Electromagnetism

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

The combined electric field and magnetic field that determine the Lorentz force on a charged particle.

An electrical circuit connects components that carry electric current and sustain electric potential difference differences. Its dynamics may couple electrical resistance, inductance and capacitance.

An RLC circuit combines electrical resistance, inductance and capacitance. Stored energy can oscillate between magnetic and electric forms, while Joule heating dissipates it.

In a series RLC circuit, the same electric current $I=q^{\prime }$ flows through a resistor, inductor and capacitor. Summing their electric potential difference drops gives $L{q}^{\prime \prime }+R{q}^{\prime }+q/C=V$, and differentiating givesThe homogeneous damped harmonic oscillator has angular frequency ${\omega }_0=1/\sqrt{LC}$ and decay rate $\gamma =R/(2L)$.

For a series RLC circuit driven by $V(t)=V_M\sin ({\omega }_{0}t)$ with $R>0$ and ${\omega }_0^2=1/(LC)$, substitution shows that $I_0=V_M\sin ({\omega }_{0}t)/R$ solves the current equation. The inductive and capacitive terms cancel at this resonance. Every homogeneous contribution decays, so every solution approaches this periodic response. Its Q factor is $L{\omega }_0/R$.

When $R^2>4L/C$, the homogeneous series RLC circuit has two distinct negative characteristic roots, $[-R\pm \sqrt{R^2-4L/C}]/(2L)$. Its current is a linear combination of the corresponding decaying exponential functions.

When $R^2=4L/C$, the homogeneous series RLC circuit has one repeated characteristic root, $-R/(2L)$. Its current is $e^{-Rt/(2L)}(A+Bt)$, with constants determined by the initial data.

When $R^2<4L/C$, the homogeneous series RLC circuit response oscillates with angular frequency $\sqrt{1/(LC)-R^2/(4L^2)}$ and envelope $e^{-Rt/(2L)}$. Its characteristic roots form a complex-conjugate pair with negative real part when $R>0$.

A capacitor stores opposite electric charges on two conductors separated by an insulator. Its capacitance is $C=Q/V$.

Two concentric conducting spherical shells of radii $a<b$ with charges $Q,-Q$ have an electric field only in the intervening vacuum: $E_r=Q/(4\pi {ϵ}_0{r}^2)$ by Gauss's law. Integrating gives a potential difference $Q(1/a-1/b)/(4\pi {ϵ}_0)$, and hence the displayed capacitance. The electrostatic energy is $Q^2/(2C)$, either by charging work or by integrating ${ϵ}_0|E{|}^2/2$ over the gap.

A capacitor of capacitance $C$ carrying charge magnitude $Q$ stores

For a quasistatic change of capacitance at fixed voltage, the capacitor energy is $U=C{V}^2/2$ and charge is $Q=CV$. Electrical work supplied to the field is $VdQ$, so mechanical work is $dU-VdQ=-V^{2}dC/2$. Increasing the separation of parallel plates makes $dC<0$: mechanical work is positive even though field energy decreases, because electrical energy returns to the source.

The electromagnetic interaction is the long-range interaction of electrically charged matter with the electromagnetic field. Its massless gauge boson is the photon.

Electromagnetic radiation consists of propagating oscillations of the electric field and magnetic field that carry energy and momentum.

The nonrelativistic radiated power of an accelerated point charge is $P=q^2{\mu }_0{a}^2/(6\pi c)$ in SI units. Integrating its dipole radiation pattern over solid angle gives this result; relativistic motion requires the Lienard generalization.

The electromagnetic scattering solution for a sphere describes the transition between Rayleigh scattering and geometrical optics. Grain-size-dependent efficiencies can produce structure in the radiation-pressure coefficient.

Bremsstrahlung is electromagnetic radiation emitted when a charged particle accelerates during interaction with another charged particle or field. The microscopic process $e^{-}+Z\to e^{-}+Z+\gamma$ creates a photon, while its inverse absorbs one. Thus it contributes to a photon-number collision source in the photon number current, unlike ordinary Compton scattering. Thermal bremsstrahlung averages this emission over a thermal distribution of charged particles.

