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Electric field

suspended over an infinite sheet of conducting material.]] An electric field is a force field that surrounds electric charges that attracts or repels other electric charges. Browne, p 225: "... around every charge there is an aura that fills all space. This aura is the electric field due to the charge. The electric field is a vector field... and has a magnitude and direction." Mathematically the electric field is a vector field that associates to each point in space the force, called the Coulomb force, that would be experienced per unit of charge, by an infinitesimal test charge at that point. The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m). Electric fields are created by electric charges, and by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On a microscopic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. The electric field and the magnetic field together form the electromagnetic force, one of the four fundamental forces of nature.

Definition of an electric field

From Coulomb's law a particle with electric charge q_1 at position \boldsymbol{x_1} exerts a force on a particle with charge q_0 at position \boldsymbol{x_0} of \boldsymbol{F} = {1\over 4\pi\varepsilon_0}{q_1q_0 \over (\boldsymbol{x_1-x_0})^2}\boldsymbol{\hat r_\text{1,0}} where \boldsymbol{\hat r_\text{1,0}} is the unit vector in the direction from point \boldsymbol{x_1} to point \boldsymbol{x_0} When the charges q_0 and 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. In order to make it easy to calculate the Coulomb force on any charge at position \boldsymbol{x_0} this expression can be divided by q_0, leaving an expression that only depends on the other charge (the source charge){{cite book | last1 = Purcell | first1 = Edward | title = Electricity and Magnetism, 2nd Ed. | publisher = Cambridge University Press | date = 2011 | location = | pages = 8-9, 15-16 | language = | url = https://books.google.com/books?id=Z3bkNh6h4WEC&pg=PA15&dq=%22electric+field%22 | doi = | id = | isbn = 1139503553 }}{{cite book | last1 = Serway | first1 = Raymond A. | last2 = Vuille | first2 = Chris | title = College Physics, 10th Ed. | publisher = Cengage Learning | date = 2014 | location = | pages = 532-533 | language = | url = https://books.google.com/books?id=xETAAgAAQBAJ&pg=PA522&dq=work+energy+capacitor | doi = | id = | isbn = 1305142829 }} \boldsymbol{E}(\boldsymbol{x_0}) = {\boldsymbol{F} \over q_0} = {1\over 4\pi\varepsilon_0}{q_1 \over (\boldsymbol{x_1-x_0})^2}\boldsymbol{\hat r_\text{1,0}} This is the electric field at point \boldsymbol{x_0} due to the point charge q_1; it is a vector equal to the Coulomb force per unit charge that a positive point charge would experience at the position \boldsymbol{x_0}. Since the formula gives the electric field magnitude and direction at any point \boldsymbol{x_0} in space (except at the location of the charge itself, \boldsymbol{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 q at any point in space is equal to the product of the charge and the electric field at that point \boldsymbol{F} = q\boldsymbol{E} The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m); in terms of the SI base units they are kg⋅m⋅s−3⋅A−1 If there are multiple charges, the resultant Coulomb force on a charge can be found by summing the vectors of the forces due to each charge. This shows the electric field obeys the superposition principle: the total electric field at a point due to a collection of charges is just equal to the vector sum of the electric fields at that point due to the individual charges.Purcell (2011) Electricity and Magnetism, 2nd Ed., p. 20-21 \boldsymbol{E}(\boldsymbol{x}) = \boldsymbol{E_1}(\boldsymbol{x}) + \boldsymbol{E_2}(\boldsymbol{x}) + \boldsymbol{E_3}(\boldsymbol{x}) + \cdots = {1\over 4\pi\varepsilon_0}{q_1 \over (\boldsymbol{x_1-x})^2}\boldsymbol{\hat r_\text{1}} + {1\over 4\pi\varepsilon_0}{q_2 \over (\boldsymbol{x_2-x})^2}\boldsymbol{\hat r_\text{2}} + {1\over 4\pi\varepsilon_0}{q_3 \over (\boldsymbol{x_3-x})^2}\boldsymbol{\hat r_\text{3}} + \cdots \boldsymbol{E}(\boldsymbol{x}) = {1\over 4\pi\varepsilon_0} \sum_{k=1}^N {q_k \over (\boldsymbol{x_k-x})^2}\boldsymbol{\hat r_\text{k}} where \boldsymbol{\hat r_\text{k}} is the unit vector in the direction from point \boldsymbol{x_k} to point \boldsymbol{x}. This is the definition of the electric field due to the point source charges q_1\cdots q_N. It diverges and becomes infinite at the locations of the charges themselves, and so is not defined there. The electric field due to a continuous distribution of charge \rho(\boldsymbol{x}) in space (where \rho is the charge density in coulombs per cubic meter) can be calculated by considering the charge \rho(\boldsymbol{x'})dV in each small volume of space dV at point \boldsymbol{x'} as a point charge, and calculating its electric field d\boldsymbol{E}(\boldsymbol{x}) at point \boldsymbol{x} d\boldsymbol{E}(\boldsymbol{x}) = {1\over 4\pi\varepsilon_0}{\rho(\boldsymbol{x'})dV \over (\boldsymbol{x'-x})^2}\boldsymbol{\hat r'} where \boldsymbol{\hat r'} is the unit vector pointing from \boldsymbol{x'} to \boldsymbol{x}, then adding up the contributions from all the increments of volume by integrating over the volume of the charge distribution V \boldsymbol{E}(\boldsymbol{x}) = {1\over 4\pi\varepsilon_0}\iiint\limits_V \,{\rho(\boldsymbol{x'})dV \over (\boldsymbol{x'-x})^2}\boldsymbol{\hat r'}

