Electric field | |
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Common symbols | E |

SI unit | volt per meter (V/m) |

In SI base units | m⋅kg⋅s^{−3}⋅A^{−1} |

Articles about |

Electromagnetism |
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An **electric field** (sometimes **E-field**^{ [1] }) 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.^{ [2] } It also refers to the physical field for a system of charged particles.^{ [3] } Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental interactions (also called forces) of nature.

- Description
- Mathematical formulation
- Electrostatics
- Superposition principle
- Continuous charge distributions
- Electric potential
- Continuous vs. discrete charge representation
- Electrostatic fields
- Parallels between electrostatic and gravitational fields
- Uniform fields
- Electrodynamic fields
- Energy in the electric field
- The electric displacement field
- Definitive equation of vector fields
- Constitutive relation
- Relativistic Effects on electric field
- Point charge in uniform motion
- Propagation of disturbances in electric fields
- Arbitrarily moving point charge
- Some Common Electric Field Values
- See also
- References
- External links

Electric fields are important in many areas of physics, and are exploited 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.

The electric field is defined 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.^{ [4] }^{ [5] }^{ [6] } The derived SI unit for the electric field is the volt per meter (V/m), which is equal to the newton per coulomb (N/C).^{ [7] }

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 stationary at that point.^{ [8] }^{: 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.^{ [8] }^{: 469–70 } Fields that may be defined in this manner 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.^{ [9] } This is the basis for Coulomb'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 of lines whose direction at each point is the same as the field's, a concept introduced by Michael Faraday,^{ [10] } 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.^{ [11] } 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.^{ [8] }^{: 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.^{ [10] } 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.^{ [12] }^{: 327 } In the absence of time-varying magnetic field, the electric field is therefore called conservative (i.e. curl-free).^{ [12] }^{: 24, 90–91 } This implies there are two kinds of electric fields: electrostatic fields and fields arising from time-varying magnetic fields.^{ [12] }^{: 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.

Electric fields are caused by electric charges, described by Gauss's law,^{ [13] } and time varying magnetic fields, described by Faraday's law of induction.^{ [14] } 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.

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 and Faraday's law with no induction term ), taken together, are equivalent to Coulomb's law, which states that a particle with electric charge at position exerts a force on a particle with charge at position of:^{ [15] }

where is the unit vector in the direction from point to point , and *ε*_{0} is the electric constant (also known as "the absolute permittivity of free space") with the unit C^{2}⋅m^{−2}⋅N^{−1}.

Note that , the vacuum electric permittivity, must be substituted with , permittivity, when charges are in non-empty media. When the charges and 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 this expression can be divided by leaving an expression that only depends on the other charge (the *source* charge)^{ [16] }^{ [6] }

This is the *electric field* at point due to the point charge ; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position . Since this formula gives the electric field magnitude and direction at any point in space (except at the location of the charge itself, , 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 at any point in space is equal to the product of the charge and the electric field at that point

The SI unit of the electric field is the newton per coulomb (N/C), or volt per meter (V/m); in terms of the SI base units it is kg⋅m⋅s^{−3}⋅A^{−1}.

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.^{ [6] } This principle is useful in calculating the field created by multiple point charges. If charges are stationary in space at points , 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:

where is the unit vector in the direction from point to point .

The superposition principle allows for the calculation of the electric field due to a continuous distribution of charge (where is the charge density in coulombs per cubic meter). By considering the charge in each small volume of space at point as a point charge, the resulting electric field, , at point can be calculated as

where is the unit vector pointing from to . 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 :

Similar equations follow for a surface charge with continuous charge distribution where is the charge density in coulombs per square meter

and for line charges with continuous charge distribution where is the charge density in coulombs per meter.

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 such that .^{ [17] } 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 , one can still define an electric potential such that:

where is the gradient of the electric potential and 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 ^{ [18] }

which justifies, a posteriori, the previous form for **E**.

