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= Coulomb's law =
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Introduction
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Coulomb's law, or Coulomb's inverse-square 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. The quantity of electrostatic force between stationary
charges is always described by Coulomb's law. The law was first
published in 1785 by French physicist Charles-Augustin de Coulomb, and
was essential to the development of the theory of electromagnetism,
maybe even its starting point, because it was now possible to discuss
quantity of electric charge in a meaningful way.
In its scalar form, the law is:
:F=k_e\frac{q_1 q_2}{r^2},
where 'k' is Coulomb's constant ('k' � ), 'q' and 'q' are the signed
magnitudes of the charges, and the scalar 'r' is the distance between
the charges. The force of the interaction between the charges is
attractive if the charges have opposite signs (i.e., 'F' is negative)
and repulsive if like-signed (i.e., 'F' is positive).
Being an inverse-square law, the law is analogous to Isaac Newton's
inverse-square law of universal gravitation, but gravitational forces
are always attractive, while electrostatic forces can be attractive or
repulsive. Coulomb's law can be used to derive Gauss's law, and vice
versa. In the case of a single stationary point charge, the two laws
are equivalent, expressing the same physical law in different ways.
The law has been tested extensively, and observations have upheld the
law on a scale from 10�16 m to 108 m.
History
======================================================================
Ancient cultures around the Mediterranean knew that certain objects,
such as rods of amber, could be rubbed with cat's fur to attract light
objects like feathers and papers. Thales of Miletus made a series of
observations on static electricity around 600 BC, from which he
believed that friction rendered amber magnetic, in contrast to
minerals such as magnetite, which needed no rubbing.
Thales was incorrect in believing the attraction was due to a
magnetic effect, but later science would prove a link between
magnetism and electricity. Electricity would remain little more than
an intellectual curiosity for millennia until 1600, when the English
scientist William Gilbert made a careful study of electricity and
magnetism, distinguishing the lodestone effect from static electricity
produced by rubbing amber. He coined the New Latin word 'electricus'
("of amber" or "like amber", from ['elektron'], the Greek word for
"amber") to refer to the property of attracting small objects after
being rubbed.
This association gave rise to the English words "electric" and
"electricity", which made their first appearance in print in Thomas
Browne's 'Pseudodoxia Epidemica' of 1646.
Early investigators of the 18th century who suspected that the
electrical force diminished with distance as the force of gravity did
(i.e., as the inverse square of the distance) included Daniel
Bernoulli
and Alessandro Volta, both of whom measured the force between plates
of a capacitor, and Franz Aepinus who supposed the inverse-square law
in 1758.
Based on experiments with electrically charged spheres, Joseph
Priestley of England was among the first to propose that electrical
force followed an inverse-square law, similar to Newton's law of
universal gravitation. However, he did not generalize or elaborate on
this.
In 1767, he conjectured that the force between charges varied as the
inverse square of the distance. May we not infer from this experiment,
that the attraction of electricity is subject to the same laws with
that of gravitation, and is therefore according to the squares of the
distances; since it is easily demonstrated, that were the earth in the
form of a shell, a body in the inside of it would not be attracted to
one side more than another?
In 1769, Scottish physicist John Robison announced that, according to
his measurements, the force of repulsion between two spheres with
charges of the same sign varied as .On
[
https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA68&redir_esc=y#v=on
epage&q&f=false
page 68], the author states that in 1769 he announced his findings
regarding the force between spheres of like charge. On
[
https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA73#v=onepage&q&
f=false
page 73], the author states the force between spheres of like charge
varies as x�2.06: The result of the whole was, that the mutual
repulsion of two spheres, electrified positively or negatively, was
very nearly in the inverse proportion of the squares of the distances
of their centres, or rather in a proportion somewhat greater,
approaching to x�2.06. When making experiments with charged spheres of
opposite charge the results were similar, as stated on
[
https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA73#v=onepage&q&
f=false
page 73]: When the experiments were repeated with balls having
opposite electricities, and which therefore attracted each other, the
results were not altogether so regular and a few irregularities
amounted to of the whole; but these anomalies were as often on one
side of the medium as on the other. This series of experiments gave a
result which deviated as little as the former (or rather less) from
the inverse duplicate ratio of the distances; but the deviation was in
defect as the other was in excess. Nonetheless, on
[
https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA74#v=onepage&q&
f=false
page 74] the author infers that the actual action is related exactly
to the inverse duplicate of the distance: We therefore think that it
may be concluded, that the action between two spheres is exactly in
the inverse duplicate ratio of the distance of their centres, and that
this difference between the observed attractions and repulsions is
owing to some unperceived cause in the form of the experiment. On
[
https://books.google.com/books?id=8pRDAAAAcAAJ&pg=PA75#v=onepage&q&
f=false
page 75], the authour compares the electric and gravitational forces:
Therefore we may conclude, that the law of electric attraction and
repulsion is similar to that of gravitation, and that each of those
forces diminishes in the same proportion that the square of the
distance between the particles increases.
