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Electric field

From Wikipedia, the free encyclopedia
(Redirected fromE-field)
Physical field surrounding an electric charge
"Electric fields" redirects here. For the Australian band, seeElectric Fields.
Electric field
Common symbols
E
SI unitvolt permeter (V/m)
InSI base unitskg⋅m⋅s−3⋅A−1
DimensionMLT−3I−1
Articles about
Electromagnetism
Solenoid

Anelectric field (sometimes calledE-field[1]) is aphysical field that surrounds electricallycharged particles such aselectrons. In classical electromagnetism, the electric field of a single charge (or group of charges) describes their capacity to exert attractive or repulsive forces on another charged object. Charged particles exert attractive forces on each other when the sign of their charges are opposite, one being positive while the other is negative, and repel each other when the signs of the charges are the same. Because these forces are exerted mutually, two charges must be present for the forces to take place. These forces are described byCoulomb's law, which says that the greater the magnitude of the charges, the greater the force, and the greater the distance between them, the weaker the force. Informally, the greater the charge of an object, the stronger its electric field. Similarly, an electric field is stronger nearer charged objects and weaker further away. Electric fields originate from electric charges and time-varyingelectric currents. Electric fields andmagnetic fields are both manifestations of theelectromagnetic field. Electromagnetism is one of the fourfundamental interactions of nature.

Electric fields are important in many areas ofphysics, and are exploited in electrical technology. For example, inatomic physics andchemistry, the interaction in the electric field between theatomic nucleus and electrons is the force that holds these particles together in atoms. Similarly, the interaction in the electric field between atoms is the force responsible forchemical bonding that result inmolecules.

The electric field is defined as avector field that associates to each point in space the force per unit ofcharge exerted on an infinitesimaltest charge at rest at that point.[2][3][4] TheSI unit for the electric field is thevolt permeter (V/m), which is equal to thenewton percoulomb (N/C).[5]

Description

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Electric field of a positive pointelectric charge suspended over an infinite sheet of conducting material. The field is depicted byelectric field lines, lines which follow the direction of the electric field in space. The induced charge distribution in the sheet is not shown.

The electric field is defined at each point in space as the force that would be experienced by aninfinitesimally small stationarytest charge at that point divided by the charge.[6]: 469–70  The electric field is defined in terms offorce, and force is avector (i.e. having bothmagnitude anddirection), so it follows that an electric field may be described by avector field.[6]: 469–70  The electric field acts between two charges similarly to the way that thegravitational field acts between twomasses, as they both obey aninverse-square law with distance.[7] This is the basis forCoulomb's law, which states that, for stationary charges, the electric field varies with the source charge and varies inversely with the square of the distance from the source. This means that if the source charge were doubled, the electric field would double, and if you move twice as far away from the source, the field at that point would be only one-quarter its original strength.

The electric field can be visualized with a set oflines whose direction at each point is the same as those of the field, a concept introduced byMichael Faraday,[8] whose term 'lines of force' is still sometimes used. This illustration has the useful property that, when drawn so that each line represents the same amount offlux, thestrength of the field is proportional to the density of the lines.[9] Field lines due to stationary charges have several important properties, including that they always originate from positive charges and terminate at negative charges, they enter all good conductors at right angles, and they never cross or close in on themselves.[6]: 479  The field lines are a representative concept; the field actually permeates all the intervening space between the lines. More or fewer lines may be drawn depending on the precision to which it is desired to represent the field.[8] The study of electric fields created by stationary charges is calledelectrostatics.

Faraday's law describes the relationship between a time-varying magnetic field and the electric field. One way of stating Faraday's law is that thecurl of the electric field is equal to the negativetime derivative of the magnetic field.[10]: 327  In the absence of time-varying magnetic field, the electric field is therefore calledconservative (i.e. curl-free).[10]: 24, 90–91  This implies there are two kinds of electric fields: electrostatic fields and fields arising from time-varying magnetic fields.[10]: 305–307  While the curl-free nature of the static electric field allows for a simpler treatment using electrostatics, time-varying magnetic fields are generally treated as a component of a unifiedelectromagnetic field. The study of magnetic and electric fields that change over time is calledelectrodynamics.

