In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
to the resulting
electric field. In its integral form, it states that the
flux of the electric field out of an arbitrary
closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.
The law was first formulated by
Joseph-Louis Lagrange in 1773, followed by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in 1835, both in the context of the attraction of ellipsoids. It is one of
Maxwell's four equations, which forms the basis of
classical electrodynamics
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fi ...
.
[The other three of ]Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside 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.
...
are: Gauss's law for magnetism
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 has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
, Faraday's law of induction, and Ampère's law with Maxwell's correction Gauss's law can be used to derive
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, and vice versa.
Qualitative description
In words, Gauss's law states:
:The net
electric flux
In electromagnetism, electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow.
The electric field E can exert a force on an electric charge at any point in space. The electric fi ...
through any hypothetical
closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
is equal to times the net
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
enclosed within that closed surface. The closed surface is also referred to as Gaussian surface.
Gauss's law has a close mathematical similarity with a number of laws in other areas of physics, such as
Gauss's law for magnetism
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 has divergence equal to zero, in other words, that it is a solenoidal vector field. It is ...
and
Gauss's law for gravity
In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux ( surface inte ...
. In fact, any
inverse-square law can be formulated in a way similar to Gauss's law: for example, Gauss's law itself is essentially equivalent to the
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, and Gauss's law for gravity is essentially equivalent to the
Newton's law of gravity, both of which are inverse-square laws.
The law can be expressed mathematically using
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
in
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
form and
differential form; both are equivalent since they are related by the
divergence theorem, also called Gauss's theorem. Each of these forms in turn can also be expressed two ways: In terms of a relation between the
electric field and the total electric charge, or in terms of the
electric displacement field
In physics, the electric displacement field (denoted by D) or electric induction 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 ...
and the
''free'' electric charge.
Equation involving the field
Gauss's law can be stated using either the
electric field or the
electric displacement field
In physics, the electric displacement field (denoted by D) or electric induction 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 ...
. This section shows some of the forms with ; the form with is below, as are other forms with .
Integral form
Gauss's law may be expressed as:
where is the
electric flux
In electromagnetism, electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow.
The electric field E can exert a force on an electric charge at any point in space. The electric fi ...
through a closed surface enclosing any volume , is the total
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* ''Charge!!'', an album by The Aqu ...
enclosed within , and is the
electric constant
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
. The electric flux is defined as a
surface integral of the
electric field:
:
where is the electric field, is a vector representing an
infinitesimal element of
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
of the surface, and represents the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
of two vectors.
In a curved spacetime, the flux of an electromagnetic field through a closed surface is expressed as
:
where
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
;
denotes the time components of the
electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. T ...
;
is the determinant of
metric tensor;
is an orthonormal element of the two-dimensional surface surrounding the charge
; indices
and do not match each other.
Since the flux is defined as an ''integral'' of the electric field, this expression of Gauss's law is called the ''integral form''.
In problems involving conductors set at known potentials, the potential away from them is obtained by solving
Laplace's equation, either analytically or numerically. The electric field is then calculated as the potential's negative gradient. Gauss's law makes it possible to find the distribution of electric charge: The charge in any given region of the conductor can be deduced by integrating the electric field to find the flux through a small box whose sides are perpendicular to the conductor's surface and by noting that the electric field is perpendicular to the surface, and zero inside the conductor.
The reverse problem, when the electric charge distribution is known and the electric field must be computed, is much more difficult. The total flux through a given surface gives little information about the electric field, and can go in and out of the surface in arbitrarily complicated patterns.
An exception is if there is some
symmetry in the problem, which mandates that the electric field passes through the surface in a uniform way. Then, if the total flux is known, the field itself can be deduced at every point. Common examples of symmetries which lend themselves to Gauss's law include: cylindrical symmetry, planar symmetry, and spherical symmetry. See the article
Gaussian surface
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, electric field, or magnetic field. It is an arbitrary closed surface (the boundary of ...
for examples where these symmetries are exploited to compute electric fields.
Differential form
By the
divergence theorem, Gauss's law can alternatively be written in the ''differential form'':
where is the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of the electric field, is the
vacuum permittivity,
is the
relative permittivity
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...