Thermal radiation is electromagnetic radiation emitted by matter because of its temperature. Matter in thermal equilibrium emits a spectrum governed by its absorptivity, approaching the Planck law for a blackbody.

Rayleigh scattering is elastic scattering by particles much smaller than the wavelength. For a fixed dielectric sphere, its scattering efficiency scales as the fourth power of the size-to-wavelength ratio.

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.

Permittivity describes a medium's response to an electric field. In a linear isotropic medium, the electric displacement is $\mathbf{D}=ϵ\mathbf{E}$; vacuum permittivity is its vacuum value ${ϵ}_0$.

Vacuum permittivity is the electromagnetic constant ${ϵ}_0$ appearing in Coulomb's law and the SI vacuum Maxwell equations.

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.

A line charge is a charge distribution idealized as lying on a curve, described by charge per unit length. An infinite straight line with constant density $\mu$ produces the radial electric field $E_r=\mu /(2\pi {\epsilon }_{0}r)$ by Gauss's law, and a logarithmic electrostatic potential.

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.

The four-current combines charge density and current density into the four-vector $J^{\mu }=(c\rho ,\mathbf{J})$. With $x^{\mu }=(ct,\mathbf{x})$, its four-divergence vanishes by the charge continuity equation: ${\partial }_{\mu }{J}^{\mu }={\partial }_t\rho +\nabla \cdot \mathbf{J}=0$. It is the source in the covariant Maxwell equations.

A surface current density is electric current per unit transverse length confined to a surface. Integrating a thin layer's current density through its thickness gives $\mathbf{K}$.

A current sheet idealizes electric current as supported on a surface. It produces a jump in the tangential magnetic field.

For a surface current density $\mathbf{K}$ on an interface with unit normal $\mathbf{n}$, Ampère's law givesin Gaussian units, or ${\mu }_0\mathbf{K}$ in SI units.

Surface conductivity is the conductivity of a thin layer integrated through its thickness. The sheet form of Ohm's law is $\mathbf{K}=\Sigma {\mathbf{E}}_{\parallel }$.

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.

Motional electromotive force is the contribution caused by a conductor moving through a magnetic field. The magnetic part of the Lorentz force separates charges within the conductor. For a bar of length $L$ moving at velocity $v$ perpendicular to a uniform field, its magnitude is $BL|v|$; the sign follows the chosen circuit orientation. Together with the electric-field contribution it enters Faraday's law for moving circuits.

For a smoothly moving circuit and a divergence-free magnetic field, the magnetic flux derivative is $d\Phi /dt={\int }_S{\partial }_t\mathbf{B}\cdot d\mathbf{S}-{\oint }_C(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{r}$. The minus sign follows by orienting the swept side surface compatibly with the two spanning surfaces. Combining this transport identity with Faraday's law yields the moving-loop electromotive-force law.

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: $V=IR$.

Electrical electrical conductivity relates current density to the local electric field in the rest frame of an isotropic ohmic medium: $\mathbf{J}=\sigma \mathbf{E}$. It is a bulk material coefficient, distinct from a sheet surface electrical conductivity. A moving conductor instead uses the moving-conductor Ohm law.

A material whose current density is proportional to its electric field, with proportionality given by its electrical conductivity.

In a homogeneous Ohmic conductor, the charge continuity equation and Gauss's law imply exponential decay of the bulk charge density, $\rho (t)=\rho (0)e^{-\sigma t/ϵ}$. For an isolated finite body, bulk charge moves to its boundary; the accumulating surface charge density preserves total electric charge. The formula uses a constant permittivity $ϵ$ and the corresponding free-charge convention, or ${ϵ}_0$ for total charge in the vacuum form of Gauss's law.

The Drude model describes classical charge carriers accelerated by an applied electric field and randomized by collisions of constant mean waiting time $\tau$. Its mean drift obeys $\dot{v}+v/\tau =eE/m$, and harmonic conductivity is $n{e}^2\tau /[m(1-i\omega \tau )]$ in the negative-frequency exponential convention.