Mathematical formulation

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Main article:Mathematical descriptions of the electromagnetic field

Electric fields are caused byelectric charges, described byGauss's law,[11] and time varyingmagnetic fields, described byFaraday's law of induction.[12] Together, these laws are enough to define the behavior of the electric field. However, since the magnetic field is described as a function of electric field, the equations of both fields are coupled and together formMaxwell's equations that describe both fields as a function of charges andcurrents.

Evidence of an electric field:styrofoam peanuts clinging to a cat's fur due tostatic electricity. Thetriboelectric effect causes anelectrostatic charge to build up on the fur due to the cat's motions. The electric field of the charge causes polarization of the molecules of the styrofoam due toelectrostatic induction, resulting in a slight attraction of the light plastic pieces to the charged fur. This effect is also the cause ofstatic cling in clothes.

Electrostatics

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Main article:Coulomb's law

In the special case of asteady state (stationary charges and currents), the Maxwell-Faraday inductive effect disappears. The resulting two equations (Gauss's law∇⋅E=ρε0{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}} and Faraday's law with no induction term∇×E=0{\displaystyle \nabla \times \mathbf {E} =0}), taken together, are equivalent toCoulomb's law, which states that a particle with electric chargeq1{\displaystyle q_{1}} at positionr1{\displaystyle \mathbf {r} _{1}} exerts a force on a particle with chargeq0{\displaystyle q_{0}} at positionr0{\displaystyle \mathbf {r} _{0}} of:[13]F01=q1q04πε0r^01|r01|2=q1q04πε0r01|r01|3{\displaystyle \mathbf {F} _{01}={\frac {q_{1}q_{0}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{01} \over {|\mathbf {r} _{01}|}^{2}}={\frac {q_{1}q_{0}}{4\pi \varepsilon _{0}}}{\mathbf {r} _{01} \over {|\mathbf {r} _{01}|}^{3}}}where

Note thatε0{\displaystyle \varepsilon _{0}} must be replaced withε{\displaystyle \varepsilon },permittivity, when charges are in non-empty media.When the chargesq0{\displaystyle q_{0}} andq1{\displaystyle q_{1}} have the same sign this force is positive, directed away from the other charge, indicating the particles repel each other. When the charges have unlike signs the force is negative, indicating the particles attract.To make it easy to calculate theCoulomb force on any charge at positionr0{\displaystyle \mathbf {r} _{0}} this expression can be divided byq0{\displaystyle q_{0}} leaving an expression that only depends on the other charge (thesource charge)[14][4]E1(r0)=F01q0=q14πε0r^01|r01|2=q14πε0r01|r01|3{\displaystyle \mathbf {E} _{1}(\mathbf {r} _{0})={\frac {\mathbf {F} _{01}}{q_{0}}}={\frac {q_{1}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{01} \over {|\mathbf {r} _{01}|}^{2}}={\frac {q_{1}}{4\pi \varepsilon _{0}}}{\mathbf {r} _{01} \over {|\mathbf {r} _{01}|}^{3}}}where:

This is theelectric field at pointr0{\displaystyle \mathbf {r} _{0}} due to the point chargeq1{\displaystyle q_{1}}; it is avector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the positionr0{\displaystyle \mathbf {r} _{0}}.Since this formula gives the electric field magnitude and direction at any pointr0{\displaystyle \mathbf {r} _{0}} in space (except at the location of the charge itself,r1{\displaystyle \mathbf {r} _{1}}, where it becomes infinite) it defines avector field.From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with theinverse square of the distance from the charge.

The Coulomb force on a charge of magnitudeq{\displaystyle q} at any point in space is equal to the product of the charge and the electric field at that pointF=qE.{\displaystyle \mathbf {F} =q\mathbf {E} .}TheSI unit of the electric field is thenewton percoulomb (N/C), orvolt permeter (V/m); in terms of theSI base units it is kg⋅m⋅s−3⋅A−1.