, and is the volume
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
(charge per unit volume).
Equivalence of integral and differential forms
The integral and differential forms are mathematically equivalent, by the
divergence theorem. Here is the argument more specifically.
Equation involving the field
Free, bound, and total charge
The electric charge that arises in the simplest textbook situations would be classified as "free charge"—for example, the charge which is transferred in
static electricity, or the charge on a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
plate. In contrast, "bound charge" arises only in the context of
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
(polarizable) materials. (All materials are polarizable to some extent.) When such materials are placed in an external electric field, the electrons remain bound to their respective atoms, but shift a microscopic distance in response to the field, so that they're more on one side of the atom than the other. All these microscopic displacements add up to give a macroscopic net charge distribution, and this constitutes the "bound charge".
Although microscopically all charge is fundamentally the same, there are often practical reasons for wanting to treat bound charge differently from free charge. The result is that the more fundamental Gauss's law, in terms of (above), is sometimes put into the equivalent form below, which is in terms of and the free charge only.
Integral form
This formulation of Gauss's law states the total charge form:
where is the
-field flux through a surface which encloses a volume , and is the free charge contained in . The flux is defined analogously to the flux of the electric field through :
:
Differential form
The differential form of Gauss's law, involving free charge only, states:
where is the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
of the electric displacement field, and is the free electric charge density.
Equivalence of total and free charge statements
Equation for linear materials
In
homogeneous,
isotropic,
nondispersive, linear materials, there is a simple relationship between and :
where is the
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of the material. For the case of
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
(aka
free space
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
), . Under these circumstances, Gauss's law modifies to
for the integral form, and
for the differential form.
Interpretations
In terms of fields of force
Gauss's theorem can be interpreted in terms of the lines of force of the field as follows:
The flux through a closed surface is dependent upon both the magnitude and direction of the electric field lines penetrating the surface. In general a positive flux is defined by these lines leaving the surface and negative flux by lines entering this surface. This results in positive charges causing a positive flux and negative charges creating a negative flux. These electric field lines will extend to infinity decreasing in strength by a factor of one over the distance from the source of the charge squared. The larger the number of field lines emanating from a charge the larger the magnitude of the charge is, and the closer together the field lines are the greater the magnitude of the electric field. This has the natural result of the electric field becoming weaker as one moves away from a charged particle, but the surface area also increases so that the net electric field exiting this particle will stay the same. In other words the closed integral of the electric field and the dot product of the derivative of the area will equal the net charge enclosed divided by permittivity of free space.
Relation to Coulomb's law
Deriving Gauss's law from Coulomb's law
Strictly speaking, Gauss's law cannot be derived from
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
alone, since Coulomb's law gives the electric field due to an individual,
electrostatic
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amb ...
point charge
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
only. However, Gauss's law ''can'' be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys the
superposition principle, and that the point charge is electrostatic. The superposition principle states that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space).
Since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law.
Deriving Coulomb's law from Gauss's law
Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the
curl of (see
Helmholtz decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
and
Faraday's law). However, Coulomb's law ''can'' be proven from Gauss's law if it is assumed, in addition, that the electric field from a
point charge
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
is spherically symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).
See also
*
Method of image charges
*
Uniqueness theorem for Poisson's equation
*
List of examples of Stigler's law
Stigler's law concerns the supposed tendency of eponymous expressions for scientific discoveries to honor people other than their respective originators.
Examples include:
A
*Aharonov–Bohm effect. Werner Ehrenberg and Raymond E. Siday first ...
Notes
Citations
References
Digital version* David J. Griffiths (6th ed.)
External links
*
MIT Video Lecture Series (30 x 50 minute lectures)- Electricity and MagnetismTaught by Professor
Walter Lewin.
section on Gauss's law in an online textbookMISN-0-132 ''Gauss's Law for Spherical Symmetry''(
PDF file
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...
) by Peter Signell fo
Project PHYSNETMISN-0-133 ''Gauss's Law Applied to Cylindrical and Planar Charge Distributions''(PDF file) by Peter Signell fo
Project PHYSNET
{{Carl Friedrich Gauss
Electrostatics
Vector calculus
Maxwell's equations
Law
Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vario ...
Electromagnetism