The Drude model relative permittivity is $1-{\omega }_p^2/[\omega (\omega +i/\tau )]$. At frequencies large compared with the collision rate but below ${\omega }_p$, its leading real part is negative and the physical spatial wave-number branch has positive imaginary part. The field therefore decays into the conductor; close to the threshold the full complex response must be retained.

Joule heating converts electrical work into internal energy. A resistor dissipates $P=IV=I^{2}R$; in an ohmic continuum the local rate per volume is $\mathbf{j}\cdot \mathbf{E}=\eta j^2$, with resistivity $\eta$. Wind-driven currents are a proposed source of heating of an inflated giant planet.

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 image charge is a fictitious point charge placed outside the physical solution region so that its potential, combined with the real sources, enforces the required conductor boundary values. The Uniqueness of the Dirichlet problem then identifies the constructed potential with the physical one. The image need not describe the actual induced surface-charge distribution.

A real charge of height $z$ has a reflected image charge at $-z$. Its acceleration is $\ddot{z}=-\kappa /z^2$, with $\kappa =q^2/(16\pi {\epsilon }_{0}m)$. Released from rest at height $d$, its ideal-model arrival time isThe image follows the instantaneous reflected position. The induced-charge energy carries one-half the full real-image pair interaction factor, consistent with this force.

For a charge $q$ at $\mathbf{p}$, $R=|\mathbf{p}|>a$, outside a grounded sphere of radius $a$, the displayed image gives zero potential on the sphere and decay at infinity. On $|\mathbf{x}|=a$, one has $|\mathbf{x}-{\mathbf{p}}^{\prime }|=(a/R)|\mathbf{x}-\mathbf{p}|$, proving boundary cancellation. The induced force is inward, of magnitude $q^{2}aR/[4\pi {\epsilon }_0(R^2-a^2{)}^2]$, from Coulomb's law applied to the image field. The self-field is excluded.

An electric potential difference compares the electric potentials at two points. In an electrical circuit it is the voltage across the terminals of a component. A vertex potential in an electrical network is specified relative to a reference vertex; terminal differences are independent of an additive change of reference.

A point charge idealizes electric charge concentrated at one point. In vacuum, a charge $Q$ at the origin has potential $Q/(4\pi {ϵ}_{0}r)$.

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.

A solenoid is a helical winding of conducting wire. For an ideal very long winding with $N$ turns per unit length and negligible pitch, Ampère's law gives a uniform axial magnetic field $B={\mu }_{0}NI$ inside and zero field outside. Finite-length end fields and the pitch correction are excluded by this approximation. The self-inductance of a length $l$ and cross-sectional area $A$ is $L={\mu }_0{N}^{2}Al$.

The magnetic field has zero divergence. The ideal magnetohydrodynamic induction equation preserves this constraint from initial data, since the divergence of its curl is zero.

A magnetic field line is an integral curve tangent to the magnetic field. In coordinates $x^i$, it obeys $d{x}^i/B^i=d{x}^j/B^j$ wherever the relevant components are nonzero.

A flux tube is a bundle of magnetic field lines bounded by a surface everywhere tangent to the magnetic field. Gauss's law for magnetism makes the magnetic flux through any transverse cross-section constant along the tube. For a narrow tube with nearly uniform field magnitude $B$ and transverse area $A$, this gives $BA=\mathrm{constant}$.

The cross-sectional area of a narrow flux tube, measured normal to its magnetic field. Constancy of the magnetic flux gives $A|\mathbf{B}|=\mathrm{constant}$ along the tube. This area controls the mass transport when flow follows the field.

The magnetic-field-line equation states that the tangent to a field line is parallel to $\mathbf{B}$. In spherical coordinates for an axisymmetric poloidal field, it can be written $dr/(rd\theta )=B_r/B_{\theta }$.

Outside a localized magnetic dipole $\mathbf{\mu }$, the vacuum field in Gaussian conventions isFor a dipole along the polar axis, $B_r=2\mu \cos \theta /r^3$ and $B_{\theta }=\mu \sin \theta /r^3$.

An axisymmetric dipole field line satisfies $r=C{\sin }^2\theta$, where $C$ is its equatorial crossing radius. This follows by integrating $dr/(rd\theta )=B_r/B_{\theta }=2\cot \theta$.