In the early 1770s, the dependence of the force between charged bodies
upon both distance and charge had already been discovered, but not
published, by Henry Cavendish of England.On
[
https://archive.org/stream/electricalresear00caveuoft#page/111/mode/2up
pages 111 and 112] the author states: We may therefore conclude that
the electric attraction and repulsion must be inversely as some power
of the distance between that of the 2 + th and that of the 2 � th,
and there is no reason to think that it differs at all from the
inverse duplicate ratio.
Finally, in 1785, the French physicist Charles-Augustin de Coulomb
published his first three reports of electricity and magnetism where
he stated his law. This publication was essential to the development
of the theory of electromagnetism. He used a torsion balance to study
the repulsion and attraction forces of charged particles, and
determined that the magnitude of the electric force between two point
charges is directly proportional to the product of the charges and
inversely proportional to the square of the distance between them.
The torsion balance consists of a bar suspended from its middle by a
thin fiber. The fiber acts as a very weak torsion spring. In Coulomb's
experiment, the torsion balance was an insulating rod with a
metal-coated ball attached to one end, suspended by a silk thread. The
ball was charged with a known charge of static electricity, and a
second charged ball of the same polarity was brought near it. The two
charged balls repelled one another, twisting the fiber through a
certain angle, which could be read from a scale on the instrument. By
knowing how much force it took to twist the fiber through a given
angle, Coulomb was able to calculate the force between the balls and
derive his inverse-square proportionality law.
The law
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Coulomb's law states that:
A graphical representation of Coulomb's law
Coulomb's law can also be stated as a simple mathematical expression.
The scalar and vector forms of the mathematical equation are
:|\mathbf F|=k_e q_1q_2|\over r^2}\qquad and \qquad\mathbf
F_1=k_e\frac{q_1q_2}{{|\mathbf r_{21}^2}
\mathbf{\widehat{r}}_{21},\qquad respectively,
where is Coulomb's constant ( = ), and are the signed magnitudes
of the charges, the scalar is the distance between the charges, the
vector is the vectorial distance between the charges, and {{math|r�
}} (a unit vector pointing from to ). The vector form of the equation
calculates the force applied on by . If is used instead, then the
effect on can be found. It can be also calculated using Newton's
third law: .
Units
=======
When the electromagnetic theory is expressed in the International
System of Units, force is measured in newtons, charge in coulombs, and
distance in meters. Coulomb's constant is given by {{math|'k' }}. The
constant is the vacuum electric permittivity (also known as "electric
constant") in C2�
m�2�
N�1. It should not be confused with , which is
the dimensionless relative permittivity of the material in which the
charges are immersed, or with their product , which is called
"absolute permittivity of the material" and is still used in
electrical engineering.
The SI derived units for the electric field are volts per meter,
newtons per coulomb, or tesla meters per second.
Coulomb's law and Coulomb's constant can also be interpreted in
various terms:
* Atomic units. In atomic units the force is expressed in hartrees per
Bohr radius, the charge in terms of the elementary charge, and the
distances in terms of the 'Bohr radius'.
* Electrostatic units or Gaussian units. In electrostatic units and
Gaussian units, the unit charge ('esu' or statcoulomb) is defined in
such a way that the Coulomb constant disappears because it has the
value of one and becomes dimensionless.
* Lorentz-Heaviside units (also called 'rationalized'). In
Lorentz-Heaviside units the Coulomb constant is and becomes
dimensionless.
Gaussian units and Lorentz-Heaviside units are both CGS unit systems.
Gaussian units are more amenable for microscopic problems such as the
electrodynamics of individual electrically charged particles. SI units
are more convenient for practical, large-scale phenomena, such as
engineering applications.
Electric field
================
An electric field is a vector field that associates to each point in
space the Coulomb force experienced by a test charge. In the simplest
case, the field is considered to be generated solely by a single
source point charge. The strength and direction of the Coulomb force
on a test charge depends on the electric field that it finds itself
in, such that . If the field is generated by a positive source point
charge , the direction of the electric field points along lines
directed radially outwards from it, i.e. in the direction that a
positive point test charge would move if placed in the field. For a
negative point source charge, the direction is radially inwards.
The magnitude of the electric field can be derived from Coulomb's
law. By choosing one of the point charges to be the source, and the
other to be the test charge, it follows from Coulomb's law that the
magnitude of the electric field created by a single source point
charge at a certain distance from it in vacuum is given by:
When it is of interest to know the magnitude of the electrostatic
force (and not its direction), it may be easiest to consider a scalar
version of the law. The scalar form of Coulomb's Law relates the
magnitude and sign of the electrostatic force acting simultaneously
on two point charges and as follows:
In the image, the vector is the force experienced by , and the
vector is the force experienced by . When the forces are repulsive
(as in the image) and when the forces are attractive (opposite to the
image). The magnitude of the forces will always be equal.
