Gauss's law


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.
The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1813, both in the context of the attraction of ellipsoids. It is one of Maxwell's four equations, which form the basis of classical electrodynamics. Gauss's law can be used to derive Coulomb's law, and vice versa.

Qualitative description

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.
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 and the total electric charge, or in terms of the electric displacement field 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. 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 through a closed surface enclosing any volume, is the total charge enclosed within, and is the electric constant. 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 of the surface, and represents the dot product of two vectors.
Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form.
If the electric field is known everywhere, Gauss's law makes it possible to find the distribution of electric charge: The charge in any given region can be deduced by integrating the electric field to find the flux.
The reverse problem 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.

Differential form

By the divergence theorem, Gauss's law can alternatively be written in the differential form:
where is the divergence of the electric field, is the electric constant, and is the total electric charge density.

Equivalence of integral and differential forms

The integral and differential forms are mathematically equivalent, by the divergence theorem. Here is the argument more specifically.
for any closed surface containing charge. By the divergence theorem, this equation is equivalent to:
for any volume containing charge. By the relation between charge and charge density, this equation is equivalent to:
for any volume. In order for this equation to be simultaneously true for every possible volume, it is necessary for the integrands to be equal everywhere. Therefore, this equation is equivalent to:
Thus the integral and differential forms are equivalent.

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 plate. In contrast, "bound charge" arises only in the context of dielectric materials. 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 , 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 of the electric displacement field, and is the free electric charge density.

Equivalence of total and free charge statements

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, which has the following relation to and :
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.

Equation for linear materials

In homogeneous, isotropic, nondispersive, linear materials, there is a simple relationship between and :
where is the permittivity of the material. For the case of vacuum,. 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 infinite 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 alone, since Coulomb's law gives the electric field due to an individual 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. The superposition principle says that the resulting field is the vector sum of fields generated by each particle.
where
Using the expression from Coulomb's law, we get the total field at by using an integral to sum the field at due to the infinitesimal charge at each other point in space, to give
where is the charge density. If we take the divergence of both sides of this equation with respect to r, and use the known theorem
where 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.

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 . 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.
By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
where is a unit vector pointing radially away from the charge. Again by spherical symmetry, 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.

Citations