In physics and electromagnetism, **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 to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface 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.

- Qualitative description
- Equation involving the E field
- Integral form
- Differential form
- Equivalence of integral and differential forms
- Equation involving the D field
- Free, bound, and total charge
- Integral form 2
- Differential form 2
- Equivalence of total and free charge statements
- Equation for linear materials
- Interpretations
- In terms of fields of force
- Relation to Coulomb's law
- Deriving Gauss's law from Coulomb's law
- Deriving Coulomb's law from Gauss's law
- See also
- Notes
- Citations
- References
- External links

The law was first^{ [1] } formulated by Joseph-Louis Lagrange in 1773,^{ [2] } followed by Carl Friedrich Gauss in 1835,^{ [3] } both in the context of the attraction of ellipsoids. It is one of Maxwell's four equations, which forms the basis of classical electrodynamics.^{ [note 1] } Gauss's law can be used to derive Coulomb's law,^{ [4] } and vice versa.

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In words, Gauss's law states that

*The net electric flux through any hypothetical closed surface is equal to **times the net electric charge within that closed surface*.^{ [5] }

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 and Gauss's law for gravity. 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 inverse-square Coulomb's law, and Gauss's law for gravity is essentially equivalent to the inverse-square Newton's law of gravity.

The law can be expressed mathematically using vector calculus in integral 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 **E** and the total electric charge, or in terms of the electric displacement field **D** and the *free* electric charge.^{ [6] }

Gauss's law can be stated using either the electric field **E** or the electric displacement field **D**. This section shows some of the forms with **E**; the form with **D** is below, as are other forms with **E**.

Gauss's law may be expressed as:^{ [6] }

where Φ_{E} is the electric flux through a closed surface S enclosing any volume V, Q is the total charge enclosed within V, and *ε*_{0} is the electric constant. The electric flux Φ_{E} is defined as a surface integral of the electric field:

where **E** is the electric field, d**A** is a vector representing an infinitesimal element of area of the surface,^{ [note 2] } and · represents the dot product of two vectors.

In a curved space-time, the flux of an electromagnetic field through a closed surface is expressed as

where is the speed of light; denotes the time components of the electromagnetic tensor; 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.^{ [8] }

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 for examples where these symmetries are exploited to compute electric fields.

By the divergence theorem, Gauss's law can alternatively be written in the *differential form*:

where ∇ · **E** is the divergence of the electric field, *ε*_{0} is the vacuum permittivity, is the relative permittivity, and ρ is the volume charge density (charge per unit volume).

The integral and differential forms are mathematically equivalent, by the divergence theorem. Here is the argument more specifically.

The integral form of Gauss' law is:

for any closed surface S containing charge Q. By the divergence theorem, this equation is equivalent to:

for any volume V containing charge Q. By the relation between charge and charge density, this equation is equivalent to:

for any volume V. In order for this equation to be *simultaneously true* for *every* possible volume V, it is necessary (and sufficient) for the integrands to be equal everywhere. Therefore, this equation is equivalent to:

Thus the integral and differential forms are equivalent.

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 plate. In contrast, "bound charge" arises only in the context of dielectric (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 **E** (above), is sometimes put into the equivalent form below, which is in terms of **D** and the free charge only.

This formulation of Gauss's law states the total charge form:

where Φ_{D} is the **D**-field flux through a surface S which encloses a volume V, and *Q*_{free} is the free charge contained in V. The flux Φ_{D} is defined analogously to the flux Φ_{E} of the electric field **E** through S:

The differential form of Gauss's law, involving free charge only, states:

where ∇ · **D** is the divergence of the electric displacement field, and *ρ*_{free} is the free electric charge density.

In this proof, we will show that the equation

is equivalent to the equation

Note that we are only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the divergence theorem.

We introduce the polarization density **P**, which has the following relation to **E** and **D**:

and the following relation to the bound charge:

Now, consider the three equations:

The key insight is that the sum of the first two equations is the third equation. This completes the proof: The first equation is true by definition, and therefore the second equation is true if and only if the third equation is true. So the second and third equations are equivalent, which is what we wanted to prove.

In homogeneous, isotropic, nondispersive, linear materials, there is a simple relationship between **E** and **D**:

where ε is the permittivity of the material. For the case of vacuum (aka free space), *ε* = *ε*_{0}. Under these circumstances, Gauss's law modifies to

for the integral form, and

for the differential form.

This section may contain content that is repetitive or redundant of text elsewhere in the article. Please help improve it by merging similar text or removing repeated statements. (September 2016) |

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.

Strictly speaking, Gauss's law cannot be derived from Coulomb's law alone, since Coulomb's law gives the electric field due to an individual, electrostatic point charge 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).

Coulomb's law states that the electric field due to a stationary point charge is:

where

**e**_{r}is the radial unit vector,- r is the radius, |
**r**|, *ε*_{0}is the electric constant,- q is the charge of the particle, which is assumed to be located at the origin.

Using the expression from Coulomb's law, we get the total field at **r** by using an integral to sum the field at **r** due to the infinitesimal charge at each other point **s** in space, to give

where ρ is the charge density. If we take the divergence of both sides of this equation with respect to **r**, and use the known theorem^{ [9] }

where *δ*(**r**) is the Dirac delta function, the result is

Using the "sifting property" of the Dirac delta function, we arrive at

which is the differential form of Gauss' law, as desired.

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.

be the electric field, with a continuous function (density of charge).

It is true for all that .

