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]

$${\displaystyle \mathbf {F} _{01}={\frac {q_{1}q_{0}}{4\pi \varepsilon _{0}}}{\frac {{\hat {\mathbf {r} }}_{01}}{r_{01}^{2}}}}$$

Where

• ${\displaystyle \mathbf {F} _{01}}$ is the force on charged particle ${\displaystyle q_{0}}$ caused by charged particle ${\displaystyle q_{1}}$.
• ε₀ is the permittivity of free space.
• ${\displaystyle {\hat {\mathbf {r} }}_{01}}$ is a unit vector directed from ${\displaystyle \mathbf {x} _{1}}$ to ${\displaystyle \mathbf {x} _{0}}$.
• ${\displaystyle r_{01}}$ is the distance from ${\displaystyle \mathbf {x} _{1}}$ to ${\displaystyle \mathbf {x} _{0}}$.

Note that ${\displaystyle \varepsilon _{0}}$ must be replaced 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]

$${\displaystyle \mathbf {E} _{1}(\mathbf {x} _{0})={\frac {\mathbf {F} _{01}}{q_{0}}}={\frac {q_{1}}{4\pi \varepsilon _{0}}}{\frac {{\hat {\mathbf {r} }}_{01}}{r_{01}^{2}}}}$$

Where

• ${\displaystyle \mathbf {E} _{1}(\mathbf {x} _{0})}$ is the component of the electric field at ${\displaystyle q_{0}}$ due to ${\displaystyle q_{1}}$.

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:

$${\displaystyle {\begin{aligned}\mathbf {E} (\mathbf {x} )=\mathbf {E} _{1}(\mathbf {x} )+\mathbf {E} _{2}(\mathbf {x} )+\mathbf {E} _{3}(\mathbf {x} )+\cdots ={\frac {1}{4\pi \varepsilon _{0}}}\sum _{k=1}^{N}q_{k}{\frac {{\hat {\mathbf {r} }}_{k}}{r_{k}^{2}}}\end{aligned}}}$$

where

• ${\displaystyle \mathbf {\hat {r}} _{k}}$ is the unit vector in the direction from point ${\displaystyle \mathbf {x} _{k}}$ to point ${\displaystyle \mathbf {x} }$
• ${\displaystyle r_{k}}$ is the distance 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 distribution of charge density${\displaystyle \rho (\mathbf {x} )}$. 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

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

where

• ${\displaystyle {\hat {\mathbf {r} }}'}$ is the unit vector pointing from ${\displaystyle \mathbf {x} '}$ to ${\displaystyle \mathbf {x} }$.
• ${\displaystyle r'}$ is the distance from ${\displaystyle \mathbf {x} '}$ to ${\displaystyle \mathbf {x} }$.

The total field is found by summing the contributions from all the increments of volume by integrating the charge density over the volume ${\displaystyle V}$:

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

Similar equations follow for a surface charge with surface charge density ${\displaystyle \sigma (\mathbf {x'} )}$ on surface ${\displaystyle S}$

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

and for line charges with linear charge density ${\displaystyle \lambda (\mathbf {x'} )}$ on line ${\displaystyle L}$

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

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 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]

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

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. 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.

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 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]

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

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.[31]

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

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

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 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.