Sources of electric field

Causes and description

Electric fields are caused by electric charges, described by Gauss's law, 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." or varying magnetic fields, described by Faraday's law of induction.Purcell, p 356: "Faraday's Law of Induction." Together, these laws are enough to define the behavior of the electric field as a function of charge repartition and magnetic 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 \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} and Faraday's law with no induction term \nabla \times \mathbf{E} = 0), taken together, are equivalent to Coulomb's law, written as \boldsymbol{E}(\boldsymbol{r}) = {1\over 4\pi\varepsilon_0}\int \rho(\boldsymbol{r'}) {\boldsymbol{r} - \boldsymbol{r'} \over |\boldsymbol{r} - \boldsymbol{r'}|^3} d^3r' for a charge density \mathbf{\rho}(\mathbf{r}) (\mathbf{r} denotes the position in space).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. Notice that \varepsilon_0, the permitivity of vacuum, must be substituted if charges are considered in non-empty media.

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 located at \mathbf{r_0} can be described mathematically as a charge density \rho(\mathbf{r})=q\delta(\mathbf{r-r_0}), where the Dirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

Superposition principle

Electric fields satisfy the superposition principle, because Maxwell's equations are linear. As a result, if \mathbf{E}_1 and \mathbf{E}_2 are the electric fields resulting from distribution of charges \rho_1 and \rho_2, a distribution of charges \rho_1+\rho_2 will create an electric field \mathbf{E}_1+\mathbf{E}_2; for instance, Coulomb's law is linear in charge density as well. This principle is useful to calculate the field created by multiple point charges. If charges q_1, q_2, ..., q_n are stationary in space at \mathbf{r}_1,\mathbf{r}_2,...\mathbf{r}_n, in the absence of currents, the superposition principle proves that the resulting field is the sum of fields generated by each particle as described by Coulomb's law: \mathbf{E}(\mathbf{r}) = \sum_{i=1}^N \mathbf{E}_i(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \sum_{i=1}^N q_i \frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3}

Electrostatic fields

connected to an electrostatic induction machine is placed in an oil-filled container. Considering that oil is a dielectric medium, when there is current through the electrode, the particles arrange themselves so as to show the force lines of the electric field.]] Electrostatic fields are E-fields which do not change with time, which happens when charges and currents are stationary. In that case, Coulomb's law fully describes the field.Purcell, p5-7.

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 \Phi such that \mathbf{E} = -\nabla \Phi . This is analogous to the gravitational potential.

Parallels between electrostatic and gravitational fields

Coulomb's law, which describes the interaction of electric charges: \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: \mathbf{F}=m\left(-GM\frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right)=m\mathbf{g} (where \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" because of that similarity. 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; 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: E = - \frac{\Delta\phi}{d} where Δϕ 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 , achieved by applying a voltage of the order of 1 volt between conductors spaced 1 µm apart.

Electrodynamic fields

Electrodynamic fields are E-fields which do change with time, for instance when charges are in motion. The electric field cannot be described independently of the magnetic field in that case. If A is the magnetic vector potential, defined so that \mathbf{B} = \nabla \times \mathbf{A} , one can still define an electric potential \Phi such that: \mathbf{E} = - \nabla \Phi - \frac { \partial \mathbf{A} } { \partial t } One can recover Faraday's law of induction by taking the curl of that equation
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