Electromagnetism

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

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

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

Electric charge obeys the continuity equationFor a steady current, this reduces to $\nabla \cdot J=0$.

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

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

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

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

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

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

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

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

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

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

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

The magnetic flux through an oriented surface $S$ is ${\Phi }_B={\int }_{S}B\cdot dS$.

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

Capacitance per unit length is the charge per unit length divided by the potential difference. A vacuum coaxial cable with radii $a<b$ has

Inductance measures magnetic flux linkage per current and gives the self-induced electromotive force $-LdI/dt$.

Polarization and magnetization produce the effective sourcesDefining $D={ϵ}_{0}E+P$ and $H=B/{\mu }_0-M$ moves these sources into the macroscopic fields.

Electric polarization is electric dipole moment per unit volume. It produces bound volume and surface chargeAcross 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.

The electric displacement is $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$ and obeys $\nabla \cdot \mathbf{D}={\rho }_{\mathrm{free}}$. In a linear isotropic dielectric, $\mathbf{D}=ϵ\mathbf{E}$.

Magnetization is magnetic dipole moment per unit volume. It produces bound volume and surface currentsAcross an interface, the net bound sheet current is $({\mathbf{M}}_1-{\mathbf{M}}_2)\times \mathbf{n}$.

The magnetic field intensity is $\mathbf{H}=\mathbf{B}/{\mu }_0-\mathbf{M}$ and obeys $\nabla \times \mathbf{H}={\mathbf{J}}_{\mathrm{free}}$ in magnetostatics. In a linear isotropic medium, $\mathbf{B}=\mu \mathbf{H}$.

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.

With electromagnetic momentum density $\mathbf{g}={ϵ}_0\mathbf{E}\times \mathbf{B}$, the momentum-flux conventionsatisfies

A massive vector field coupled to a conserved current obeysThe mass term breaks electromagnetic gauge invariance.

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.

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 $m\to 0$.

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

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.

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

ChooseA particle released from rest at the origin has the proper-time trajectory

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

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

Electrostatics studies time-independent electric fields and charge distributions.

Point charges $q_1,q_2$ separated by a displacement $r$ exert the force

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.

A Gaussian pillbox givesSince $E_{\mathrm{in}}=0$, the surface charge density is $\sigma ={ϵ}_0{E}_{\mathrm{out}}\cdot n$.

Surface charge density is electric charge per unit area. At a conductor in electrostatic equilibrium it equals ${ϵ}_0{\mathbf{E}}_{\mathrm{out}}\cdot \mathbf{n}$ for the normal pointing out of the conductor.

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

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.

A neutral conducting sphere of radius $R$ in the field $E\hat{z}$ has exterior potential and induced surface chargeThe surface integral of $\sigma$ is zero.

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

For coaxial cylinders of radii $a<b$ and length $L\gg b$, carrying charges per unit length $\pm \lambda$,The field energy is $U={\lambda }^{2}L\log (b/a)/(4\pi {ϵ}_0)=QV/2$.

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

For a localized charge and current distribution satisfying charge conservation from Maxwell equations,

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

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.

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 $r^{-3}$.

For charges $-Q,NQ,-MQ$ at $z=d,0,-d$, respectively,The monopole vanishes when $N=M+1$, the dipole vanishes when $M=1$, and both vanish when $(N,M)=(2,1)$.

Subject to $N=M+1$, keeping $Qd$ fixed as $d\to 0$ gives a finite point-dipole limit when $M\ne 1$. Keeping $Q{d}^2$ fixed gives a finite pure-quadrupole limit only when $M=1$; otherwise the dipole diverges.

For charges $-Q,2Q,-Q$ at $z=d,0,-d$, the exact force on a charge $-Q$ at $(x,0,0)$ isIt points toward the origin for every nonzero $x$.

Maxwell’s equations relate electric and magnetic fields to charge and current:

In macroscopic matter with free charge density $\rho$ and free current density $J$,The electric displacement $D$ and magnetic field strength $H$ encode the material response.

A linear anisotropic medium has constitutive relations $D_i={\epsilon }_{ij}{E}_j$ and $B_i={\mu }_{ij}{H}_j$. Symmetric, time-independent tensors make the field-energy density a quadratic form.

The Ampère-Maxwell equationrelates magnetic circulation to conduction and displacement current.

Faraday's law is

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

Gauss's law states

Taking the divergence of the Ampère-Maxwell equation and using Gauss's law givesHence the charge in a fixed volume changes only through current crossing its boundary.

With metric signature $(-,+,+,+)$, $A_0=-\Phi /c$, and $j^{\mu }=(c\rho ,J)$, the field tensor satisfiesIts components are $F_{0i}=-E_i/c$ and $F_{ij}={ϵ}_{ijk}{B}_k$, so these tensor equations reproduce all four Maxwell equations.

