Gauss's
law

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

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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 law was formulated by Carl Friedrich
Gauss in 1835, but was not published until 1867.[1]
It is one of the four Maxwell's
equations which form the basis of classical
electrodynamics, the other three being Gauss's law for
magnetism, Faraday's law
of induction, and Ampère's law
with Maxwell's correction. Gauss's law can be used to derive Coulomb's law,[2]
and vice versa.

Contents

1 Qualitative description of
the law

2 Equation involving E
field

2.1 Integral
form

2.1.1 Applying the
integral form

2.2 Differential
form

2.3 Equivalence of integral
and differential forms

3 Equation involving D
field

3.1 Free, bound, and total
charge

3.2 Integral
form

3.3 Differential
form

4 Equivalence of total and
free charge statements

5 Equation for linear
materials

6 Relation to Coulomb's
law

6.1 Deriving Gauss's law
from Coulomb's law

6.2 Deriving Coulomb's law
from Gauss's law

7 See also

8 Notes

9 References

10 External
links

Qualitative
description of the law[edit]

In words, Gauss's law states that:

The net outward normal electric flux through
any closed surface is
proportional to the total electric charge
enclosed within that closed surface.[3]

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.

Gauss's law is something of an electrical analogue of Ampère's law, which
deals with magnetism.

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.[4]

Equation involving E
field[edit]

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.

Integral form[edit]

Gauss's law may be expressed as:[5]

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

where E is the electric field, dA is a vector representing an
infinitesimal element of
area,[note
1] 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.

Applying the integral
form[edit]

Main article: Gaussian surface

See also Capacitance
(Gauss's law)

If the electric field is known everywhere, Gauss's 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 you 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'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[edit]

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

where ∇•E is the divergence of the electric
field, and ρ is the total electric charge density.

Equivalence of integral and differential forms[edit]

Main article: Divergence
theorem

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

Outline of proof

The integral form of Gauss's 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.

Equation involving D
field[edit]

See also: Maxwell's
equations

Free, bound,
and total charge[edit]

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'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.

Integral form[edit]

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

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:

Differential form[edit]

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

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

Equivalence of total and free charge statements[edit]

Proof that the formulations of Gauss's law in terms of free charge
are equivalent to the formulations involving total charge.

In this proof, we will show that the equation

is equivalent to the equation

Note that we're 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.

Equation for linear
materials[edit]

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 = ε0. Under these circumstances, Gauss's
law modifies to

for the integral form, and

for the differential form.

Relation to Coulomb's
law[edit]

Deriving Gauss's law from Coulomb's law[edit]

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

Outline of proof

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

where

er is the radial unit vector,

r is the radius, |r|,

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
is the charge density. If we take the divergence of both sides of this
equation with respect to r, and use the known theorem[7]

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's law, as
desired.

Note that 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[edit]

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).

Outline of proof

Taking S in the integral form of Gauss's 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
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's law.

See also[edit]

Book:
Maxwell's equations

Method of image
charges

Uniqueness
theorem for Poisson's equation

Notes[edit]

^
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.[6]

References[edit]

^
Bellone, Enrico (1980).
A World on Paper: Studies on the Second Scientific
Revolution.

^
Halliday, David;
Resnick, Robert (1970). Fundamentals of Physics. John Wiley & Sons,
Inc. pp. 452–53.

^
Serway, Raymond A.
(1996). Physics for Scientists and Engineers with Modern Physics, 4th
edition. p. 687.

^
I.S. Grant, W.R.
Phillips (2008). Electromagnetism (2nd ed.). Manchester Physics, John
Wiley & Sons. ISBN 978-0-471-92712-9.

^
I.S. Grant, W.R.
Phillips (2008). Electromagnetism (2nd ed.). Manchester Physics, John
Wiley & Sons. ISBN 978-0-471-92712-9.

^
Matthews, Paul (1998).
Vector Calculus. Springer. ISBN 3-540-76180-2.

^
See, for example, Griffiths, David J. (1998). Introduction to
Electrodynamics (3rd ed.). Prentice Hall. p. 50. ISBN 0-13-805326-X.

Jackson, John David (1999). Classical Electrodynamics, 3rd ed., New York:
Wiley. ISBN
0-471-30932-X.

External links[edit]

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.

Retrieved from "http://en.wikipedia.org/w/index.php?title=Gauss%27s_law&oldid=556907053"

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