions}

Equality of the boundary fields for every tangential position and time forces equal frequency and equal tangential wavevector for incident, reflected, and transmitted plane waves.

= Snell law for electromagnetic waves
{parent=Phase matching at a planar wave interface}
{c}
{wiki=Snell%27s_law}

For refractive indices $n_-$ and $n_+$,
$$
\theta_R=\theta_I,
\qquad
n_-\sin\theta_I=n_+\sin\theta_T.
$$

= Transverse-electric Fresnel reflection coefficient
{parent=Dielectric-interface boundary conditions}
{wiki=Fresnel_equations}

When the permeabilities agree, the signed reflected-to-incident electric-field ratio for polarization normal to the plane of incidence is
$$
r_{\rm TE}
=\frac{n_-\cos\theta_I-n_+\cos\theta_T}
{n_-\cos\theta_I+n_+\cos\theta_T}.
$$
For incidence from lower to higher refractive index it never vanishes.

= Plane wave in an anisotropic dielectric
{parent=Plane electromagnetic wave}

For $D_i=\varepsilon_{ij}E_j$, $B=\mu H$, and a plane wave with electric polarization $e$, the source-free Maxwell equations imply
$$
k\times(k\times e)+\omega^2\mu\varepsilon e=0.
$$
If $e$ is an eigenvector of $\varepsilon$, then $e\perp k$, the dispersion relation is $|k|^2=\omega^2\mu\varepsilon_e$, and the <Poynting vector> is parallel to $k$.

= Poynting vector
{parent=Electromagnetic wave}
{c}
{wiki}

The Poynting vector $S=E\times B/\mu_0$ is electromagnetic energy flux.

= Poynting theorem
{parent=Poynting vector}
{c}
{wiki=Poynting%27s_theorem}

For electromagnetic energy density
$$
u=\frac12\left(\epsilon_0E^2+\frac{B^2}{\mu_0}\right),
$$
Maxwell's equations imply
$$
\partial_tu+\nabla\cdot S=-J\cdot E.
$$
The field energy lost from a volume becomes outward electromagnetic flux or work on charges.

= Poynting's theorem
{c}
{synonym}

= Energy density and flux of a plane electromagnetic wave
{parent=Poynting theorem}

For a vacuum plane wave, $B=\widehat k\times E/c$, so
$$
S=\epsilon_0cE^2\widehat k.
$$
The electric and magnetic energy densities are equal, and the time averages satisfy $\langle S\rangle=c\langle u\rangle\widehat k$.

= Poynting theorem in a linear anisotropic medium
{c}
{parent=Poynting theorem}

For symmetric time-independent constitutive tensors,
$$
u=\frac12(\varepsilon_{ij}E_iE_j+\mu_{ij}H_iH_j),
\qquad
S=E\times H,
$$
and the macroscopic Maxwell equations give $\partial_tu+\nabla\cdot S=-E\cdot J$.

= Radiation zone
{parent=Electromagnetic wave}
{wiki=Near_and_far_field}

In the radiation zone of a localized source, the leading electromagnetic fields decay as $1/R$, are transverse to the observation direction, and satisfy $B=\widehat R\times E/c$. Terms decaying faster than $1/R$ do not contribute to the limiting radiated power.

= Intensity of an isotropic radiator
{parent=Radiation zone}

An isotropic source of average power $P$ has intensity
$$
\langle S\rangle=\frac{P}{4\pi r^2}
$$
at distance $r$. For monochromatic vacuum radiation, its electric-field amplitude is $E_0=\sqrt{2\langle S\rangle/(\epsilon_0c)}$.

= Electric dipole radiation
{parent=Radiation zone}
{wiki=Multipole_radiation\#Electric_dipole_radiation}

For a localized source whose size is small compared with its radiation wavelength, the radiation fields of its electric dipole moment $p$ are
$$
E_{\rm rad}=\frac{\mu_0}{4\pi R}\widehat R\times(\widehat R\times\ddot p(t-R/c)),
\qquad
B_{\rm rad}=\frac1c\widehat R\times E_{\rm rad}.
$$
Their total instantaneous power is
$$
\mathcal P=\frac{\mu_0}{6\pi c}|\ddot p(t-R/c)|^2.
$$

= Magnetic dipole radiation
{parent=Radiation zone}
{wiki=Multipole_radiation\#Magnetic_dipole_radiation}

Electromagnetic duality replaces the electric dipole $p$ by $m/c$ for a magnetic dipole moment $m$. Therefore
$$
\mathcal P=\frac{\mu_0}{6\pi c^3}|\ddot m|^2.
$$

= Magnetic-dipole spin-down
{parent=Magnetic dipole radiation}

If a rigid body's inclined magnetic dipole rotates at angular speed $\Omega$, its radiated power is $C\Omega^4$. Conservation of rotational energy $I\Omega^2/2$ then gives
$$
\dot\Omega=-\frac CI\Omega^3,
\qquad
\Omega(t)^{-2}=\Omega_0^{-2}+\frac{2C}{I}t.
$$