Electric field

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Electric field produced by a set of punctual loads. The vectorial sum of the individual load fields is shown in pink;

{displaystyle {vec {E}}={vec {E}}_{1}+{vec {E}}_{2}+{vec {E}}_{3}}

The electric field (region of space in which electrical force interacts) is a physical field that is represented by a model that describes the interaction between bodies and systems with properties of an electrical nature. It can be described as a vector field in which a punctual electrical charge of value $q$ suffers the effects of an electrical force ${mathbf {F}}$ given by the following equation:

(1) ${displaystyle mathbf {F} =qmathbf {E} }$

In current relativistic models, the electric field is incorporated, together with the magnetic field, in a four-dimensional tensor field, called the F^μν electromagnetic field.

Electric fields can originate from both electric charges and changing magnetic fields. The first descriptions of electrical phenomena, such as Coulomb's law, only took electric charges into account, but the investigations of Michael Faraday and the later studies of James Clerk Maxwell made it possible to establish the complete laws that also take into account the electrical charge. variation of the magnetic field.

This general definition indicates that the field is not directly measurable, but what is observable is its effect on some charge placed within it. The idea of electric field was proposed by Faraday when demonstrating the principle of electromagnetic induction in the year 1832.

The SI unit of electric field is Newton per Coulomb (N/C), Volt per meter (V/m), or, in base units, kg m s⁻³ A⁻¹, and the dimensional equation is MLT^-3I^-1.

Definition

The presence of electric charge in a region of space modifies the characteristics of that space, giving rise to an electric field. Thus, we can consider an electric field as a region of space whose properties have been modified by the presence of an electric charge, in such a way that when a new electric charge is introduced into said electric field, it will experience a force.

The electric field is represented mathematically by the electric field vector, defined as the quotient between the electric force experienced by a witness charge and the value of that witness charge (a positive witness charge).

The most intuitive definition of the electric field can be given by Coulomb's law. This law, once generalized, makes it possible to express the field between charge distributions at relative rest. However, for moving charges a more formal and complete definition is required; the use of quadrivectors and the principle of least action are required. Both are described below.

It must be kept in mind, however, that from the relativistic point of view, the definition of electric field is relative and not absolute, since observers moving relative to each other will measure electric fields or "electrical parts" of the electromagnetic field are different, so the measured electric field will depend on the chosen reference system.

Definition using Coulomb's law

Electric field of a linear load distribution. A punctual charge P is subjected to a force in radial direction

{displaystyle {vec {u}}_{r}}

by a load distribution

lambda

in line differential form (

{displaystyle dL}

), which produces an electric field

{displaystyle d{vec {E}}}

Starting from Coulomb's law, which expresses that the force between two charges at relative rest depends on the square of the distance, mathematically it is equal to:

${displaystyle mathbf {F} _{12}={frac {1}{4pi epsilon _{0}}}{Bigl [}{frac {q_{1} q_{2}}{(r_{12})^{2}}}{Bigr ]}{hat {mathbf {r} }}_{12}}$

Where:

{displaystyle scriptstyle epsilon _{0}}

is the electric permitivity of the vacuum, constantly defined in the international system;

{displaystyle q_{1}, q_{2}}

are the burdens that interact;

{displaystyle r_{12}=|mathbf {r} _{12}|}

is the distance between both loads;

{displaystyle mathbf {r} _{12}}

is the relative position vector of load 2 compared to load 1; and

{displaystyle {hat {mathbf {r} }}}

is the unit in the direction

{displaystyle {vec {mathbf {r} }}}

Note that the formula is being used $epsilon _{0}$ ; this is the allowivity in the void. To calculate the interaction in another medium it is necessary to change the allowivity of that medium. ( ${displaystyle epsilon =epsilon _{r}epsilon _{0}}$ )

The previous law presupposed that the position of a particle at a given instant causes its electric field to affect at the same instant any other charge. That type of interactions in which the effect on the rest of the particles seems to depend only on the position of the causing particle, regardless of the distance between the particles, is called action at a distance in physics. Although the notion of action at a distance was initially accepted by Newton himself, more careful experiments throughout the 19th century led to his dismissing the notion as unrealistic. In this context, it was thought that the electric field was not only a mathematical artifice but a physical entity that propagates at a finite speed (the speed of light) until it affects other particles. This idea entailed modifying Coulomb's law in accordance with the requirements of the theory of relativity and endowing the electric field with a physical entity. Thus, the electric field is an electromagnetic distortion suffered by space-time due to the presence of a charge. Considering this, an expression of the electric field can be obtained when it only depends on the distance between the charges (electrostatic case):