Superposition principle

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Due to thelinearity ofMaxwell's equations, electric fields satisfy thesuperposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges.[4] This principle is useful in calculating the field created by multiple point charges. If chargesq1,q2,…,qn{\displaystyle q_{1},q_{2},\dots ,q_{n}} are stationary in space at pointsr1,r2,…,rn{\displaystyle \mathbf {r} _{1},\mathbf {r} _{2},\dots ,\mathbf {r} _{n}}, in the absence of currents, the superposition principle says that the resulting field is the sum of fields generated by each particle as described by Coulomb's law:E(r)=E1(r)+E2(r)+⋯+En(r)=14πε0∑i=1nqir^i|ri|2=14πε0∑i=1nqiri|ri|3{\displaystyle {\begin{aligned}\mathbf {E} (\mathbf {r} )=\mathbf {E} _{1}(\mathbf {r} )+\mathbf {E} _{2}(\mathbf {r} )+\dots +\mathbf {E} _{n}(\mathbf {r} )={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {|\mathbf {r} _{i}|}^{3}}\end{aligned}}}where

Continuous charge distributions

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The superposition principle allows for the calculation of the electric field due to a distribution ofcharge densityρ(r){\displaystyle \rho (\mathbf {r} )}. By considering the chargeρ(r′)dv{\displaystyle \rho (\mathbf {r} ')dv} in each small volume of spacedv{\displaystyle dv} at pointr′{\displaystyle \mathbf {r} '} as a point charge, the resulting electric field,dE(r){\displaystyle d\mathbf {E} (\mathbf {r} )}, at pointr{\displaystyle \mathbf {r} } can be calculated asdE(r)=ρ(r′)4πε0r^′|r′|2dv=ρ(r′)4πε0r′|r′|3dv{\displaystyle d\mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} ')}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}' \over {|\mathbf {r} '|}^{2}}dv={\frac {\rho (\mathbf {r} ')}{4\pi \varepsilon _{0}}}{\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}dv}where

The total field is found by summing the contributions from all the increments of volume byintegrating the charge density over the volumeV{\displaystyle V}:E(r)=14πε0∭Vρ(r′)r′|r′|3dv{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\iiint _{V}\,\rho (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}dv}

Similar equations follow for a surface charge withsurface charge densityσ(r′){\displaystyle \sigma (\mathbf {r} ')} on surfaceS{\displaystyle S}E(r)=14πε0∬Sσ(r′)r′|r′|3da,{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\iint _{S}\,\sigma (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}da,}and for line charges withlinear charge densityλ(r′){\displaystyle \lambda (\mathbf {r} ')} on lineL{\displaystyle L}E(r)=14πε0∫Lλ(r′)r′|r′|3dℓ.{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{L}\,\lambda (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}d\ell .}

Electric potential

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Main article:Electric potential
See also:Helmholtz decomposition andConservative vector field § Irrotational vector fields

If a system is static, such that magnetic fields are not time-varying, then by Faraday's law, the electric field iscurl-free. In this case, one can define anelectric potential, that is, a functionφ{\displaystyle \varphi } such thatE=−∇φ{\displaystyle \mathbf {E} =-\nabla \varphi }.[15] This is analogous to thegravitational potential. The difference between the electric potential at two points in space is called thepotential difference (or voltage) between the two points.

In general, however, the electric field cannot be described independently of the magnetic field. Given themagnetic vector potential,A, defined so that⁠B=∇×A{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }⁠, one can still define an electric potentialφ{\displaystyle \varphi } such that:E=−∇φ−∂A∂t,{\displaystyle \mathbf {E} =-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}},}where∇φ{\displaystyle \nabla \varphi } is thegradient of the electric potential and∂A∂t{\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}} is thepartial derivative ofA with respect to time.

Faraday's law of induction can be recovered by taking thecurl of that equation[16]∇×E=−∂(∇×A)∂t=−∂B∂t,{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial (\nabla \times \mathbf {A} )}{\partial t}}=-{\frac {\partial \mathbf {B} }{\partial t}},}which justifies, a posteriori, the previous form forE.

Continuous vs. discrete charge representation

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Main article:Charge density

The equations of electromagnetism are best described in a continuous description. However, charges are sometimes best described as discrete points; for example, some models may describeelectrons as point sources where charge density is infinite on an infinitesimal section of space.

