# Electric flux

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In electromagnetism, electric flux is the measure of the electric field through a given surface, [1] although an electric field in itself cannot flow. It is a way of describing the electric field strength at any distance from the charge causing the field.

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields, is responsible for electromagnetic radiation such as light, and is one of the four fundamental interactions in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force.

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically 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 electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic 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. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

## Contents

The electric field E can exert a force on an electric charge at any point in space. The electric field is proportional to the gradient of the voltage.

In vector calculus, the gradient is a multi-variable generalization of the derivative. Whereas the ordinary derivative of a function of a single variable is a scalar-valued function, the gradient of a function of several variables is a vector-valued function. Specifically, the gradient of a differentiable function of several variables, at a point , is the vector whose components are the partial derivatives of at .

## Overview

An electric "charge," such as a single electron in space, has an electric field surrounding it. In pictorial form, this electric field is shown as a dot, the charge, radiating "lines of flux". These are called Gauss lines. [2] The density of these lines corresponds to the electric field strength, which could also be called the electric flux density: the number of "lines" per unit area. Electric flux is proportional to the total number of electric field lines going through a surface. For simplicity in calculations, it is often convenient to consider a surface perpendicular to the flux lines. If the electric field is uniform, the electric flux passing through a surface of vector area S is

A field line is a graphical visual aid for visualizing vector fields. It consists of a directed line which is tangent to the field vector at each point along its length. A drawing showing a representative set of neighboring field lines is a common way of depicting a vector field in scientific and mathematical literature; this is called a field line diagram. They are used to show electric fields, magnetic fields, and gravitational fields among many other types. In fluid mechanics field lines showing the velocity field of a fluid flow are called streamlines.

In 3-dimensional geometry, for a finite planar surface of scalar area S and unit normal , the vector area S is defined as the unit normal scaled by the area:

${\displaystyle \Phi _{E}=\mathbf {E} \cdot \mathbf {S} =ES\cos \theta ,}$

where E is the electric field (having units of V/m), E is its magnitude, S is the area of the surface, and θ is the angle between the electric field lines and the normal (perpendicular) to S.

For a non-uniform electric field, the electric flux dΦE through a small surface area dS is given by

${\displaystyle d\Phi _{E}=\mathbf {E} \cdot d\mathbf {S} }$

(the electric field, E, multiplied by the component of area perpendicular to the field). The electric flux over a surface S is therefore given by the surface integral:

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over its scalar fields, and vector fields.

${\displaystyle \Phi _{E}=\iint _{S}\mathbf {E} \cdot d\mathbf {S} }$

where E is the electric field and dS is a differential area on the closed surface S with an outward facing surface normal defining its direction.

For a closed Gaussian surface, electric flux is given by:

A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, the electric field, or magnetic field. It is an arbitrary closed surface S = ∂V used in conjunction with Gauss's law for the corresponding field by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution.

${\displaystyle \Phi _{E}=\,\!}$${\displaystyle \scriptstyle S}$${\displaystyle \mathbf {E} \cdot d\mathbf {S} ={\frac {Q}{\varepsilon _{0}}}\,\!}$

where

E is the electric field,
S is any closed surface,
Q is the total electric charge inside the surface S,
ε0 is the electric constant (a universal constant, also called the "permittivity of free space") (ε0 ≈ 8.854 187 817... x 10−12 farads per meter (F·m−1)).

This relation is known as Gauss' law for electric field in its integral form and it is one of the four Maxwell's equations.

While the electric flux is not affected by charges that are not within the closed surface, the net electric field, E, in the Gauss' Law equation, can be affected by charges that lie outside the closed surface. While Gauss' Law holds for all situations, it is most useful for "by hand" calculations when high degrees of symmetry exist in the electric field. Examples include spherical and cylindrical symmetry.

Electrical flux has SI units of volt meters (V m), or, equivalently, newton meters squared per coulomb (N m2 C−1). Thus, the SI base units of electric flux are kg·m3·s−3·A−1. Its dimensional formula is [L3MT−3I−1].

## Notes

• Purcell, Edward, Morin, David; Electricity and Magnetism, 3rd Edition; Cambridge University Press, New York. 2013 ISBN   9781107014022.
• Browne, Michael, PhD; Physics for Engineering and Science, 2nd Edition; McGraw Hill/Schaum, New York; 2010. ISBN   0071613994

## Related Research Articles

In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the "speed of light". Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.

Flux describes any effect that appears to pass or travel through a surface or substance. A flux is either a concept based in physics or used with applied mathematics. Both concepts have mathematical rigor, enabling comparison of the underlying mathematics when the terminology is unclear. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component of a vector field perpendicular to the surface at each point.

In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B passing through that surface. The SI unit of magnetic flux is the weber (Wb), and the CGS unit is the maxwell. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and electronics, that evaluates the change of voltage in the measuring coils to calculate the magnetic flux.

In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson.

"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers, relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.

The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, or simply vector potential, A; and the magnetic scalar potentialψ. Both quantities can be used in certain circumstances to calculate the magnetic field B.

In physics, the electric displacement field, denoted by D, is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter square (C⋅m−2).

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted or received, per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant flux is the watt (W), that is the joule per second in SI base units, while that of spectral flux in frequency is the watt per hertz and that of spectral flux in wavelength is the watt per metre —commonly the watt per nanometre.

The method of image charges is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem.

The uniqueness theorem for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. In the case of electrostatics, this means that there is a unique electric field derived from a potential function satisfying Poisson's equation under the boundary conditions.

In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field B has divergence equal to zero, in other words, that it is a solenoidal vector field. It is equivalent to the statement that magnetic monopoles do not exist. Rather than "magnetic charges", the basic entity for magnetism is the magnetic dipole.

In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is essentially equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. Although Gauss's law for gravity is equivalent to Newton's law, there are many situations where Gauss's law for gravity offers a more convenient and simple way to do a calculation than Newton's law.

The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, the most commonly used unit in atomic physics and chemistry is the debye (D).

## References

1. Purcell, p22-26
2. Purcell, p5-6.