# Electric field

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Electric field
Effects of an electric field. The girl is touching an electrostatic generator, which charges her body with a high voltage. Her hair, which is charged with the same polarity, is repelled by the electric field of her head and stands out from her head.
Common symbols
E
SI unit volts per meter (V/m)
In SI base units m⋅kg⋅s−3⋅A−1
Behaviour under
vector
Derivations from
other quantities
F / q

An electric field (sometimes E-field  ) is the physical field that surrounds electrically-charged particles and exerts force on all other charged particles in the field, either attracting or repelling them.  It also refers to the physical field for a system of charged particles.  Electric fields originate from electric charges, or from time-varying magnetic fields. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces (or interactions) of nature.

## Contents

Electric fields are important in many areas of physics, and are exploited practically in electrical technology. In atomic physics and chemistry, for instance, the electric field is the attractive force holding the atomic nucleus and electrons together in atoms. It is also the force responsible for chemical bonding between atoms that result in molecules.

Other applications of electric fields include motion detection via electric field proximity sensing and an increasing number of diagnostic and therapeutic medical uses.

The electric field is defined mathematically as a vector field that associates to each point in space the (electrostatic or Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point.    The derived SI units for the electric field are volts per meter (V/m), exactly equivalent to newtons per coulomb (N/C). 

## Description Electric field of a positive point electric charge suspended over an infinite sheet of conducting material. The field is depicted by electric field lines, lines which follow the direction of the electric field in space.

The electric field is defined at each point in space as the force (per unit charge) that would be experienced by a vanishingly small positive test charge if held at that point.  :469–70 As the electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), it follows that an electric field is a vector field.  :469–70 Vector fields of this form are sometimes referred to as force fields. The electric field acts between two charges similarly to the way the gravitational field acts between two masses, as they both obey an inverse-square law with distance.  This is the basis for Coulomb's law, which states that, for 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 of lines whose direction at each point is the same as the field's, a concept introduced by Michael Faraday,  whose term 'lines of force' is still sometimes used. This illustration has the useful property that the field's strength is proportional to the density of the lines.  The field lines are the paths that a point positive charge would follow as it is forced to move within the field, similar to trajectories that masses follow within a gravitational field. Field lines due to stationary charges have several important properties, including always originating from positive charges and terminating at negative charges, they enter all good conductors at right angles, and they never cross or close in on themselves.  :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.  The study of electric fields created by stationary charges is called electrostatics.

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 the curl of the electric field is equal to the negative time derivative of the magnetic field.  :327 In the absence of time-varying magnetic field, the electric field is therefore called conservative (i.e. curl-free).  :24,90–91 This implies there are two kinds of electric fields: electrostatic fields and fields arising from time-varying magnetic fields.  :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 unified electromagnetic field. The study of time varying magnetic and electric fields is called electrodynamics.

## Mathematical formulation

Electric fields are caused by electric charges, described by Gauss's law,  and time varying magnetic fields, described by Faraday's law of induction.  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 form Maxwell's equations that describe both fields as a function of charges and currents. Evidence of an electric field: styrofoam peanuts clinging to a cat's fur due to static electricity. The triboelectric effect causes an electrostatic 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 to electrostatic induction, resulting in a slight attraction of the light plastic pieces to the charged fur. This effect is also the cause of static cling in clothes.

### Electrostatics

In the special case of a steady state (stationary charges and currents), the Maxwell-Faraday inductive effect disappears. The resulting two equations (Gauss's law $\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$ and Faraday's law with no induction term $\nabla \times \mathbf {E} =0$ ), taken together, are equivalent to Coulomb's law, which states that a particle with electric charge $q_{1}$ at position ${\boldsymbol {x}}_{1}$ exerts a force on a particle with charge $q_{0}$ at position ${\boldsymbol {x}}_{0}$ of: 

${\boldsymbol {F}}={1 \over 4\pi \varepsilon _{0}}{q_{1}q_{0} \over ({\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{0})^{2}}{\hat {\boldsymbol {r}}}_{1,0}\,,$ where ${\boldsymbol {r}}_{1,0}$ is the unit vector in the direction from point ${\boldsymbol {x}}_{1}$ to point ${\boldsymbol {x}}_{0}$ , and ε0 is the electric constant (also known as "the absolute permittivity of free space") with units C2⋅m−2⋅N−1.