A chargeq{\displaystyle q} located atr0{\displaystyle \mathbf {r} _{0}} can be described mathematically as a charge density⁠ρ(r)=qδ(r−r0){\displaystyle \rho (\mathbf {r} )=q\delta (\mathbf {r} -\mathbf {r} _{0})}⁠, where theDirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

Electrostatic fields

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Main article:Electrostatics
Illustration of the electric field surrounding a positive (red) and a negative (blue) charge

Electrostatic fields are electric fields that do not change with time. Such fields are present when systems of charged matter are stationary, or when electric currents are unchanging. In that case,Coulomb's law fully describes the field.[17]

Parallels between electrostatic and gravitational fields

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See also:Gravitoelectromagnetism

Coulomb's law, which describes the interaction of electric charges:F=q(Q4πε0r^|r|2)=qE{\displaystyle \mathbf {F} =q\left({\frac {Q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {\hat {r}} }{|\mathbf {r} |^{2}}}\right)=q\mathbf {E} }is similar toNewton's law of universal gravitation:F=m(−GMr^|r|2)=mg{\displaystyle \mathbf {F} =m\left(-GM{\frac {\mathbf {\hat {r}} }{|\mathbf {r} |^{2}}}\right)=m\mathbf {g} }(wherer^=r|r|{\textstyle \mathbf {\hat {r}} =\mathbf {\frac {r}{|r|}} }).

This suggests similarities between the electric fieldE and the gravitational fieldg, or their associated potentials. Mass is sometimes called "gravitational charge".[18]

Electrostatic and gravitational forces both arecentral,conservative and obey aninverse-square law.

Uniform fields

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Illustration of the electric field between two parallelconductive plates of finite size (known as aparallel plate capacitor). In the middle of the plates, far from any edges, the electric field is very nearly uniform.

A uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conductingplates parallel to each other and maintaining avoltage (potential difference) between them; it is only an approximation because of boundary effects (near the edge of the planes, the electric field is distorted because the plane does not continue). Assuming infinite planes, the magnitude of the electric fieldE is:E=−ΔVd,{\displaystyle E=-{\frac {\Delta V}{d}},}whereΔV is thepotential difference between the plates andd is the distance separating the plates. The negative sign arises as positive charges repel, so a positive charge will experience a force away from the positively charged plate, in the opposite direction to that in which the voltage increases. In micro- and nano-applications, for instance in relation to semiconductors, a typical magnitude of an electric field is in the order of106 V⋅m−1, achieved by applying a voltage of the order of 1 volt between conductors spaced 1 μm apart.

Electromagnetic fields

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The electric field(lines with arrows) of a charge(+) induces surface charges(red andblue areas) on metal objects due toelectrostatic induction.
Main article:Electromagnetic field

Electromagnetic fields are electric and magnetic fields, which may change with time, for instance when charges are in motion. Moving charges produce a magnetic field in accordance withAmpère's circuital law (with Maxwell's addition), which, along with Maxwell's other equations, defines the magnetic field,B{\displaystyle \mathbf {B} }, in terms of its curl:∇×B=μ0(J+ε0∂E∂t),{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right),}whereJ{\displaystyle \mathbf {J} } is thecurrent density,μ0{\displaystyle \mu _{0}} is thevacuum permeability, andε0{\displaystyle \varepsilon _{0}} is thevacuum permittivity.

Both theelectric current density and thepartial derivative of the electric field with respect to time, contribute to the curl of the magnetic field. In addition, theMaxwell–Faraday equation states∇×E=−∂B∂t.{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}.}These represent two ofMaxwell's four equations and they intricately link the electric and magnetic fields together, resulting in theelectromagnetic field. The equations represent a set of four coupled multi-dimensional partial differential equations which, when solved for a system, describe the combined behavior of the electromagnetic fields. In general, the force experienced by a test charge in an electromagnetic field is given by theLorentz force law:F=qE+qv×B.{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} .}

Energy in the electric field

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The total energy per unit volume stored by theelectromagnetic field is[19]uEM=ε2|E|2+12μ|B|2{\displaystyle u_{\text{EM}}={\frac {\varepsilon }{2}}|\mathbf {E} |^{2}+{\frac {1}{2\mu }}|\mathbf {B} |^{2}}whereε is thepermittivity of the medium in which the field exists,μ{\displaystyle \mu } itsmagnetic permeability, andE andB are the electric and magnetic field vectors.

AsE andB fields are coupled, it would be misleading to split this expression into "electric" and "magnetic" contributions. In particular, an electrostatic field in any given frame of reference in general transforms into a field with a magnetic component in a relatively moving frame. Accordingly, decomposing the electromagnetic field into an electric and magnetic component is frame-specific, and similarly for the associated energy.