Coulomb's law states that the electrostatic force experienced by a
charge, at position , in the vicinity of another charge, at position
, in a vacuum is equal to:
where , the unit vector {{math|**r�** }}, and is the electric
constant.
The vector form of Coulomb's law is simply the scalar definition of
the law with the direction given by the unit vector, , parallel with
the line 'from' charge 'to' charge . If both charges have the same
sign (like charges) then the product is positive and the direction of
the force on is given by ; the charges repel each other. If the
charges have opposite signs then the product is negative and the
direction of the force on is given by ; the charges attract each
other.
The electrostatic force experienced by , according to Newton's third
law, is .
The force on a small charge at position , due to a system of
discrete charges in vacuum is:
where and are the magnitude and position respectively of the th
charge, is a unit vector in the direction of (a vector pointing from
charges to ).
For a linear charge distribution (a good approximation for charge in
a wire) where gives the charge per unit length at position , and is
an infinitesimal element of length,
For a surface charge distribution (a good approximation for charge on
a plate in a parallel plate capacitor) where gives the charge per
unit area at position , and is an infinitesimal element of area,
For a volume charge distribution (such as charge within a bulk metal)
where gives the charge per unit volume at position , and is an
infinitesimal element of volume,
The force on a small test charge at position in vacuum is given by
the integral over the distribution of charge:
Coulomb's constant
====================
: is the radial unit vector,
: is the radius, ,
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
|Coulomb's law states that the electric field due to a stationary
point charge is: :\mathbf{E}(\mathbf{r}) = \frac{q}{4\pi
\varepsilon_0} \frac{\mathbf{e}_r}{r^2} where : is the electric
constant, : is the charge of the particle, which is assumed to be
located at the origin. :\mathbf{E}(\mathbf{r}) =
\frac{1}{4\pi\varepsilon_0} \int
\frac{\rho(\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3}
\, \mathrm{d}^3 \mathbf{s} :\nabla \cdot
\left(\frac{\mathbf{r}}{|\mathbf{r}|^3}\right) = 4\pi
\delta(\mathbf{r}) :\nabla\cdot\mathbf{E}(\mathbf{r}) =
\frac{1}{\varepsilon_0} \int \rho(\mathbf{s})\,
\delta(\mathbf{r}-\mathbf{s})\, \mathrm{d}^3 \mathbf{s} Using the
"sifting property" of the Dirac delta function, we arrive at
:\nabla\cdot\mathbf{E}(\mathbf{r}) =
\frac{\rho(\mathbf{r})}{\varepsilon_0}, which is the differential form
of Gauss' 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
=========================================
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 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
where is a unit vector pointing radially away from the charge. Again
by spherical symmetry, points in the radial direction, and so we get
|Taking in the integral form of Gauss' law to be a spherical surface
of radius , centered at the point charge , we have
:\oint_S\mathbf{E}\cdot d\mathbf{A} = \frac{Q}{\varepsilon_0} By the
assumption of spherical symmetry, the integrand is a constant which
can be taken out of the integral. The result is : 4\pi
r^2\hat{\mathbf{r}}\cdot\mathbf{E}(\mathbf{r}) =
\frac{Q}{\varepsilon_0} : \mathbf{E}(\mathbf{r}) = \frac{Q}{4\pi
\varepsilon_0} \frac{\hat{\mathbf{r}}}{r^2} 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.
See also
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* Biot-Savart law
* Darwin Lagrangian
* Electromagnetic force
* Gauss's law
* Method of image charges
* Molecular modelling
* Newton's law of universal gravitation, which uses a similar
structure, but for mass instead of charge
* Static forces and virtual-particle exchange
External links
======================================================================
* [
http://www.physnet.org/modules/pdf_modules/m114.pdf 'Coulomb's
Law'] on [
http://www.physnet.org Project PHYSNET]
* [
http://www.lightandmatter.com/html_books/4em/ch01/ch01.html
Electricity and the Atom]�a chapter from an online textbook
* [
http://mw2.concord.org/public/student/game/electrostatic_maze5.html
A maze game for teaching Coulomb's Law]�a game created by the
Molecular Workbench software
*
[
https://web.archive.org/web/20090314044312/http://ocw.mit.edu/OcwWeb/Physics/8-
02Electricity-and-MagnetismSpring2002/VideoAndCaptions/detail/embed01.htm
Electric Charges, Polarization, Electric Force, Coulomb's Law] Walter
Lewin, '8.02 Electricity and Magnetism, Spring 2002: Lecture 1'
(video). MIT OpenCourseWare. License: Creative Commons
Attribution-Noncommercial-Share Alike.
License
=========
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Original Article:
http://en.wikipedia.org/wiki/Coulomb's_law