Note that $\varepsilon _{0}$ , the vacuum electric permittivity, must be substituted with $\varepsilon$ , permittivity, when charges are in non-empty media. When the charges $q_{0}$ and $q_{1}$ 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 the Coulomb force on any charge at position ${\boldsymbol {x}}_{0}$ this expression can be divided by $q_{0}$ leaving an expression that only depends on the other charge (the source charge)  

${\boldsymbol {E}}({\boldsymbol {x}}_{0})={{\boldsymbol {F}} \over q_{0}}={1 \over 4\pi \varepsilon _{0}}{q_{1} \over ({\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{0})^{2}}{\hat {\boldsymbol {r}}}_{1,0}$ This is the electric field at point ${\boldsymbol {x}}_{0}$ due to the point charge $q_{1}$ ; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position ${\boldsymbol {x}}_{0}$ . Since this formula gives the electric field magnitude and direction at any point ${\boldsymbol {x}}_{0}$ in space (except at the location of the charge itself, ${\boldsymbol {x}}_{1}$ , where it becomes infinite) it defines a vector 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 the inverse square of the distance from the charge.

The Coulomb force on a charge of magnitude $q$ at any point in space is equal to the product of the charge and the electric field at that point

${\boldsymbol {F}}=q{\boldsymbol {E}}$ The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m); in terms of the SI base units they are kg⋅m⋅s−3⋅A−1.

### Superposition principle

Due to the linearity of Maxwell's equations, electric fields satisfy the superposition 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.  This principle is useful in calculating the field created by multiple point charges. If charges $q_{1},q_{2},...,q_{n}$ are stationary in space at points $\mathbf {x} _{1},\mathbf {x} _{2},...\mathbf {x} _{n}$ , 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:

${\boldsymbol {E}}({\boldsymbol {x}})={\boldsymbol {E}}_{1}({\boldsymbol {x}})+{\boldsymbol {E}}_{2}({\boldsymbol {x}})+{\boldsymbol {E}}_{3}({\boldsymbol {x}})+\cdots ={1 \over 4\pi \varepsilon _{0}}{q_{1} \over ({\boldsymbol {x}}_{1}-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}_{1}+{1 \over 4\pi \varepsilon _{0}}{q_{2} \over ({\boldsymbol {x}}_{2}-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}_{2}+{1 \over 4\pi \varepsilon _{0}}{q_{3} \over ({\boldsymbol {x}}_{3}-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}_{3}+\cdots$ ${\boldsymbol {E}}({\boldsymbol {x}})={1 \over 4\pi \varepsilon _{0}}\sum _{k=1}^{N}{q_{k} \over ({\boldsymbol {x}}_{k}-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}_{k}$ where ${\boldsymbol {{\hat {r}}_{k}}}$ is the unit vector in the direction from point ${\boldsymbol {x}}_{k}$ to point ${\boldsymbol {x}}$ .

### Continuous charge distributions

The superposition principle allows for the calculation of the electric field due to a continuous distribution of charge $\rho ({\boldsymbol {x}})$ (where $\rho$ is the charge density in coulombs per cubic meter). By considering the charge $\rho ({\boldsymbol {x}}')dV$ in each small volume of space $dV$ at point ${\boldsymbol {x}}'$ as a point charge, the resulting electric field, $d{\boldsymbol {E}}({\boldsymbol {x}})$ , at point ${\boldsymbol {x}}$ can be calculated as

$d{\boldsymbol {E}}({\boldsymbol {x}})={1 \over 4\pi \varepsilon _{0}}{\rho ({\boldsymbol {x}}')dV \over ({\boldsymbol {x}}'-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}'$ where ${\hat {\boldsymbol {r}}}'$ is the unit vector pointing from ${\boldsymbol {x}}'$ to ${\boldsymbol {x}}$ . The total field is then found by "adding up" the contributions from all the increments of volume by integrating over the volume of the charge distribution $V$ :