The total energyUEM stored in the electromagnetic field in a given volumeV isUEM=12∫V(ε|E|2+1μ|B|2)dV.{\displaystyle U_{\text{EM}}={\frac {1}{2}}\int _{V}\left(\varepsilon |\mathbf {E} |^{2}+{\frac {1}{\mu }}|\mathbf {B} |^{2}\right)dV\,.}

Electric displacement field

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Main article:electric displacement field

Definitive equation of vector fields

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See also:List of electromagnetism equations

In the presence of matter, it is helpful to extend the notion of the electric field into three vector fields:[20]D=ε0E+P{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }whereP is theelectric polarization – the volume density ofelectric dipole moments, andD is theelectric displacement field. SinceE andP are defined separately, this equation can be used to defineD. The physical interpretation ofD is not as clear asE (effectively the field applied to the material) orP (induced field due to the dipoles in the material), but still serves as a convenient mathematical simplification, since Maxwell's equations can be simplified in terms offree charges and currents.

Constitutive relation

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Main article:Constitutive equation

TheE andD fields are related by thepermittivity of the material,ε.[21][20]

For linear,homogeneous,isotropic materialsE andD are proportional and constant throughout the region, there is no position dependence:D(r)=εE(r).{\displaystyle \mathbf {D} (\mathbf {r} )=\varepsilon \mathbf {E} (\mathbf {r} ).}

For inhomogeneous materials, there is a position dependence throughout the material:[22]D(r)=ε(r)E(r){\displaystyle \mathbf {D} (\mathbf {r} )=\varepsilon (\mathbf {r} )\mathbf {E} (\mathbf {r} )}

For anisotropic materials theE andD fields are not parallel, and soE andD are related by thepermittivity tensor (a 2nd ordertensor field), in component form:Di=εijEj{\displaystyle D_{i}=\varepsilon _{ij}E_{j}}

For non-linear media,E andD are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.

Relativistic effects on electric field

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Point charge in uniform motion

[edit]

The invariance of the form ofMaxwell's equations underLorentz transformation can be used to derive the electric field of a uniformly moving point charge. The charge of a particle is considered frame invariant, as supported by experimental evidence.[23] Alternatively the electric field of uniformly moving point charges can be derived from theLorentz transformation offour-force experienced by test charges in the source's rest frame given byCoulomb's law and assigning electric field and magnetic field by their definition given by the form ofLorentz force.[24] However the following equation is only applicable when no acceleration is involved in the particle's history whereCoulomb's law can be considered or symmetry arguments can be used for solvingMaxwell's equations in a simple manner. The electric field of such a uniformly moving point charge is hence given by:[25]E=q4πε0r31−β2(1−β2sin2⁡θ)3/2r,{\displaystyle \mathbf {E} ={\frac {q}{4\pi \varepsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}\mathbf {r} ,}whereq{\displaystyle q} is the charge of the point source,r{\displaystyle \mathbf {r} } is the position vector from the point source to the point in space,β{\displaystyle \beta } is the ratio of observed speed of the charge particle to the speed of light andθ{\displaystyle \theta } is the angle betweenr{\displaystyle \mathbf {r} } and the observed velocity of the charged particle.

The above equation reduces to that given by Coulomb's law for non-relativistic speeds of the point charge. Spherical symmetry is not satisfied due to breaking of symmetry in the problem by specification of direction of velocity for calculation of field. To illustrate this, field lines of moving charges are sometimes represented as unequally spaced radial lines which would appear equally spaced in a co-moving reference frame.[23]

Propagation of disturbances in electric fields

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See also:Paradox of radiation of charged particles in a gravitational field

Special theory of relativity imposes theprinciple of locality, that requires cause and effect to be time-like separated events where thecausal efficacy does not travel faster than thespeed of light.[26]Maxwell's laws are found to confirm to this view since the general solutions of fields are given in terms of retarded time which indicate thatelectromagnetic disturbances travel at thespeed of light. Advanced time, which also provides a solution forMaxwell's law are ignored as an unphysical solution.

An illustrative example showing bremsstrahlung radiation: Field lines and modulus of the electric field generated by a (negative) charge first moving at a constant speed and then stopping quickly to show the electromagnetic wave generated and propagation of disturbances in electromagnetic field.