For $D=ϵE$ and $B=\mu H$ with no free charge or current,

A plane wave in a homogeneous linear medium satisfies

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

For thin closed wire loops $C_1,C_2$, the mutual inductance is

The Coulomb gauge imposeson the magnetic vector potential. A gauge transformation $A\mapsto A+\nabla \chi$ reaches this gauge when $\chi$ solves

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

For a localized steady current density,It has dimensions $\mathrm{A}{\mathrm{m}}^2$.

The Bohr magneton,is the natural unit of an electron's orbital and spin magnetic moment.

If $\nabla \cdot J=0$ and the current decays enough for boundary terms to vanish, thensatisfies $\nabla \cdot A=0$. Its far field is

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 $m=\pi \sigma \omega R^4\hat{n}/4$.

For a steady current density $J$, the magnetic field is

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

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.

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

Relativistic electromagnetism combines electric and magnetic fields into Lorentz tensors.

For metric signature $(-,+,+,+)$, a particle of mass $m$ and charge $q$ coupled to a four-potential has the reparametrization-invariant action

Under $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\chi$, the interaction changes byFixed-endpoint equations of motion are therefore gauge invariant.

Varying the relativistic charged-particle action with fixed endpoints givesor the equivalent raised-index equation.

The four-potential combines the electric potential and magnetic vector potential as $A^{\mu }=(\varphi /c,\mathbf{A})$. The electromagnetic four-potential $A_{\mu }$ transforms underwhile the antisymmetric field tensor $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is unchanged.

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.

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.

For a source point $y(s)$ and field event $(x,t)$, the retarded time solves

For $R(s)=|x-y(s)|$ and $|v|<c$, the function $F(s)=s+R(s)/c$ hasIt is strictly increasing, so the retarded-time equation has at most one solution.

The potentials of a point charge $q$ arewith every source quantity evaluated at the retarded time.

The Lorenz gauge condition isor ${\partial }_{\mu }{A}^{\mu }=0$ in covariant notation.

The electromagnetic field tensor isIt packages the electric and magnetic fields into an antisymmetric rank-two Lorentz tensor and transforms as

For four-momentum $p^{\mu }$ and four-velocity $u^{\mu }$, the covariant Lorentz-force law isIts spatial and temporal components are

In uniform perpendicular electric and magnetic fields, a charged particle's cyclotron orbit has guiding-centre velocityindependent of its charge and mass.

For $\mathcal{E}=\gamma m{c}^2$, the temporal component of the covariant Lorentz-force equation givesThe magnetic field does no work because $\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})=0$.

Combining $d(\gamma m\mathbf{v})/dt=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ with $m{c}^{2}d\gamma /dt=q\mathbf{E}\cdot \mathbf{v}$ gives

For signature $(-,+,+,+)$ and a consistent orientation,Overall signs depend on tensor and orientation conventions, while their vanishing does not.

A null field has both electromagnetic invariants zero:For a nonzero constant null field, the mixed tensor is nilpotent of index three,

For metric signature $(-,+,+,+)$,In particular, $T^{00}=\frac{1}{2}({ϵ}_0{E}^2+B^2/{\mu }_0)$.

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

Lorentz boosts mix transverse electric and magnetic field components.

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

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

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

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.

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.

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.

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.

For refractive indices $n_{-}$ and $n_{+}$,

When the permeabilities agree, the signed reflected-to-incident electric-field ratio for polarization normal to the plane of incidence isFor incidence from lower to higher refractive index it never vanishes.

For $D_i={\epsilon }_{ij}{E}_j$, $B=\mu H$, and a plane wave with electric polarization $e$, the source-free Maxwell equations implyIf $e$ is an eigenvector of $\epsilon$, then $e\perp k$, the dispersion relation is $|k{|}^2={\omega }^2\mu {\epsilon }_e$, and the Poynting vector is parallel to $k$.

The Poynting vector $S=E\times B/{\mu }_0$ is electromagnetic energy flux.

For electromagnetic energy densityMaxwell's equations implyThe field energy lost from a volume becomes outward electromagnetic flux or work on charges.

For a vacuum plane wave, $B=\hat{k}\times E/c$, soThe electric and magnetic energy densities are equal, and the time averages satisfy $⟨S⟩=c⟨u⟩\hat{k}$.

For symmetric time-independent constitutive tensors,and the macroscopic Maxwell equations give ${\partial }_{t}u+\nabla \cdot S=-E\cdot J$.

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

An isotropic source of average power $P$ has intensityat distance $r$. For monochromatic vacuum radiation, its electric-field amplitude is $E_0=\sqrt{2⟨S⟩/({ϵ}_{0}c)}$.

For a localized source whose size is small compared with its radiation wavelength, the radiation fields of its electric dipole moment $p$ areTheir total instantaneous power is

Electromagnetic duality replaces the electric dipole $p$ by $m/c$ for a magnetic dipole moment $m$. Therefore

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