An electric field (sometimes called E-field) is a physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) describes their capacity to exert attractive or repulsive forces on another charged object. Charged particles exert attractive forces on each other when the sign of their charges are opposite, one being positive while the other is negative, and repel each other when the signs of the charges are the same. Because these forces are exerted mutually, two charges must be present for the forces to take place. These forces are described by Coulomb's law, which says that the greater the magnitude of the charges, the greater the force, and the greater the distance between them, the weaker the force. Informally, the greater the charge of an object, the stronger its electric field. Similarly, an electric field is stronger nearer charged objects and weaker further away. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field. Electromagnetism is one of the four fundamental interactions of nature. Electric fields are important in many areas of physics, and are exploited in electrical technology. For example, in atomic physics and chemistry, the interaction in the electric field between the atomic nucleus and electrons is the force that holds these particles together in atoms. Similarly, the interaction in the electric field between atoms is the force responsible for chemical bonding that result in molecules. The electric field is defined as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for the electric field is the volt per meter (V/m), which is equal to the newton per coulomb (N/C).
Description
The electric field is defined at each point in space as the force that would be experienced by an infinitesimally small stationary positive test charge at that point divided by the charge. The electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), so it follows that an electric field may be described by a vector field. The electric field acts between two charges similarly to the way that 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 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 those of the field, a concept introduced by Michael Faraday, whose term 'lines of force' is still sometimes used. This illustration has the useful property that, when drawn so that each line represents the same amount of flux, the strength of the field is proportional to the density of the lines. Field lines due to stationary charges have several important properties, including that they always originate from positive charges and terminate at negative charges, they enter all good conductors at right angles, and they never cross or close in on themselves. 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. In the absence of time-varying magnetic field, the electric field is therefore called conservative (i.e. curl-free). This implies there are two kinds of electric fields: electrostatic fields and fields arising from time-varying magnetic fields. 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 magnetic and electric fields that change over time 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.
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 ∇ ⋅ E = ρ ε 0 {\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}} and Faraday's law with no induction term ∇ × E = 0 {\displaystyle \nabla \times \mathbf {E} =0} ), taken together, are equivalent to Coulomb's law, which states that a particle with electric charge q 1 {\displaystyle q_{1}} at position r 1 {\displaystyle \mathbf {r} _{1}} exerts a force on a particle with charge q 0 {\displaystyle q_{0}} at position r 0 {\displaystyle \mathbf {r} _{0}} of:
F 01 = q 1 q 0 4 π ε 0 r ^ 01 | r 01 | 2 = q 1 q 0 4 π ε 0 r 01 | r 01 | 3 {\displaystyle \mathbf {F} _{01}={\frac {q_{1}q_{0}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{01} \over {|\mathbf {r} _{01}|}^{2}}={\frac {q_{1}q_{0}}{4\pi \varepsilon _{0}}}{\mathbf {r} _{01} \over {|\mathbf {r} _{01}|}^{3}}}
where
F 01 {\displaystyle \mathbf {F} _{01}} is the force on charged particle q 0 {\displaystyle q_{0}} caused by charged particle q 1 {\displaystyle q_{1}} . ε0 is the permittivity of free space.
r ^ 01 {\displaystyle {\hat {\mathbf {r} }}_{01}} is a unit vector directed from r 1 {\displaystyle \mathbf {r} _{1}} to r 0 {\displaystyle \mathbf {r} _{0}} .
r 01 {\displaystyle \mathbf {r} _{01}} is the displacement vector from r 1 {\displaystyle \mathbf {r} _{1}} to r 0 {\displaystyle \mathbf {r} _{0}} . Note that ε 0 {\displaystyle \varepsilon _{0}} must be replaced with ε {\displaystyle \varepsilon } , permittivity, when charges are in non-empty media. When the charges q 0 {\displaystyle q_{0}} and q 1 {\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 r 0 {\displaystyle \mathbf {r} _{0}} this expression can be divided by q 0 {\displaystyle q_{0}} leaving an expression that only depends on the other charge (the source charge)
E 1 ( r 0 ) = F 01 q 0 = q 1 4 π ε 0 r ^ 01 | r 01 | 2 = q 1 4 π ε 0 r 01 | r 01 | 3 {\displaystyle \mathbf {E} _{1}(\mathbf {r} _{0})={\frac {\mathbf {F} _{01}}{q_{0}}}={\frac {q_{1}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{01} \over {|\mathbf {r} _{01}|}^{2}}={\frac {q_{1}}{4\pi \varepsilon _{0}}}{\mathbf {r} _{01} \over {|\mathbf {r} _{01}|}^{3}}}
where E 1 ( r 0 ) {\displaystyle \mathbf {E} _{1}(\mathbf {r} _{0})} is the component of the electric field at q 0 {\displaystyle q_{0}} due to q 1 {\displaystyle q_{1}} . This is the electric field at point r 0 {\displaystyle \mathbf {r} _{0}} due to the point charge q 1 {\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 r 0 {\displaystyle \mathbf {r} _{0}} . Since this formula gives the electric field magnitude and direction at any point r 0 {\displaystyle \mathbf {r} _{0}} in space (except at the location of the charge itself, r 1 {\displaystyle \mathbf {r} _{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 {\displaystyle q} at any point in space is equal to the product of the charge and the electric field at that point
F = q E . {\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.