For the motion of acharged particle, considering for example the case of a moving particle with the above described electric field coming to an abrupt stop, the electric fields at points far from it do not immediately revert to that classically given for a stationary charge. On stopping, the field around the stationary points begin to revert to the expected state and this effect propagates outwards at thespeed of light while the electric field lines far away from this will continue to point radially towards an assumed moving charge. This virtual particle will never be outside the range of propagation of the disturbance inelectromagnetic field, since charged particles are restricted to have speeds slower than that of light, which makes it impossible to construct aGaussian surface in this region that violatesGauss's law. Another technical difficulty that supports this is that charged particles travelling faster than or equal to speed of light no longer have a unique retarded time. Since electric field lines are continuous, anelectromagnetic pulse of radiation is generated that connects at the boundary of this disturbance travelling outwards at thespeed of light.[27] In general, any accelerating point charge radiateselectromagnetic waves however,non-radiating acceleration is possible in a systems of charges.

Arbitrarily moving point charge

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See also:Jefimenko's equations § Heaviside–Feynman formula

For arbitrarily moving point charges, propagation of potential fields such as Lorenz gauge fields at the speed of light needs to be accounted for by usingLiénard–Wiechert potential.[28] Since the potentials satisfyMaxwell's equations, the fields derived for point charge also satisfyMaxwell's equations. The electric field is expressed as:[29]E(r,t)=14πε0(q(ns−βs)γ2(1−ns⋅βs)3|r−rs|2+qns×((ns−βs)×βs˙)c(1−ns⋅βs)3|r−rs|)t=tr{\displaystyle \mathbf {E} (\mathbf {r} ,\mathbf {t} )={\frac {1}{4\pi \varepsilon _{0}}}\left({\frac {q(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})}{\gamma ^{2}(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|^{2}}}+{\frac {q\mathbf {n} _{s}\times {\big (}(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})\times {\dot {{\boldsymbol {\beta }}_{s}}}{\big )}}{c(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}}whereq{\displaystyle q} is the charge of the point source,tr{\textstyle {t_{r}}} isretarded time or the time at which the source's contribution of the electric field originated,rs(t){\textstyle {r}_{s}(t)} is the position vector of the particle,ns(r,t){\textstyle {n}_{s}(\mathbf {r} ,t)} is a unit vector pointing from charged particle to the point in space,βs(t){\textstyle {\boldsymbol {\beta }}_{s}(t)} is the velocity of the particle divided by the speed of light, andγ(t){\textstyle \gamma (t)} is the correspondingLorentz factor. The retarded time is given as solution of:tr=t−|r−rs(tr)|c{\displaystyle t_{r}=\mathbf {t} -{\frac {|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}{c}}}

The uniqueness of solution fortr{\textstyle {t_{r}}} for givent{\displaystyle \mathbf {t} },r{\displaystyle \mathbf {r} } andrs(t){\displaystyle r_{s}(t)} is valid for charged particles moving slower than speed of light.Electromagnetic radiation of accelerating charges is known to be caused by the acceleration dependent term in the electric field from which relativistic correction forLarmor formula is obtained.[29]

There exist yet another set of solutions for Maxwell's equation of the same form but for advanced timeta{\textstyle {t_{a}}} instead of retarded time given as a solution of:

ta=t+|r−rs(ta)|c{\displaystyle t_{a}=\mathbf {t} +{\frac {|\mathbf {r} -\mathbf {r} _{s}(t_{a})|}{c}}}

Since the physical interpretation of this indicates that the electric field at a point is governed by the particle's state at a point of time in the future, it is considered as an unphysical solution and hence neglected. However, there have been theories exploring the advanced time solutions ofMaxwell's equations, such asFeynman Wheeler absorber theory.

The above equation, although consistent with that of uniformly moving point charges as well as its non-relativistic limit, are not corrected for quantum-mechanical effects.

Common formulæ

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Charge configurationFigureElectric field
Infinite wireE=λ2πε0xx^,{\displaystyle \mathbf {E} ={\frac {\lambda }{2\pi \varepsilon _{0}x}}{\hat {\mathbf {x} }},}

whereλ{\displaystyle \lambda } is uniform linear charge density.

Infinitely large surfaceE=σ2ε0x^,{\displaystyle \mathbf {E} ={\frac {\sigma }{2\varepsilon _{0}}}{\hat {\mathbf {x} }},}

whereσ{\displaystyle \sigma } is uniform surface charge density.

Infinitely long cylindrical volumeE=λ2πε0xx^,{\displaystyle \mathbf {E} ={\frac {\lambda }{2\pi \varepsilon _{0}x}}{\hat {\mathbf {x} }},}

whereλ{\displaystyle \lambda } is uniform linear charge density.