Near the polar axis of a magnetic dipole field, $B\propto r^{-3}$. Conservation of magnetic flux in a flux tube consequently gives $A(r)\propto r^3$. A dipole magnetic-field line intersecting a sphere of radius $R_{\star }$ at small polar angle ${\theta }_{\star }$ obeysThis area is for one polar tube. The approximation requires ${\theta }_{\star }^{2}r/R_{\star }\ll 1$; the full magnetic dipole field does not remain a narrow polar tube at arbitrarily large radius.

A magnetostatic equilibrium is a time-independent configuration in which magnetic, pressure, gravitational, and any other body-force densities balance.

A plane-parallel magnetostatic equilibrium with constant downward gravitational acceleration $g$, horizontal magnetic field, constant isothermal sound speed $c_s$, and constant Alfvén speed $v_a$ hasIndeed the magnetohydrodynamic total pressure is $(c_s^2+v_a^2/2)\rho$, and its vertical derivative is $-g\rho$. The magnetic pressure therefore increases the scale height above that of an unmagnetized atmosphere.

A cylindrical magnetostatic equilibrium is an axisymmetric vector field and pressure configuration independent of the axial coordinate. Its radial Lorentz force density balances the pressure gradient, as expressed by cylindrical magnetostatic pressure balance.

In a Z-pinch, an axial electric current produces a toroidal magnetic field whose inward magnetic tension confines a plasma. A cylindrical magnetostatic equilibrium balances the resulting Lorentz force density against a pressure gradient.

For an axisymmetric vector field independent of the axial coordinate and regular on the axis, Gauss's law for magnetism gives $B_R=0$. The magnetostatic Ampère-Maxwell equation then gives $j_R=0$, $j_{\varphi }=-c{B}_z^{\prime }/(4\pi )$, and $j_z=c(R{B}_{\varphi }{)}^{\prime }/(4\pi R)$ in Gaussian units. Radial magnetostatic equilibrium isFor enclosed axial electric current $I(R)$, the magnetostatic Ampère-Maxwell equation gives $B_{\varphi }=2I/(cR)$, henceThe first term contains pressure and axial magnetic pressure; the second combines toroidal magnetic pressure with the hoop force from magnetic tension.

For a regular cylindrical flux tube with $\mathbf{B}=(0,B_{\varphi }(s),B_z(s))$, vanishing field beyond radius $s_0$, and exterior pressure $p_e$, define $⟨h⟩=2s_0^{-2}{\int }_0^{s_0}sh(s)ds$ and $B_p^2=2{\mu }_0{p}_e$. Integrating cylindrical magnetostatic pressure balance against $s^2$ gives the displayed identity. The toroidal magnetic pressure and hoop magnetic tension cancel in this cross-sectional axial-field average. Nonnegative gas pressure gives $⟨B_z^2⟩\le B_p^2$, with strict inequality precisely when $⟨p⟩>0$.

For a flux tube with $B_{\varphi }=\gamma (s/s_0)B_z$, regularity at the axis and cylindrical magnetostatic pressure balance give $B_z(0{)}^2=B_p^2-2{\mu }_{0}p(0)+{\gamma }^2⟨B_z^2⟩$. The flux-tube axial field virial identity and $p\ge 0$ give the bound. A twisted toroidal magnetic field can support a central axial field above the untwisted magnetic pressure limit even though its cross-sectional mean obeys that limit.

For constant axial magnetic field and $p=p_0(1+R^2/a^2{)}^{-k}$ with $p_0>0$ and $k>0$, cylindrical magnetostatic pressure balance gives, with $x=R^2/a^2$,The nontrivial electric current is finite at infinity exactly when $k>1$, in which case $I^2(\infty )=2\pi c^2{p}_0{a}^2/(k-1)$. Either sign of $I$ is possible because the pressure balance fixes its square.