(7) ${displaystyle mathbf {E} ={frac {1}{4pi epsilon _{0}}}{frac {q}{r^{2}}}{hat {mathbf {r} }}}$

Where clearly you have to ${displaystyle scriptstyle mathbf {F} =qmathbf {E} }$ , which is one of the best known definitions about the electric field. For a continuous distribution of loads the electric field is given by:

${displaystyle mathbf {E} (mathbf {r})={frac {1}{4pi epsilon _{0}}}int _{V}{frac {rho (mathbf {r} ')}{|mathbf {r} -mathbf {r} '|^{3}}}(mathbf {r} -mathbf {r} '){text{d}}^{3}mathbf {r} '}$

Formal definition

The most formal definition of electric field, also valid for loads moving at speeds close to that of light, arises from calculating the action of a particle loaded in motion through an electromagnetic field. This field is part of a single tensorial electromagnetic field ${displaystyle F^{mu nu }}$ defined by a quadritorial potential of the form:

(1) ${displaystyle F^{mu nu }=partial ^{mu }A^{nu }-partial ^{nu }A^{mu }quad;qquad A^{i}=left({frac {phi }{c}},mathbf {A} right)}$

where $phi$ is the potential to scale and ${displaystyle scriptstyle mathbf {A} }$ is the three-dimensional vector potential. Thus, according to the principle of minimum action, it is proposed for a motion particle in a four-dimensional space:

(2) ${displaystyle S=-int _{a}^{b}(mc {text{d}}s+{frac {e}{c}}A_{i}{text{d}}x^{i})}$

where $e$ is the load of the particle, $m$ It's his mass. $c$ the speed of light. Replacement (1) in (2) and knowing that ${displaystyle {text{d}}x^{i}=u^{i}{text{d}}s}$ Where ${displaystyle {text{d}}x^{i}}$ is the differential of the defined position ${displaystyle {text{d}}x^{i}=(c{text{d}}t,{text{d}}x,{text{d}}y,{text{d}}z)}$ and ${displaystyle u^{i}}$ is the velocity of the particle, you get:

(3) ${displaystyle S=-int _{a}^{b}(mc {text{d}}s+{frac {e}{c}}mathbf {A} cdot {text{d}}mathbf {text{r}} -ephi {text{d}}t)}$

The term inside the integral is known as the Lagrangian of the system; Deriving this expression with respect to velocity, the momentum of the particle is obtained, and applying the Euler-Lagrange equations, it is found that the temporal variation of the momentum of the particle is:

(4) ${displaystyle {frac {{text{d}}mathbf {p} }{{text{d}}t}}=-{frac {e}{c}}{frac {partial mathbf {A} }{partial t}}-e{boldsymbol {nabla }}phi +{frac {e}{c}}mathbf {v} times ({boldsymbol {nabla }}times mathbf {A})}$

From where the total force of the particle is obtained. The first two terms are independent of the particle's velocity, while the last one depends on it. Then the first two are associated with the electric field and the third with the magnetic field. Here is the most general definition for the electric field:

(5) ${displaystyle mathbf {E} =-{frac {1}{c}}{frac {partial mathbf {A} }{partial t}}-{boldsymbol {nabla }}phi }$

The equation (5) provides a lot of information about the electric field. On the one hand, the first term indicates that an electric field is produced by the temporal variation of a vector potential described as ${displaystyle scriptstyle mathbf {B} ={boldsymbol {nabla }}times mathbf {A} }$ Where ${displaystyle scriptstyle mathbf {B} }$ is the magnetic field; and on the other, the second represents the well-known description of the field as the gradient of a potential.

Description of the electric field

Mathematically, a field is described by two of its properties: its divergence and its curl. The equation that describes the divergence of the electric field is known as Gauss's law and that of its curl is Faraday's law.