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 located at can be described mathematically as a charge density , where the Dirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

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.^{ [19] }

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

is similar to Newton's law of universal gravitation:

(where ).

This suggests similarities between the electric field **E** and the gravitational field **g**, or their associated potentials. Mass is sometimes called "gravitational charge".^{ [20] }

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

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:

where Δ*V* 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 10^{6} V⋅m^{−1}, achieved by applying a voltage of the order of 1 volt between conductors spaced 1 µm apart.

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, , in terms of its curl:

where is the current density, is the vacuum permeability, and 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

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:

The total energy per unit volume stored by the electromagnetic field is^{ [21] }

where ε is the permittivity of the medium in which the field exists, 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 *U*_{EM} stored in the electromagnetic field in a given volume *V* is

In the presence of matter, it is helpful to extend the notion of the electric field into three vector fields:^{ [22] }

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.

The **E** and **D** fields are related by the permittivity of the material, *ε*.^{ [23] }^{ [22] }

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

For inhomogeneous materials, there is a position dependence throughout the material:^{ [24] }

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:

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

The invariance of the form of Maxwell's equations under Lorentz 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.^{ [25] } Alternatively the electric field of uniformly moving point charges can be derived from the Lorentz transformation of four-force experienced by test charges in the source's rest frame given by Coulomb's law and assigning electric field and magnetic field by their definition given by the form of Lorentz force.^{ [26] } However the following equation is only applicable when no acceleration is involved in the particle's history where Coulomb's law can be considered or symmetry arguments can be used for solving Maxwell's equations in a simple manner. The electric field of such a uniformly moving point charge is hence given by:^{ [27] }

where is the charge of the point source, is the position vector from the point source to the point in space, is the ratio of observed speed of the charge particle to the speed of light and is the angle between 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. Spherically 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.^{ [25] }

Special theory of relativity imposes the principle of locality, that requires cause and effect to be time-like separated events where the causal efficacy does not travel faster than the speed of light.^{ [28] } 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 that electromagnetic disturbances travel at the speed of light. Advanced time, which also provides a solution for maxwell's law are ignored as an unphysical solution.

For the motion of a charged 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 the speed 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 in electromagnetic field, since charged particles are restricted to have speeds slower than that of light, which makes it impossible to construct a gaussian surface in this region that violates gauss' 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, an electromagnetic pulse of radiation is generated that connects at the boundary of this disturbance travelling outwards at the speed of light.^{ [29] } In general, any accelerating point charge radiates electromagnetic waves however, non radiating acceleration is possible in a systems of charges.

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 using Liénard–Wiechert potential.^{ [30] } Since the potentials satisfy maxwell's equations, the fields derived for point charge also satisfy Maxwell's equations. The electric field is expressed as:^{ [31] }

where is the charge of the point source, is retarded time or the time at which the source's contribution of the electric field originated, is the position vector of the particle, is a unit vector pointing from charged particle to the point in space, is the velocity of the particle divided by the speed of light, and is the corresponding Lorentz factor. The retarded time is given as solution of:

The uniqueness of solution for for given , and 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 for Larmor formula is obtained.^{ [31] }

There exist yet another set of solutions for maxwell's equation of the same form but for advanced time instead of retarded time given as a solution of:

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 of maxwell's equations, such as Feynman 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.

- Infinite Wire having Uniform charge density has Electric Field at a distance from it as
- Infinitely large surface having charge density has Electric Field at a distance from it as
- Infinitely long cylinder having Uniform charge density that is charge contained along unit length of the cylinder has Electric Field at a distance from it as while it is everywhere inside the cylinder
- Uniformly Charged non-conducting sphere of radius , volume charge density and total charge has Electric Field at a distance from it as while the electric field at a point inside sphere from its center is given by
- Uniformly Charged conducting sphere of radius , surface charge density and total charge has Electric Field at a distance from it as while the electric field inside is
- Electric field infinitely close to a conducting surface in electrostatic equilibrium having charge density at that point is
- Uniformly Charged Ring having total charge has Electric Field at a distance along its axis as '
- Uniformly charged disc of radius and charge density has Electric Field at a distance along its axis from it as
- Electric field due to dipole of dipole moment at a distance from their center along equatorial plane is given as and the same along the axial line is approximated to for much bigger than the distance between dipoles. Further generalization is given by multipole expansion.