${\boldsymbol {E}}({\boldsymbol {x}})={1 \over 4\pi \varepsilon _{0}}\iiint _{V}\,{\rho ({\boldsymbol {x}}')dV \over ({\boldsymbol {x}}'-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}'$ Similar equations follow for a surface charge with continuous charge distribution $\sigma ({\boldsymbol {x}})$ where $\sigma$ is the charge density in coulombs per square meter

${\boldsymbol {E}}({\boldsymbol {x}})={1 \over 4\pi \varepsilon _{0}}\iint _{S}\,{\sigma ({\boldsymbol {x}}')dA \over ({\boldsymbol {x}}'-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}'$ and for line charges with continuous charge distribution $\lambda ({\boldsymbol {x}})$ where $\lambda$ is the charge density in coulombs per meter.

${\boldsymbol {E}}({\boldsymbol {x}})={1 \over 4\pi \varepsilon _{0}}\int _{P}\,{\lambda ({\boldsymbol {x}}')dL \over ({\boldsymbol {x}}'-{\boldsymbol {x}})^{2}}{\hat {\boldsymbol {r}}}'$ ### Electric potential

If a system is static, such that magnetic fields are not time-varying, then by Faraday's law, the electric field is curl-free. In this case, one can define an electric potential, that is, a function $\Phi$ such that $\mathbf {E} =-\nabla \Phi$ .  This is analogous to the gravitational potential. The difference between the electric potential at two points in space is called the potential difference (or voltage) between the two points.

In general, however, the electric field cannot be described independently of the magnetic field. Given the magnetic vector potential, A, defined so that $\mathbf {B} =\nabla \times \mathbf {A}$ , one can still define an electric potential $\Phi$ such that:

$\mathbf {E} =-\nabla \Phi -{\frac {\partial \mathbf {A} }{\partial t}}$ Where $\nabla \Phi$ is the gradient of the electric potential and ${\frac {\partial \mathbf {A} }{\partial t}}$ is the partial derivative of A with respect to time.

Faraday's law of induction can be recovered by taking the curl of that equation 

$\nabla \times \mathbf {E} =-{\frac {\partial (\nabla \times \mathbf {A} )}{\partial t}}=-{\frac {\partial \mathbf {B} }{\partial t}}$ which justifies, a posteriori, the previous form for E.

### Continuous vs. discrete charge representation

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 describe electrons as point sources where charge density is infinite on an infinitesimal section of space.

A charge $q$ located at $\mathbf {r_{0}}$ can be described mathematically as a charge density $\rho (\mathbf {r} )=q\delta (\mathbf {r-r_{0}} )$ , where the Dirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

## Electrostatic fields

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. 

### Parallels between electrostatic and gravitational fields

Coulomb's law, which describes the interaction of electric charges:

$\mathbf {F} =q\left({\frac {Q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {\hat {r}} }{|\mathbf {r} |^{2}}}\right)=q\mathbf {E}$ is similar to Newton's law of universal gravitation:

$\mathbf {F} =m\left(-GM{\frac {\mathbf {\hat {r}} }{|\mathbf {r} |^{2}}}\right)=m\mathbf {g}$ (where ${\textstyle \mathbf {\hat {r}} =\mathbf {\frac {r}{|r|}} }$ ).

This suggests similarities between the electric field E and the gravitational field g, or their associated potentials. Mass is sometimes called "gravitational charge". 

Electrostatic and gravitational forces both are central, conservative and obey an inverse-square law.

### Uniform fields Illustration of the electric field between two parallel conductive plates of finite size (known as a parallel 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 conducting plates parallel to each other and maintaining a voltage (potential difference) between them; it is only an approximation because of boundary effects (near the edge of the planes, electric field is distorted because the plane does not continue). Assuming infinite planes, the magnitude of the electric field E is:

$E=-{\frac {\Delta V}{d}}$ where ΔV is the potential difference between the plates and d 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 of 106 V⋅m−1, achieved by applying a voltage of the order of 1 volt between conductors spaced 1 µm apart.

