# Gauss' law

(Redirected from Gauss's Law)
This article is about Gauss' law concerning the electric field. For analogous law concerning different fields, see Gauss' law for magnetism and Gauss' law for gravity. For the Ostrogradsky-Gauss theorem, a mathematical theorem relevant to all of these laws, see Divergence theorem.

In physics, Gauss' law, also known as Gauss' flux theorem, is a law relating the distribution of electric charge to the resulting electric field.

The law was first[1] formulated by Joseph-Louis Lagrange in 1773,[2] followed by Carl Friedrich Gauss in 1813,[3] both in the context of the attraction of ellipsoids. It is one of Maxwell's four equations, which form the basis of classical electrodynamics.[note 1] Gauss' law can be used to derive Coulomb's law,[4] and vice versa.

## Qualitative description

In words, Gauss' law states that:

The net electric flux through any closed surface is equal to 1/ε times the net electric charge within that closed surface.[5]

Gauss' law has a close mathematical similarity with a number of laws in other areas of physics, such as Gauss' law for magnetism and Gauss' law for gravity. In fact, any inverse-square law can be formulated in a way similar to Gauss' law: for example, Gauss' law itself is essentially equivalent to the inverse-square Coulomb's law, and Gauss' 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' 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]

## Equation involving the E field

Gauss' 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.

### Integral form

Gauss' law may be expressed as:[6]

${\displaystyle \Phi _{E}={\frac {Q}{\varepsilon _{0}}}}$

where ΦE is the electric flux through a closed surface S enclosing any volume V, Q is the total charge enclosed within V, and ε0 is the electric constant. The electric flux ΦE is defined as a surface integral of the electric field:

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

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

Since the flux is defined as an integral of the electric field, this expression of Gauss' law is called the integral form.

#### Applying the integral form

Main article: Gaussian surface

If the electric field is known everywhere, Gauss' law makes it quite easy, in principle, 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.

However, much more often, it is the reverse problem that needs to be solved: The electric charge distribution is known, and the electric field needs to be computed. This is much more difficult, since if you know the total flux through a given surface, that gives almost no information about the electric field, which (for all we know) could go in and out of the surface in arbitrarily complicated patterns.

An exception is if there is some symmetry in the situation, 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' 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' law can alternatively be written in the differential form:

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}$

where ∇ · E is the divergence of the electric field, ε0 is the electric constant, and ρ is the total electric charge density (charge per unit volume).

### Equivalence of integral and differential forms

Main article: Divergence theorem

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

## Equation involving the D field

### Free, bound, and total charge

Main article: Electric polarization

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' 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.

### Integral form

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

${\displaystyle \Phi _{D}=Q_{\mathrm {free} }}$

where ΦD is the D-field flux through a surface S which encloses a volume V, and Qfree is the free charge contained in V. The flux ΦD is defined analogously to the flux ΦE of the electric field E through S:

${\displaystyle \Phi _{D}=}$ ${\displaystyle {\scriptstyle _{S}}}$ ${\displaystyle \mathbf {D} \cdot \mathrm {d} \mathbf {A} }$

### Differential form

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

${\displaystyle \nabla \cdot \mathbf {D} =\rho _{\mathrm {free} }}$

where ∇ · D is the divergence of the electric displacement field, and ρfree is the free electric charge density.

## Equation for linear materials

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

${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} }$

where ε is the permittivity of the material. For the case of vacuum (aka free space), ε = ε0. Under these circumstances, Gauss' law modifies to

${\displaystyle \Phi _{E}={\frac {Q_{\mathrm {free} }}{\varepsilon }}}$

for the integral form, and

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{\mathrm {free} }}{\varepsilon }}}$

for the differential form.

## Interpretations

### In terms of fields of force

Gauss' theorem can be interpreted in terms of the lines of force of the field as follows:

The flow field through the surface is the number of field lines penetrating the surface. This takes into account the direction of – field lines penetrating the surface considered with a minus sign in the opposite direction. Force lines begin or end only on charges (start on the positive end to the negative), or may even go to infinity. The number of lines of force emanating from the charge (starting it) anyway, the magnitude of this charge (the charge is defined in the model). (For all the negative charges of the same, only the charge is equal to minus the number of its member (it ends) lines. On the basis of these two provisions of the Gauss theorem is evident in the statement: the number of lines emanating from a closed surface is equal to the total number of charges inside it – that is, the number of lines that appear within it. Of course, meant keeping signs, in particular, the line, which began within the surface on the positive charge can end on a negative charge and within it (if there is), then it does not give a contribution to the flux through this surface, as, or even before it not reach, or be released, and then enters back (or, in general, the surface intersects an even number of times equal to the forward and the opposite direction) that gives zero contribution to the flow in the summation with the correct sign.[clarification needed].The same can be said about the lines begin and end outside the given surface – for the same reason, they also give a zero contribution to flow through it. [clarification needed].

## Relation to Coulomb's law

### Deriving Gauss' law from Coulomb's law

Gauss' law can be derived from Coulomb's law.

Note that since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss' law to hold for moving charges based on this derivation alone. In fact, Gauss' law does hold for moving charges, and in this respect Gauss' law is more general than Coulomb's law.

### Deriving Coulomb's law from Gauss' law

Strictly speaking, Coulomb's law cannot be derived from Gauss' law alone, since Gauss' 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' 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).

## Notes

1. ^ The other three of Maxwell's equations are: Gauss' law for magnetism, Faraday's law of induction, and "Ampère"'s law with Maxwell's correction
2. ^ More specifically, the infinitesimal area is thought of as planar and with area dA. The vector dA is normal to this area element and has magnitude dA.[7]

## Citations

1. ^ Duhem, Pierre. Leçons sur l'électricité et le magnétisme (in French). vol. 1, ch. 4, p. 22–23. shows that Lagrange has priority over Gauss. Others after Gauss discovered "Gauss' Law", too.
2. ^ Lagrange, Joseph-Louis (1773). "Sur l'attraction des sphéroïdes elliptiques". Mémoires de l'Académie de Berlin (in French): 125.
3. ^ Gauss, Carl Friedrich. 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.
4. ^ Halliday, David; Resnick, Robert (1970). Fundamentals of Physics. John Wiley & Sons. pp. 452–453.
5. ^ Serway, Raymond A. (1996). Physics for Scientists and Engineers with Modern Physics (4th ed.). p. 687.
6. ^ a b Grant, I. S.; Phillips, W. R. (2008). Electromagnetism. Manchester Physics (2nd ed.). John Wiley & Sons. ISBN 978-0-471-92712-9.
7. ^ Matthews, Paul (1998). Vector Calculus. Springer. ISBN 3-540-76180-2.
8. ^ See, for example, Griffiths, David J. (2013). Introduction to Electrodynamics (4th ed.). Prentice Hall. p. 50.

## References

• Jackson, John David (1998). Classical Electrodynamics (3rd ed.). New York: Wiley. ISBN 0-471-30932-X.