Gauss's Law

Main article: Law of Gauss

To know one of the properties of the electric field is studied what happens with the flow of this by crossing a closed "gaussian" surface, that is to say a surface such that in every infinitesimal piece of surface is well defined its orientation. The flow of a field $Phi$ is obtained as follows:

(8) ${displaystyle Phi _{E}=oint _{S}{vec {E}}cdot {text{d}}{vec {a}}}$

where ${displaystyle d{vec {a}}}$ is the area differential in the normal direction to the surface. Applying the equation (7) in (8) and analyzing the flow through a closed surface is found that:

(9) ${displaystyle oint _{S}{vec {E}}cdot {text{d}}{vec {a}}={frac {1}{epsilon _{0}}}Q_{enc}}$

where ${displaystyle Q_{enc}}$ It's the load locked on that surface. The equation (9) is known as the integral law of Gauss and its derivative form is:

(10) ${displaystyle {vec {nabla }}cdot {vec {E}}={frac {rho }{epsilon _{0}}}}$

where $rho$ is the volumetric density of load. This indicates that the electric field dives into a load distribution; in other words, that the electric field begins in one load and ends in another.

This idea can be visualized through the concept of field lines. If you have a charge at one point, the electric field would be directed toward the other charge.

Faraday's Law

Main article: Faraday Law

In 1821, Michael Faraday conducted a series of experiments that led him to determine that temporary changes in the magnetic field induce an electric field. This is known as Faraday's law. The electromotive force, defined as the curl across a line differential is determined by:

(11) ${displaystyle {mathcal {E}}=oint {vec {E}}cdot {text{d}}{vec {text{l}}}=-{frac {dPhi }{dt}}}$

where the less sign indicates the Law of Lenz and $Phi$ is the magnetic flow on a surface determined by:

(12) ${displaystyle Phi =int {vec {B}}cdot {text{d}}{vec {text{a}}}}$

replacing (12) in (11) yields the integral equation of Faraday's law:

(13) ${displaystyle oint {vec {E}}cdot {text{d}}{vec {text{l}}}=-int {frac {d{vec {B}}}{dt}}cdot {text{d}}{vec {text{a}}}}$

Applying Stokes' theorem we find the differential form:

(14) ${displaystyle {vec {nabla }}times {vec {E}}=-{frac {partial {vec {B}}}{partial {text{t}}}}}$

The equation (14) completes the description of the electric field, indicating that the temporal variation of the magnetic field induces an electric field.

Electric field expressions

Electrostatic field (charges at rest)

Main article: Electrostatic field

A special case of the electric field is called electrostatic. An electrostatic field does not depend on time, that is, it is stationary. For this type of fields, Gauss's Law is still valid because it does not have any temporal consideration, however, Faraday's Law must be modified. If the field is stationary, the right hand side of the equation (13) and (14 ) does not make sense, so it is overridden:

(15) ${displaystyle {vec {nabla }}times {vec {E}}=0}$

This equation along with (10) define an electrostatic field. In addition, by differential calculation, it is known that a field whose rotation is zero can be described by the gradient of a scaling function $V$ , known as electric potential:

(16) ${displaystyle {vec {E}}=-{vec {nabla }}V}$

The importance of (15) is that since the curl of the electric field is zero, the superposition principle can be applied to this type of fields. For various charges, the electric field is defined as the vector sum of their individual fields:

(17) ${displaystyle {vec {E}}={vec {E}}_{1}+{vec {E}}_{2}+{vec {E}}_{3}+...}$

then

(18) ${displaystyle {vec {nabla }}times {vec {E}}={vec {nabla }}times ({vec {E}}_{1}+{vec {E}}_{2}+{vec {E}}_{3}+dots)=({vec {nabla }}times {vec {E}}_{1})+({vec {nabla }}times {vec {E}}_{2})+({vec {nabla }}times {vec {E}}_{3})+dots =0}$

Field lines

Electric field lines corresponding to equal and opposite loads, respectively.

A static electric field can be represented geometrically with lines such that at each point the vector field is tangent to these lines, these lines are known as "field lines". Mathematically the field lines are the integral curves of the vector field. Field lines are used to create a graphical representation of the field, and can be as many as needed to be displayed.

Field lines are lines perpendicular to the surface of the body, such that their geometric tangent at a point coincides with the direction of the field at that point. This is a direct consequence of Gauss's law, that is, we find that the greatest directional variation in the field is directed perpendicular to the charge. By joining the points in which the electric field is of equal magnitude, what is known as equipotential surfaces is obtained, they are those where the potential has the same numerical value. In the static case, since the electric field is an irrotational field, the field lines will never be closed (which can happen in the dynamic case, where the curl of the electric field is equal to the temporal variation of the magnetic field changed sign, for therefore a closed electric field line requires a variable magnetic field, which is impossible in the static case).

In the dynamic case, the lines can be defined equally, except that the pattern of lines will vary from one instant to another in time, that is, the field lines, like the charges, will be mobile.