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 {\displaystyle q_{1},q_{2},\dots ,q_{n}} are stationary in space at points R 1 , R 2 , … , R n {\displaystyle \mathbf {R} _{1},\mathbf {R} _{2},\dots ,\mathbf {R} _{n}} , in the absence of currents, the superposition principle says that the resulting field is the sum of fields generated by each particle as described by Coulomb's law:
E ( r ) = E 1 ( r ) + E 2 ( r ) + ⋯ + E n ( r ) = 1 4 π ε 0 ∑ i = 1 n q i r ^ i | r i | 2 = 1 4 π ε 0 ∑ i = 1 n q i r i | r i | 3 {\displaystyle {\begin{aligned}\mathbf {E} (\mathbf {r} )=\mathbf {E} _{1}(\mathbf {r} )+\mathbf {E} _{2}(\mathbf {r} )+\dots +\mathbf {E} _{n}(\mathbf {r} )={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {|\mathbf {r} _{i}|}^{3}}\end{aligned}}}
where
r ^ i {\displaystyle {\hat {\mathbf {r} }}_{i}} is the unit vector in the direction from point R i {\displaystyle \mathbf {R} _{i}} to point r {\displaystyle \mathbf {r} }
r i {\displaystyle \mathbf {r} _{i}} is the displacement vector from point R i {\displaystyle \mathbf {R} _{i}} to point r {\displaystyle \mathbf {r} } .
Continuous charge distributions The superposition principle allows for the calculation of the electric field due to a distribution of charge density ρ ( r ) {\displaystyle \rho (\mathbf {r} )} . By considering the charge ρ ( r ′ ) d v {\displaystyle \rho (\mathbf {r} ')dv} in each small volume of space d v {\displaystyle dv} at point r ′ {\displaystyle \mathbf {r} '} as a point charge, the resulting electric field, d E ( r ) {\displaystyle d\mathbf {E} (\mathbf {r} )} , at point r {\displaystyle \mathbf {r} } can be calculated as
d E ( r ) = ρ ( r ′ ) 4 π ε 0 r ^ ′ | r ′ | 2 d v = ρ ( r ′ ) 4 π ε 0 r ′ | r ′ | 3 d v {\displaystyle d\mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} ')}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}' \over {|\mathbf {r} '|}^{2}}dv={\frac {\rho (\mathbf {r} ')}{4\pi \varepsilon _{0}}}{\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}dv}
where
r ^ ′ {\displaystyle {\hat {\mathbf {r} }}'} is the unit vector pointing from r ′ {\displaystyle \mathbf {r} '} to r {\displaystyle \mathbf {r} } .
r ′ {\displaystyle \mathbf {r} '} is the displacement vector from r ′ {\displaystyle \mathbf {r} '} to r {\displaystyle \mathbf {r} } . The total field is found by summing the contributions from all the increments of volume by integrating the charge density over the volume V {\displaystyle V} :
E ( r ) = 1 4 π ε 0 ∭ V ρ ( r ′ ) r ′ | r ′ | 3 d v {\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\iiint _{V}\,\rho (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}dv}
Similar equations follow for a surface charge with surface charge density σ ( r ′ ) {\displaystyle \sigma (\mathbf {r} ')} on surface S {\displaystyle S}
E ( r ) = 1 4 π ε 0 ∬ S σ ( r ′ ) r ′ | r ′ | 3 d a , {\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\iint _{S}\,\sigma (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}da,}
and for line charges with linear charge density λ ( r ′ ) {\displaystyle \lambda (\mathbf {r} ')} on line L {\displaystyle L}
E ( r ) = 1 4 π ε 0 ∫ L λ ( r ′ ) r ′ | r ′ | 3 d ℓ . {\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{L}\,\lambda (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}d\ell .}
Electric potential
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 φ {\displaystyle \varphi } such that E = − ∇ φ {\displaystyle \mathbf {E} =-\nabla \varphi } . 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 B = ∇ × A {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} } , one can still define an electric potential φ {\displaystyle \varphi } such that:
E = − ∇ φ − ∂ A ∂ t , {\displaystyle \mathbf {E} =-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}},}
where ∇ φ {\displaystyle \nabla \varphi } is the gradient of the electric potential and ∂ A ∂ t {\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
∇ × E = − ∂ ( ∇ × A ) ∂ t = − ∂ B ∂ t , {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial (\nabla \times \mathbf {A} )}{\partial t}}=-{\frac {\partial \mathbf {B} }{\partial t}},}
which justifies, a posteriori, the previous form 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 {\displaystyle q} located at r 0 {\displaystyle \mathbf {r} _{0}} can be described mathematically as a charge density ρ ( r ) = q δ ( r − r 0 ) {\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.
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:
F = q ( Q 4 π ε 0 r ^ | r | 2 ) = q E {\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:
F = m ( − G M r ^ | r | 2 ) = m g {\displaystyle \mathbf {F} =m\left(-GM{\frac {\mathbf {\hat {r}} }{|\mathbf {r} |^{2}}}\right)=m\mathbf {g} }
(where r ^ = r | r | {\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
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; i