Spherical volumeE=Q4πε0x2x^,{\displaystyle \mathbf {E} ={\frac {Q}{4\pi \varepsilon _{0}x^{2}}}{\hat {\mathbf {x} }},}

outside the sphere, whereQ{\displaystyle Q} is the total charge uniformly distributed in the volume.

E=Qr4πε0R3r^,{\displaystyle \mathbf {E} ={\frac {Qr}{4\pi \varepsilon _{0}R^{3}}}{\hat {\mathbf {r} }},}

inside the sphere, whereQ{\displaystyle Q} is the total charge uniformly distributed in the volume.

Spherical surfaceE=Q4πε0x2x^,{\displaystyle \mathbf {E} ={\frac {Q}{4\pi \varepsilon _{0}x^{2}}}{\hat {\mathbf {x} }},}

outside the sphere, whereQ{\displaystyle Q} is the total charge uniformly distributed on the surface.

E=0,{\displaystyle \mathbf {E} =0,}

inside the sphere for uniform charge distribution.

Charged RingE=Qx4πε0(R2+x2)3/2x^,{\displaystyle \mathbf {E} ={\frac {Qx}{4\pi \varepsilon _{0}(R^{2}+x^{2})^{3/2}}}{\hat {\mathbf {x} }},}

on the axis, whereQ{\displaystyle Q} is the total charge uniformly distributed on the ring.

Charged DiscE=σ2ε0[1−xx2+R2]x^,{\displaystyle \mathbf {E} ={\frac {\sigma }{2\varepsilon _{0}}}\left[1-{\frac {x}{\sqrt {x^{2}+R^{2}}}}\right]{\hat {\mathbf {x} }},}

on the axis, whereσ{\displaystyle \sigma } is the uniform surface charge density.

Electric DipoleE=−p4πε0r3,{\displaystyle \mathbf {E} =-{\frac {\mathbf {p} }{4\pi \varepsilon _{0}r^{3}}},}

on the equatorial plane, wherep{\displaystyle \mathbf {p} } is the electric dipole moment.

E=p2πε0x3,{\displaystyle \mathbf {E} ={\frac {\mathbf {p} }{2\pi \varepsilon _{0}x^{3}}},}

on the axis (given thatx≫d{\displaystyle x\gg d}), wherex{\displaystyle x} can also be negative to indicate position at the opposite direction on the axis, andp{\displaystyle \mathbf {p} } is the electric dipole moment.

Electric field infinitely close to a conducting surface in electrostatic equilibrium having charge densityσ{\displaystyle \sigma } at that point isσε0x^{\textstyle {\frac {\sigma }{\varepsilon _{0}}}{\hat {\mathbf {x} }}} since charges are only formed on the surface and the surface at the infinitesimal scale resembles an infinite 2D plane. In the absence of external fields, spherical conductors exhibit a uniform charge distribution on the surface and hence have the same electric field as that of uniform spherical surface distribution.