## Electrodynamic fields The electric field (lines with arrows) of a charge (+) induces surface charges (red and blue areas) on metal objects due to electrostatic induction.

Electrodynamic fields are electric fields which do change with time, for instance when charges are in motion. In this case, a magnetic field is produced in accordance with Ampère's circuital law (with Maxwell's addition), which, along with Maxwell's other equations, defines the magnetic field, $\mathbf {B}$ , in terms of its curl:

$\nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right),$ where $\mathbf {J}$ is the current density, $\mu _{0}$ is the vacuum permeability, and $\varepsilon _{0}$ is the vacuum permittivity.

That is, both electric currents (i.e. charges in uniform motion) and the (partial) time derivative of the electric field directly contributes to the magnetic field. In addition, the Maxwell–Faraday equation states

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}.$ These represent two of Maxwell's four equations and they intricately link the electric and magnetic fields together, resulting in the electromagnetic 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 the Lorentz force law:

$\mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B}$ ## Energy in the electric field

The total energy per unit volume stored by the electromagnetic field is 

$u_{\text{EM}}={\frac {\varepsilon }{2}}|\mathbf {E} |^{2}+{\frac {1}{2\mu }}|\mathbf {B} |^{2}$ where ε is the permittivity of the medium in which the field exists, $\mu$ its magnetic permeability, and E and B are the electric and magnetic field vectors.

As E and B 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 energy UEM stored in the electromagnetic field in a given volume V is

$U_{\text{EM}}={\frac {1}{2}}\int _{V}\left(\varepsilon |\mathbf {E} |^{2}+{\frac {1}{\mu }}|\mathbf {B} |^{2}\right)\mathrm {d} V\,.$ ## The electric displacement field

### Definitive equation of vector fields

In the presence of matter, it is helpful to extend the notion of the electric field into three vector fields: 

$\mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} \!$ where P is the electric polarization – the volume density of electric dipole moments, and D is the electric displacement field. Since E and P are defined separately, this equation can be used to define D. The physical interpretation of D is not as clear as E (effectively the field applied to the material) or P (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 of free charges and currents.

### Constitutive relation

The E and D fields are related by the permittivity of the material, ε.  

For linear, homogeneous, isotropic materials E and D are proportional and constant throughout the region, there is no position dependence:

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

$\mathbf {D} (\mathbf {r} )=\varepsilon (\mathbf {r} )\mathbf {E} (\mathbf {r} )$ For anisotropic materials the E and D fields are not parallel, and so E and D are related by the permittivity tensor (a 2nd order tensor field), in component form:

$D_{i}=\varepsilon _{ij}E_{j}$ For non-linear media, E and D are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.

## Related Research Articles In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The electric potential is the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field with negligible acceleration of the test charge to avoid producing kinetic energy or radiation by test charge. Typically, the reference point is the Earth or a point at infinity, although any point can be used. More precisely it is the energy per unit charge for a small test charge that does not disturb significantly the field and the charge distribution producing the field under consideration. In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge. In physics, specifically electromagnetism, the Biot–Savart law is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot–Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820. Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson. In classical electromagnetism, Ampère's circuital law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell derived it using hydrodynamics in his 1861 published paper "On Physical Lines of Force" In 1865 he generalized the equation to apply to time-varying currents by adding the displacement current term, resulting in the modern form of the law, sometimes called the Ampère–Maxwell law, which is one of Maxwell's equations which form the basis of classical electromagnetism. Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

In physics, chemistry and biology, a potential gradient is the local rate of change of the potential with respect to displacement, i.e. spatial derivative, or gradient. This quantity frequently occurs in equations of physical processes because it leads to some form of flux. In electromagnetism, displacement current density is the quantity D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials, there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave. Magnetic vector potential, A, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field: . Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Therefore, many equations of electromagnetism can be written either in terms of the fields E and B, or equivalently in terms of the potentials φ and A. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

In physics, the electric displacement field or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding of Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter square (C⋅m−2).