With no axial electric current, cylindrical magnetostatic pressure balance reduces to constant $p+B_z^2/(8\pi )$. For $p=p_0{e}^{-R^2/a^2}$ and $B_z(0)=0$, its radial solutions areAs $R\downarrow 0$, $B_z\sim \sigma \sqrt{8\pi p_0}R/a$ and $j_{\varphi }\to -\sigma c\sqrt{8\pi p_0}/(4\pi a)$. These finite scalar limits do not give a smooth function of Cartesian position: $R=\sqrt{x^2+y^2}$ has a cusp at the axis, and the azimuthal unit vector has no unique limit there. Thus these data describe radial equilibrium away from the axis, but cannot satisfy smooth vector-field regularity on the axis without modifying the data.

A conductor or plasma moving with nonrelativistic velocity $\mathbf{u}$ through a magnetic field experiences the motional electric field $\mathbf{E}=-\mathbf{u}\times \mathbf{B}/c$ in Gaussian units when the electric field vanishes in its rest frame.

The London equation relates supercurrent to the electromagnetic vector potential. In a gauge with constant condensate phase it has the form $\mathbf{J}=-\mathbf{A}/({\mu }_0{\lambda }_L^2)$ and implies exponential magnetic-field screening.

The Meissner effect is the expulsion of magnetic field from the bulk of a superconductor. A static field decays from its surface over the London penetration depth.

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

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

An isolated conducting sphere of radius $R$ in a medium of permittivity $ϵ$ has surface electrostatic potential $V=Q/(4\pi ϵR)$ relative to infinity. Thus its capacitance is $C=Q/V=4\pi ϵR$.

For a conducting sphere of radius $R$ at center height $h>R$ above a grounded infinite plane,The distance $h$ is measured to the center, so the surface gap is $h-R$. Relative to the capacitance of an isolated conducting sphere, the enhancement is $c(R/h)=C(R,h)/(4\pi ϵR)$.

The method of images gives a constructive proof. At fixed sphere potential $V_s$, start with $q_0=4\pi ϵR{V}_s$ at height $z_0=h$ and place its opposite image below the plane. Cancel that image's nonconstant sphere potential by the successive chargeswith opposite mirror charges below the plane at every step. The resulting sphere potential is $V_s$ and the plane potential is zero. Since $q_n=4\pi ϵR{V}_s\sinh \alpha /\sinh ((n+1)\alpha )$, summing the sphere charges gives the displayed capacitance. In particular $c(y)=1+y/2+O(y^2)$ for small $y$.

The exact sphere-plane boundary-value problem is also treated in Behunin and colleagues' analysis of sphere-plane electrostatics.

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 $-LdI/dt$.

For a circuit whose flux linkage $\Lambda$ is proportional to its current, the self-inductance is the coefficient $L$ in $\Lambda =LI$. Faraday's law gives the induced electromotive force $-L\dot{I}$ when $L$ is constant. In series with a resistance $R$ and a constant applied electromotive force, the current obeys $L\dot{I}+RI={\mathcal{E}}_0$, with time constant $L/R$.

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.

The current associated with electric polarization and magnetization in matter. If the magnetic response is neglected, it is ${\mathbf{J}}_b={\partial }_t\mathbf{P}$. It is distinguished from a free conduction current in the Maxwell equations in matter.

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 $({\mathbf{M}}_1-{\mathbf{M}}_2)\times \mathbf{n}$.

A relativistic magnetization current has the form $J_{\mathrm{mag}}^{\mu }={\partial }_{\nu }{M}^{\nu \mu }$ for an antisymmetric polarization-magnetization tensor. It is identically conserved because commuting derivatives contract an antisymmetric tensor.

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

The electromagnetic momentum density in vacuum is $\mathbf{g}={ϵ}_0\mathbf{E}\times \mathbf{B}$. It equals the Poynting vector divided by $c^2$, expressing the relativistic relation between energy flux and momentum density.

With the momentum-flux sign convention used for the Maxwell stress tensor, Maxwell's equations implywhere $\mathbf{g}$ is the electromagnetic momentum density and ${\mathbf{f}}_{\mathrm{L}}$ is the Lorentz force density exerted by the field on matter. Thus total field-plus-matter momentum is locally conserved.

A normally incident vacuum wave of time-averaged intensity $I$ transfers momentum $I/c$ per unit area and time to a perfectly absorbing surface, so its radiation pressure isA perfect reflector reverses the normal momentum and gives $2I/c$.