Consider now a compact set having a piecewise smooth boundary such that . It follows that and so, for the divergence theorem:

But because ,

for the argument above ( and then )

Therefore the flux through a closed surface generated by some charge density outside (the surface) is null.

Now consider , and as the sphere centered in having as radius (it exists because is an open set).

Let and be the electric field created inside and outside the sphere respectively. Then,

- , and

The last equality follows by observing that , and the argument above.

The RHS is the electric flux generated by a charged sphere, and so:

with

Where the last equality follows by the mean value theorem for integrals. Using the squeeze theorem and the continuity of , one arrives at:

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 **E** (see Helmholtz decomposition 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 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).

Taking S in the integral form of Gauss' law to be a spherical surface of radius r, centered at the point charge Q, we have

By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is

where **r̂** is a unit vector pointing radially away from the charge. Again by spherical symmetry, **E** points in the radial direction, and so we get

which is essentially equivalent to Coulomb's law. Thus the inverse-square law dependence of the electric field in Coulomb's law follows from Gauss' law.

- ↑ The other three of Maxwell's equations are: Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction
- ↑ More specifically, the infinitesimal area is thought of as planar and with area d
*N*. The vector d**R**is normal to this area element and has magnitude d*A*.^{ [7] }

- ↑ Duhem, Pierre (1891).
*Leçons sur l'électricité et le magnétisme*(in French). Paris Gauthier-Villars. vol. 1, ch. 4, p. 22–23. shows that Lagrange has priority over Gauss. Others after Gauss discovered "Gauss' Law", too. - ↑ Lagrange, Joseph-Louis (1773). "Sur l'attraction des sphéroïdes elliptiques".
*Mémoires de l'Académie de Berlin*(in French): 125. - ↑ Gauss, Carl Friedrich (1877).
*Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata*(in Latin). (Gauss,*Werke*, vol. V, p. 1). Gauss mentions Newton's*Principia*proposition XCI regarding finding the force exerted by a sphere on a point anywhere along an axis passing through the sphere. - ↑ Halliday, David; Resnick, Robert (1970).
*Fundamentals of Physics*. John Wiley & Sons. pp. 452–453. - ↑ Serway, Raymond A. (1996).
*Physics for Scientists and Engineers with Modern Physics*(4th ed.). p. 687. - 1 2 Grant, I. S.; Phillips, W. R. (2008).
*Electromagnetism*. Manchester Physics (2nd ed.). John Wiley & Sons. ISBN 978-0-471-92712-9. - ↑ Matthews, Paul (1998).
*Vector Calculus*. Springer. ISBN 3-540-76180-2. - ↑ Fedosin, Sergey G. (2019). "On the Covariant Representation of Integral Equations of the Electromagnetic Field".
*Progress in Electromagnetics Research C*.**96**: 109–122. arXiv: 1911.11138 . Bibcode:2019arXiv191111138F. doi:10.2528/PIERC19062902. S2CID 208095922. - ↑ See, for example, Griffiths, David J. (2013).
*Introduction to Electrodynamics*(4th ed.). Prentice Hall. p. 50.

**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. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.

An **electric field** is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental interactions of nature.

The **electric potential** is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. Furthermore, the motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used.

In physics, **screening** is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

**Poisson's equation** is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.

**Electrostatics** is a branch of physics that studies electric charges at rest.

A **continuity equation** or **transport equation** is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

**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. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

In electromagnetism, **displacement current density** is the quantity ∂**D**/∂*t* appearing in Maxwell's equations that is defined in terms of the rate of change of **D**, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials, there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

In classical electromagnetism, **polarization density** is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.

In classical electromagnetism, **magnetic vector potential** is the vector quantity defined so that its curl is equal to the magnetic field: . Together with the electric potential *φ*, the magnetic vector potential can be used to specify the electric field **E** as well. Therefore, many equations of electromagnetism can be written either in terms of the fields **E** and **B**, or equivalently in terms of the potentials *φ* and **A**. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

In physics, the **electric displacement field** 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 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 of Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter square (C⋅m^{−2}).

A **classical field theory** is a physical theory that predicts how one or more physical fields interact with matter through **field equations**, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically meant to describe electromagnetism and gravitation, two of the fundamental forces of nature.

**Electric potential energy** is a potential energy that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An *object* may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged *objects*.

In electromagnetism, **charge density** is the amount of electric charge per unit length, surface area, or volume. **Volume charge density** is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m^{−3}), at any point in a volume. **Surface charge density** (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m^{−2}), at any point on a surface charge distribution on a two dimensional surface. **Linear charge density** (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

The **Maxwell stress tensor** is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

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 *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.

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.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

**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 conventionally called *electrostatic force* or **Coulomb force**. Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb, hence the name. Coulomb's law was essential to the development of the theory of electromagnetism, maybe even its starting point, as it made it possible to discuss the quantity of electric charge in a meaningful way.

- Gauss, Carl Friedrich (1867).
*Werke Band 5*. Digital version - Jackson, John David (1998).
*Classical Electrodynamics*(3rd ed.). New York: Wiley. ISBN 0-471-30932-X. David J. Griffiths (6th ed.)

- Media related to Gauss' Law at Wikimedia Commons
- MIT Video Lecture Series (30 x 50 minute lectures)- Electricity and Magnetism Taught by Professor Walter Lewin.
- section on Gauss's law in an online textbook
- MISN-0-132
*Gauss's Law for Spherical Symmetry*(PDF file) by Peter Signell for Project PHYSNET. - MISN-0-133
*Gauss's Law Applied to Cylindrical and Planar Charge Distributions*(PDF file) by Peter Signell for Project PHYSNET.

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