Electric field

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 ${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}$ and Faraday's law with no induction term ${\displaystyle \nabla \times \mathbf {E} =0}$), taken together, are equivalent to Coulomb's law, which states that a particle with electric charge ${\displaystyle q_{1}}$ at position ${\displaystyle \mathbf {x} _{1}}$ exerts a force on a particle with charge ${\displaystyle q_{0}}$ at position ${\displaystyle \mathbf {x} _{0}}$ of:[15]

$${\displaystyle \mathbf {F} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{0}}{(\mathbf {x} _{1}-\mathbf {x} _{0})^{2}}}{\hat {\mathbf {r} }}_{1,0}\,,}$$

where ${\displaystyle {\hat {\mathbf {r} }}_{1,0}}$ is the unit vector in the direction from point ${\displaystyle \mathbf {x} _{1}}$ to point ${\displaystyle \mathbf {x} _{0}}$, and ε₀ is the electric constant (also known as "the absolute permittivity of free space") with the unit C²⋅m⁻²⋅N⁻¹.

Note that ${\displaystyle \varepsilon _{0}}$, the vacuum electric permittivity, must be substituted with ${\displaystyle \varepsilon }$, permittivity, when charges are in non-empty media. When the charges ${\displaystyle q_{0}}$ and ${\displaystyle q_{1}}$ 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 the Coulomb force on any charge at position ${\displaystyle \mathbf {x} _{0}}$ this expression can be divided by ${\displaystyle q_{0}}$ leaving an expression that only depends on the other charge (the source charge)[16][6]

$${\displaystyle \mathbf {E} (\mathbf {x} _{0})={\frac {\mathbf {F} }{q_{0}}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}}{(\mathbf {x} _{1}-\mathbf {x} _{0})^{2}}}{\hat {\mathbf {r} }}_{1,0}}$$

This is the electric field at point ${\displaystyle \mathbf {x} _{0}}$ due to the point charge ${\displaystyle q_{1}}$; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position ${\displaystyle \mathbf {x} _{0}}$. Since this formula gives the electric field magnitude and direction at any point ${\displaystyle \mathbf {x} _{0}}$ in space (except at the location of the charge itself, ${\displaystyle \mathbf {x} _{1}}$, 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 ${\displaystyle q}$ at any point in space is equal to the product of the charge and the electric field at that point

$${\displaystyle \mathbf {F} =q\mathbf {E} }$$

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 ${\displaystyle q_{1},q_{2},\dots ,q_{n}}$ are stationary in space at points ${\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\dots ,\mathbf {x} _{n}}$, 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:

$${\displaystyle {\begin{aligned}\mathbf {E} (\mathbf {x} )&=\mathbf {E} _{1}(\mathbf {x} )+\mathbf {E} _{2}(\mathbf {x} )+\mathbf {E} _{3}(\mathbf {x} )+\cdots \\[2pt]&={1 \over 4\pi \varepsilon _{0}}{q_{1} \over (\mathbf {x} _{1}-\mathbf {x} )^{2}}{\hat {\mathbf {r} }}_{1}+{1 \over 4\pi \varepsilon _{0}}{q_{2} \over (\mathbf {x} _{2}-\mathbf {x} )^{2}}{\hat {\mathbf {r} }}_{2}+{1 \over 4\pi \varepsilon _{0}}{q_{3} \over (\mathbf {x} _{3}-\mathbf {x} )^{2}}{\hat {\mathbf {r} }}_{3}+\cdots \\[2pt]&={1 \over 4\pi \varepsilon _{0}}\sum _{k=1}^{N}{q_{k} \over (\mathbf {x} _{k}-\mathbf {x} )^{2}}{\hat {\mathbf {r} }}_{k}\end{aligned}}}$$

where ${\displaystyle \mathbf {{\hat {r}}_{k}} }$ is the unit vector in the direction from point ${\displaystyle \mathbf {x} _{k}}$ to point ${\displaystyle \mathbf {x} }$.

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

$${\displaystyle d\mathbf {E} (\mathbf {x} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {\rho (\mathbf {x} ')dV}{(\mathbf {x} '-\mathbf {x} )^{2}}}{\hat {\mathbf {r} }}'}$$

where ${\displaystyle {\hat {\mathbf {r} }}'}$ is the unit vector pointing from ${\displaystyle \mathbf {x} '}$ to ${\displaystyle \mathbf {x} }$. 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 ${\displaystyle V}$:

$${\displaystyle \mathbf {E} (\mathbf {x} )={\frac {1}{4\pi \varepsilon _{0}}}\iiint _{V}\,{\rho (\mathbf {x} ')dV \over (\mathbf {x} '-\mathbf {x} )^{2}}{\hat {\mathbf {r} }}'}$$