See also

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References

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  1. ^Roche, John (2016). "Introducing electric fields".Physics Education.51 (5): 055005.Bibcode:2016PhyEd..51e5005R.doi:10.1088/0031-9120/51/5/055005.S2CID 125014664.
  2. ^Feynman, Richard (1970).The Feynman Lectures on Physics Vol II. Addison Wesley Longman. pp. 1–3,1–4.ISBN 978-0-201-02115-8.
  3. ^Purcell, Edward M.; Morin, David J. (2013).Electricity and Magnetism (3rd ed.). New York: Cambridge University Press. pp. 15–16.ISBN 978-1-107-01402-2.
  4. ^abcSerway, Raymond A.; Vuille, Chris (2014).College Physics (10th ed.). Cengage Learning. pp. 532–533.ISBN 978-1305142824.
  5. ^The International System of Units(PDF) (9th ed.), International Bureau of Weights and Measures, Dec 2022,ISBN 978-92-822-2272-0, p. 23
  6. ^abcSears, Francis; et al. (1982),University Physics (6th ed.), Addison Wesley,ISBN 0-201-07199-1
  7. ^Umashankar, Korada (1989),Introduction to Engineering Electromagnetic Fields, World Scientific, pp. 77–79,ISBN 9971-5-0921-0
  8. ^abMorely & Hughes (1970),Principles of Electricity (5th ed.), Longman, p. 73,ISBN 0-582-42629-4
  9. ^Tou, Stephen (2011).Visualization of Fields and Applications in Engineering. John Wiley and Sons. p. 64.ISBN 9780470978467.
  10. ^abcGriffiths, David J. (1999).Introduction to electrodynamics (3rd ed.). Upper Saddle River, NJ: Prentice Hall.ISBN 0-13-805326-X.OCLC 40251748.
  11. ^Purcell, p. 25: "Gauss's Law: the flux of the electric field E through any closed surface ... equals 1/e times the total charge enclosed by the surface."
  12. ^Purcell, p 356: "Faraday's Law of Induction."
  13. ^Purcell, p7: "... the interaction between electric chargesat rest is described by Coulomb's Law: two stationary electric charges repel or attract each other with a force proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them.
  14. ^Purcell, Edward (2011).Electricity and Magnetism (2nd ed.). Cambridge University Press. pp. 8–9.ISBN 978-1139503556.
  15. ^gwrowe (8 October 2011)."Curl & Potential in Electrostatics"(PDF).physicspages.com. Archived fromthe original(PDF) on 22 March 2019. Retrieved2 November 2020.
  16. ^Huray, Paul G. (2009).Maxwell's Equations. Wiley-IEEE. p. 205.ISBN 978-0-470-54276-7.[permanent dead link‍]
  17. ^Purcell, pp. 5–7.
  18. ^Salam, Abdus (16 December 1976)."Quarks and leptons come out to play".New Scientist.72: 652.[permanent dead link‍]
  19. ^Griffiths, D.J. (2017).Introduction to Electrodynamics (3 ed.). Cambridge University Press. p. 357, eq. 8.5.ISBN 9781108420419.
  20. ^abGrant, I.S.; Phillips, W.R. (2008).Electromagnetism (2 ed.). John Wiley & Sons.ISBN 978-0-471-92712-9.
  21. ^Bennet, G.A.G.; Arnold, Edward (1974).Electricity and Modern Physics (2 ed.). Edward Arnold.ISBN 0-7131-2459-8.
  22. ^Landau, Lev Davidovich;Lifshitz, Evgeny M. (1963). "68 the propagation of waves in an inhomogeneous medium".Electrodynamics of Continuous Media.Course of Theoretical Physics. Vol. 8. Pergamon. p. 285.ISBN 978-0-7581-6499-5.In Maxwell's equations…ε is a function of the co-ordinates.
  23. ^abPurcell, Edward M.; Morin, David J. (2013-01-21).Electricity and Magnetism. pp. 241–251.doi:10.1017/cbo9781139012973.ISBN 9781139012973. Retrieved2022-07-04.{{cite book}}:|website= ignored (help)
  24. ^Rosser, W. G. V. (1968).Classical Electromagnetism via Relativity. pp. 29–42.doi:10.1007/978-1-4899-6559-2.ISBN 978-1-4899-6258-4.
  25. ^Heaviside, Oliver.Electromagnetic waves, the propagation of potential, and the electromagnetic effects of a moving charge.
  26. ^Naber, Gregory L. (2012).The Geometry of Minkowski spacetime: an introduction to the mathematics of the special theory of relativity. Springer. pp. 4–5.ISBN 978-1-4419-7837-0.OCLC 804823303.
  27. ^Purcell, Edward M.; David J. Morin (2013).Electricity and Magnetism (Third ed.). Cambridge. pp. 251–255.ISBN 978-1-139-01297-3.OCLC 1105718330.{{cite book}}: CS1 maint: location missing publisher (link)
  28. ^Griffiths, David J. (2017).Introduction to electrodynamics (4th ed.). United Kingdom:Cambridge University Press. p. 454.ISBN 978-1-108-42041-9.OCLC 1021068059.
  29. ^abJackson, John David (1999).Classical electrodynamics (3rd ed.). New York: Wiley. pp. 664–665.ISBN 0-471-30932-X.OCLC 38073290.
  • Purcell, Edward; Morin, David (2013).Electricity and Magnetism (3rd ed.). Cambridge University Press, New York.ISBN 978-1-107-01402-2.
  • Browne, Michael (2011).Physics for Engineering and Science (2nd ed.). McGraw-Hill, Schaum, New York.ISBN 978-0-07-161399-6.

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