- Classical electromagnetism
- Relativistic Electromagnetism
- Electricity
- History of electromagnetic theory
- Optical field
- Magnetism
- Teltron tube
- Teledeltos, a conductive paper that may be used as a simple analog computer for modelling fields

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**, or **Maxwell–Heaviside 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 modern form of the equations in their most common formulation is credited to Oliver Heaviside.

**Flux** describes any effect that appears to pass or travel through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.

The **electric potential** is defined as 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. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. Furthermore, the motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used.

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.

In physics, **screening** is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

**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.

**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 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.

**Gaussian units** constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the **Gaussian unit system**, **Gaussian-cgs units**, or often just **cgs units**. The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of cgs with conflicting definitions of electromagnetic quantities and units.

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 intended to describe electromagnetism and gravitation, two of the fundamental forces of nature.

**Vacuum permittivity**, commonly denoted ** ε_{0}**, is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the

In electromagnetism, **charge density** is the amount of electric charge per unit length, surface area, or volume. **Volume charge density** is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m^{−3}), at any point in a volume. **Surface charge density** (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m^{−2}), at any point on a surface charge distribution on a two dimensional surface. **Linear charge density** (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

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.

In electromagnetism, **Jefimenko's equations** give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay of the fields due to the finite speed of light and relativistic effects. Therefore they can be used for *moving* charges and currents. They are the particular solutions to Maxwell's equations for any arbitrary distribution of charges and currents.

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.

**Coulomb's inverse-square law**, or simply **Coulomb's 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 unit for electric dipole moment is the coulomb-meter (C⋅m). The debye (D) is another unit of measurement used in atomic physics and chemistry.

- ↑ Roche, John (2016). "Introducing electric fields".
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- ↑ 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.* - ↑ Richard Feynman (1970).
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*Introduction to electrodynamics*(3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-805326-X. OCLC 40251748. - ↑ 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." - ↑ Purcell, p 356: "Faraday's Law of Induction."
- ↑ 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. - ↑ Purcell, Edward (2011).
*Electricity and Magnetism*(2nd ed.). Cambridge University Press. pp. 8–9. ISBN 978-1139503556. - ↑ gwrowe (8 October 2011). "Curl & Potential in Electrostatics" (PDF).
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*New Scientist*.**72**: 652. - ↑ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
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- ↑
*Electricity and Modern Physics (2nd Edition)*, G.A.G. Bennet, Edward Arnold (UK), 1974, ISBN 0-7131-2459-8 - ↑ 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. - 1 2 Purcell, Edward M.; Morin, David J. (2013-01-21).
*Electricity and Magnetism*.*Higher Education from Cambridge University Press*. pp. 241–251. doi:10.1017/cbo9781139012973. ISBN 9781139012973 . Retrieved 2022-07-04. - ↑ 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. - ↑ Heaviside, Oliver.
*Electromagnetic waves, the propagation of potential, and the electromagnetic effects of a moving charge*. - ↑ 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. - ↑ Purcell, Edward M.; David J. Morin (2013).
*Electricity and Magnetism*(Third ed.). Cambridge. pp. 251–255. ISBN 978-1-139-01297-3. OCLC 1105718330. - ↑ Griffiths, David J. (2017).
*Introduction to electrodynamics*(4th ed.). United Kingdom: Cambridge University Press. p. 454. ISBN 978-1-108-42041-9. OCLC 1021068059. - 1 2 Jackson, 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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