Electrodynamic field (uniform motion)

The electric field created by a punctual charge presents spatial isotropy, instead, the field created by a moving load has a more intense field in the plane perpendicular to the speed according to the predictions of the theory of relativity. This is because for a resting observer regarding a load that moves at even speed the distance in the direction of the load movement will be less than the measures by an observer at rest regarding the load, by effect of the contraction of Lorentz, assuming that the load moves along the X axis of observer we would have the following relation of coordinates between the measure of the observer in movement regarding the load ${displaystyle scriptstyle ({bar {x}},{bar {y}},{bar {z}})}$ and the observer at rest regarding the burden ${displaystyle scriptstyle (x,y,z)}$ :

${displaystyle {bar {x}}={frac {x-Vt}{sqrt {1-{frac {V^{2}}{c^{2}}}}}},quad {bar {y}}=y,quad {bar {z}}=z}$

Being V the speed of the load with respect to the observer, thus the effective distance to the load measured by the observer moving with respect to the load will fulfill that:

${displaystyle {bar {r}}^{2}={frac {(x-Vt)^{2}+(1-{frac {V^{2}}{c^{2}}})(y^{2}+z^{2})}{1-{frac {V^{2}}{c^{2}}}}}}$

And therefore the electric field measured by an observer moving with respect to the charge will be:

(19) ${displaystyle mathbf {E} ={frac {1}{4pi epsilon _{0}}}{frac {q}{|{bar {r}}|^{3}}}{bar {mathbf {r} }}={frac {1}{4pi epsilon _{0}}}{frac {qleft(1-{frac {V^{2}}{c^{2}}}right)}{r^{3}left(1-{frac {V^{2}}{c^{2}}}sin ^{2}theta right)^{3/2}}}mathbf {r} }$

Where $scriptstyle theta$ is the angle formed by the position vector of the point where the field is measured (respect to the load) and the speed of the movement. From this last expression it is observed that if it is considered a radio sphere r around the load the field is more intense in the "Equator", taking as north and south poles the intersection of the sphere with the trajectory of the particle, it can be seen that the field on the sphere varies between a maximum ${displaystyle scriptstyle E_{bot }}$ and a minimum ${displaystyle scriptstyle E_{|}}$ given by:

(20) ${displaystyle E_{|}={frac {1}{4pi epsilon _{0}}}{frac {qleft(1-{frac {V^{2}}{c^{2}}}right)}{r^{2}}},qquad E_{bot }={frac {1}{4pi epsilon _{0}}}{frac {q}{r^{2}{sqrt {1-{frac {V^{2}}{c^{2}}}}}}}}$

This loss of spherical symmetry is barely noticeable for speeds small compared to the speed of light and becomes very marked at speeds close to light.

Electrodynamic field (accelerated motion)

The field of a charge in motion with respect to an observer is notably complicated with respect to the case of uniform motion if, in addition to a relative motion, the charge presents an accelerated motion with respect to an inertial observer. From the Lienard-Wiechert potentials it is obtained that the field created by a moving charge is given by:

(21) ${displaystyle mathbf {E} ={frac {1}{4pi epsilon }}left[{frac {q(1-{frac {v^{2}}{c^{2}}})}{(r-{frac {mathbf {r} cdot mathbf {v} }{c}})^{3}}}(mathbf {r} -{frac {mathbf {v} }{c}}r)+{frac {q}{c^{2}(R-{frac {mathbf {r} cdot mathbf {v} }{c}})^{3}}}left[mathbf {r} times left((mathbf {r} -{frac {mathbf {v} }{c}}r)times {dot {mathbf {v} }}right)right]right]}$

The first member only depends on the speed and matches the electric field caused by a uniform moving load, at large distances varies according to a reverse law of square 1/R² and, therefore, does not imply energy emission, the second member depends on acceleration ${displaystyle {dot {mathbf {v} }}}$ and has a 1/R variation that represents the decreasing intensity of a spherical wave of electromagnetic radiation, since the loads in accelerated motion emit radiation.

Electric field energy

Main article: Electromagnetic energy

A field in general stores energy and in the case of accelerated charges it can also transmit energy (principle used in telecommunications antennas). The volumetric energy density of an electric field is given by the following expression:

(22) ${displaystyle u={frac {1}{2}}epsilon _{0}E^{2}}$

So the total energy in a volume V is given by:

(23) ${displaystyle U={frac {epsilon _{0}}{2}}int _{V}E^{2} dV}$

where $dV$ It's the volume differential.

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