Similar equations follow for a surface charge with continuous charge distribution ${\displaystyle \sigma (\mathbf {x} )}$ where ${\displaystyle \sigma }$ is the charge density in coulombs per square meter

$${\displaystyle \mathbf {E} (\mathbf {x} )={\frac {1}{4\pi \varepsilon _{0}}}\iint _{S}\,{\sigma (\mathbf {x} ')dA \over (\mathbf {x} '-\mathbf {x} )^{2}}{\hat {\mathbf {r} }}'}$$

and for line charges with continuous charge distribution ${\displaystyle \lambda (\mathbf {x} )}$ where ${\displaystyle \lambda }$ is the charge density in coulombs per meter.

$${\displaystyle \mathbf {E} (\mathbf {x} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{P}\,{\lambda (\mathbf {x} ')dL \over (\mathbf {x} '-\mathbf {x} )^{2}}{\hat {\mathbf {r} }}'}$$

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 ${\displaystyle \Phi }$ such that ${\displaystyle \mathbf {E} =-\nabla \Phi }$.[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 ${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$, one can still define an electric potential ${\displaystyle \Phi }$ such that:

$${\displaystyle \mathbf {E} =-\nabla \Phi -{\frac {\partial \mathbf {A} }{\partial t}}}$$

where ${\displaystyle \nabla \Phi }$ is the gradient of the electric potential and ${\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}}$ 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 [18]

$${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial (\nabla \times \mathbf {A} )}{\partial t}}=-{\frac {\partial \mathbf {B} }{\partial t}}}$$

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

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

$${\displaystyle \mathbf {F} =q\left({\frac {Q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {\hat {r}} }{|\mathbf {r} |^{2}}}\right)=q\mathbf {E} }$$

is similar to Newton's law of universal gravitation:

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

(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".[20]

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

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:

$${\displaystyle E=-{\frac {\Delta V}{d}}}$$

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.

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

$${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }$$

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.

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:

$${\displaystyle \mathbf {D} (\mathbf {r} )=\varepsilon \mathbf {E} (\mathbf {r} )}$$

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

$${\displaystyle \mathbf {D} (\mathbf {r} )=\varepsilon (\mathbf {r} )\mathbf {E} (\mathbf {r} )}$$

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:

$${\displaystyle D_{i}=\varepsilon _{ij}E_{j}}$$

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

The electric field given by Coulomb's law for a point charge particle at rest is found to preserve the form of Gauss's law under Lorentz transformation consistent with the first postulate of relativity. Hence the invariance of form of Maxwell's equations under Lorentz transformation can be used to derive the electric field of uniformly moving point charges from that of a stationary charge.[25] Alternatively the electric field of uniformly moving point charges can be derived from the Lorentz transformation of four-force experienced by 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] Either method is only considered when no acceleration is involved in the particle's history. This limited range of application can be attributed to the fact that Maxwell's equations are considered as physical laws for classical electrodynamics and are more universal than relativistic transformations of Coulomb's law alone. The particular solution can be shown to satisfy maxwell's equations indicating the validity of Coulomb's law within its specific range of application. Compatibility of Maxwell's equations with relativity however, are given by their covariant formulation. The charge of the particle is also considered invariant of reference frame, as supported by experimental evidence.[25] The electric field of such a uniformly moving point charge is given by:[27]

$${\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}\mathbf {r} }$$

where ${\displaystyle q}$ is the charge of the point source, ${\displaystyle \mathbf {r} }$ is the position vector from the point source to the point in space, ${\displaystyle \beta }$ is the ratio of observed speed of the charge particle to the speed of light and ${\displaystyle \theta }$ is the angle between ${\displaystyle \mathbf {r} }$ 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.

For arbitrarily moving point charges, propagation of potential fields such as Lorenz gauge fields at the speed of light needs to be accounted for, using Liénard–Wiechert potential.[28] Since the electric and magnetic fields may be expressed in terms of retarded potentials, the fields derived for point charge also satisfy Maxwell's equations. The electric field is expressed as:[29]

$${\displaystyle \mathbf {E} (\mathbf {r} ,\mathbf {t} )={\frac {1}{4\pi \varepsilon _{0}}}\left({\frac {q(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})}{\gamma ^{2}(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|^{2}}}+{\frac {q\mathbf {n} _{s}\times {\big (}(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})\times {\dot {{\boldsymbol {\beta }}_{s}}}{\big )}}{c(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}}$$

where ${\displaystyle q}$ is the charge of the point source, ${\textstyle {t_{r}}}$ is retarded time or the time at which the source's contribution of the electric field originated, ${\textstyle {r}_{s}(t)}$ is the position vector of the particle, ${\textstyle {n}_{s}(\mathbf {r} ,t)}$ is a unit vector pointing from charged particle to the point in space, ${\textstyle {\boldsymbol {\beta }}_{s}(t)}$ is the velocity of the particle divided by the speed of light, and ${\textstyle \gamma (t)}$ is the corresponding Lorentz factor. The retarded time is given as solution of:

${\displaystyle t_{r}=\mathbf {t} -{\frac {|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}{c}}}$

The uniqueness of solution for ${\textstyle {t_{r}}}$ for given ${\displaystyle \mathbf {t} }$, ${\displaystyle \mathbf {r} }$ and ${\displaystyle r_{s}(t)}$ 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 for Larmor formula is obtained.[29]

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