A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically meant to describe electromagnetism and gravitation, two of the fundamental forces of nature.

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form: The Maxwell stress tensor is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand. There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking. The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898 and independently by Emil Wiechert in 1900. Coulomb's law, or Coulomb's inverse-square law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force or Coulomb force. Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb, hence the name. Coulomb's law was essential to the development of the theory of electromagnetism, maybe even its starting point, as it made it possible to discuss the quantity of electric charge in a meaningful way. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, a commonly used unit in atomic physics and chemistry is the debye (D).

1. Roche, John (2016). "Introducing electric fields". Physics Education. 51 (5): 055005. Bibcode:2016PhyEd..51e5005R. doi:10.1088/0031-9120/51/5/055005.
2. Browne, p 225: "... around every charge there is an aura that fills all space. This aura is the electric field due to the charge. The electric field is a vector field... and has a magnitude and direction."
3. Purcell, Edward M.; Morin, David J. (2013). Electricity and Magnetism (3rd ed.). New York: Cambridge University Press. pp. 16–20. ISBN   978-1-107-01402-2.
4. Richard Feynman (1970). The Feynman Lectures on Physics Vol II. Addison Wesley Longman. pp. 1–3, 1–4. ISBN   978-0-201-02115-8.
5. 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.
6. Serway, Raymond A.; Vuille, Chris (2014). College Physics, 10th Ed. Cengage Learning. pp. 532–533. ISBN   978-1305142824.
7. International Bureau of Weights and Measures (2019-05-20), SI Brochure: The International System of Units (SI) (PDF) (9th ed.), ISBN   978-92-822-2272-0 , p. 23
8. Sears, Francis; et al. (1982), University Physics, Sixth Edition, Addison Wesley, ISBN   0-201-07199-1
9. Umashankar, Korada (1989), Introduction to Engineering Electromagnetic Fields, World Scientific, pp. 77–79, ISBN   9971-5-0921-0
10. Morely & Hughes, Principles of Electricity, Fifth edition, p. 73, ISBN   0-582-42629-4
11. Tou, Stephen (2011). Visualization of Fields and Applications in Engineering. John Wiley and Sons. p. 64. ISBN   9780470978467.
12. Griffiths, David J. (David Jeffery), 1942- (1999). Introduction to electrodynamics (3rd ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN   0-13-805326-X. OCLC   40251748.CS1 maint: multiple names: authors list (link)
13. 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."
14. Purcell, p 356: "Faraday's Law of Induction."
15. Purcell, p7: "... the interaction between electric charges at 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.
16. Purcell, Edward (2011). Electricity and Magnetism, 2nd Ed. Cambridge University Press. pp. 8–9. ISBN   978-1139503556.
17. gwrowe (8 October 2011). "Curl & Potential in Electrostatics" (PDF). physicspages.com. Archived from the original (PDF) on 22 March 2019. Retrieved 2 November 2020.
18. Huray, Paul G. (2009). Maxwell's Equations. Wiley-IEEE. p. 205. ISBN   978-0-470-54276-7.
19. Purcell, pp. 5-7.
20. Salam, Abdus (16 December 1976). "Quarks and leptons come out to play". New Scientist. 72: 652.
21. Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN   81-7758-293-3
22. Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN   978-0-471-92712-9
23. Electricity and Modern Physics (2nd Edition), G.A.G. Bennet, Edward Arnold (UK), 1974, ISBN   0-7131-2459-8
24. Landau, Lev Davidovich; Lifshitz, Evgeny M. (1963). "68 the propagation of waves in an inhomogeneous medium". Electrodynamics of Continuous Media. Course of Theoretical Physics. 8. Pergamon. p. 285. ISBN   978-0-7581-6499-5. In Maxwell's equations… ε is a function of the co-ordinates.
• 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.