Study of stationary or slow-moving electric charges
Electrostatics
is a branch of
physics
that studies slow-moving or stationary
electric charges
.
Since
classical times
, it has been known that some materials, such as
amber
, attract lightweight particles after
rubbing
. The
Greek
word for amber,
?λεκτρον
(
?lektron
), was thus the source of the word
electricity
. Electrostatic phenomena arise from the
forces
that electric charges exert on each other. Such
forces
are described by
Coulomb's law
.
There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and
photocopier
and
laser printer
operation.
The electrostatic model accurately predicts electrical phenomena in "classical" cases where the velocities are low and the system is macroscopic so no quantum effects are involved. It also plays a role in quantum mechanics, where additional terms also need to be included.
Coulomb's law
[
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]
Coulomb's law
states that:
[5]
The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.
The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive.
If
is the distance (in
meters
) between two charges, then the force between two point charges
and
is:
where
ε
0
=
8.854
187
8188
(14)
×
10
?12
F?m
?1
[6]
is the
vacuum permittivity
.
[7]
The
SI unit
of
ε
0
is equivalently
A
2
?
s
4
?kg
?1
?m
?3
or
C
2
?
N
?1
?m
?2
or
F
?m
?1
.
Electric field
[
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]
The electric field,
, in units of
Newtons
per
Coulomb
or
volts
per meter, is a
vector field
that can be defined everywhere, except at the location of point charges (where it diverges to infinity).
[8]
It is defined as the electrostatic force
on a hypothetical small
test charge
at the point due to Coulomb's law, divided by the charge
Electric field lines
are useful for visualizing the electric field. Field lines begin on positive charge and terminate on negative charge. They are parallel to the direction of the electric field at each point, and the density of these field lines is a measure of the magnitude of the electric field at any given point.
Consider a collection of
particles of charge
, located at points
(called
source points
), the electric field at
(called the
field point
) is:
[8]
where
is the
displacement vector
from a
source point
to the
field point
, and
is a
unit vector
that indicates the direction of the field. For a single point charge at the origin, the magnitude of this electric field is
and points away from that charge if it is positive. The fact that the force (and hence the field) can be calculated by summing over all the contributions due to individual source particles is an example of the
superposition principle
. The electric field produced by a distribution of charges is given by the
volume charge density
and can be obtained by converting this sum into a
triple integral
:
Gauss's law
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]
Gauss's law
[9]
[10]
states that "the total
electric flux
through any closed surface in free space of any shape drawn in an electric field is proportional to the total
electric charge
enclosed by the surface." Many numerical problems can be solved by considering a
Gaussian surface
around a body. Mathematically, Gauss's law takes the form of an integral equation:
where
is a volume element. If the charge is distributed over a surface or along a line, replace
by
or
. The
divergence theorem
allows Gauss's Law to be written in differential form:
where
is the
divergence operator
.
Poisson and Laplace equations
[
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]
The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential Φ and the charge density
ρ
:
This relationship is a form of
Poisson's equation
.
[11]
In the absence of unpaired electric charge, the equation becomes
Laplace's equation
:
Electrostatic approximation
[
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]
The validity of the electrostatic approximation rests on the assumption that the electric field is
irrotational
:
From
Faraday's law
, this assumption implies the absence or near-absence of time-varying magnetic fields:
In other words, electrostatics does not require the absence of magnetic fields or electric currents. Rather, if magnetic fields or electric currents
do
exist, they must not change with time, or in the worst-case, they must change with time only
very slowly
. In some problems, both electrostatics and
magnetostatics
may be required for accurate predictions, but the coupling between the two can still be ignored. Electrostatics and magnetostatics can both be seen as non-relativistic
Galilean limits
for electromagnetism.
[12]
In addition, conventional electrostatics ignore quantum effects which have to be added for a complete description.
[8]
: 2
Electrostatic potential
[
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]
As the electric field is
irrotational
, it is possible to express the electric field as the
gradient
of a scalar function,
, called the
electrostatic potential
(also known as the
voltage
). An electric field,
, points from regions of high electric potential to regions of low electric potential, expressed mathematically as
The
gradient theorem
can be used to establish that the electrostatic potential is the amount of
work
per unit charge required to move a charge from point
to point
with the following
line integral
:
From these equations, we see that the electric potential is constant in any region for which the electric field vanishes (such as occurs inside a conducting object).
Electrostatic energy
[
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]
A
test particle
's potential energy,
, can be calculated from a
line integral
of the work,
. We integrate from a point at infinity, and assume a collection of
particles of charge
, are already situated at the points
. This potential energy (in
Joules
) is:
where
is the distance of each charge
from the
test charge
, which situated at the point
, and
is the electric potential that would be at
if the
test charge
were not present. If only two charges are present, the potential energy is
. The total
electric potential energy
due a collection of
N
charges is calculating by assembling these particles
one at a time
:
where the following sum from,
j
= 1 to
N
, excludes
i
=
j
:
This electric potential,
is what would be measured at
if the charge
were missing. This formula obviously excludes the (infinite) energy that would be required to assemble each point charge from a disperse cloud of charge. The sum over charges can be converted into an integral over charge density using the prescription
:
This second expression for
electrostatic energy
uses the fact that the electric field is the negative
gradient
of the electric potential, as well as
vector calculus identities
in a way that resembles
integration by parts
. These two integrals for electric field energy seem to indicate two mutually exclusive formulas for electrostatic energy density, namely
and
; they yield equal values for the total electrostatic energy only if both are integrated over all space.
Electrostatic pressure
[
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]
On a
conductor
, a surface charge will experience a force in the presence of an
electric field
. This force is the average of the discontinuous electric field at the surface charge. This average in terms of the field just outside the surface amounts to:
This pressure tends to draw the conductor into the field, regardless of the sign of the surface charge.
See also
[
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]
References
[
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]
- ^
Ling, Samuel J.; Moebs, William; Sanny, Jeff (2019).
University Physics, Vol. 2
. OpenStax.
ISBN
9781947172210
.
Ch.30: Conductors, Insulators, and Charging by Induction
- ^
Bloomfield, Louis A. (2015).
How Things Work: The Physics of Everyday Life
. John Wiley and Sons. p. 270.
ISBN
9781119013846
.
- ^
"Polarization"
.
Static Electricity ? Lesson 1 ? Basic Terminology and Concepts
. The Physics Classroom. 2020
. Retrieved
18 June
2021
.
- ^
Thompson, Xochitl Zamora (2004).
"Charge It! All About Electrical Attraction and Repulsion"
.
Teach Engineering: Stem curriculum for K-12
. University of Colorado
. Retrieved
18 June
2021
.
- ^
J, Griffiths (2017).
Introduction to Electrodynamics
. Cambridge University Press. pp. 296?354.
doi
:
10.1017/9781108333511.008
.
ISBN
978-1-108-33351-1
. Retrieved
2023-08-11
.
- ^
"2022 CODATA Value: vacuum electric permittivity"
.
The NIST Reference on Constants, Units, and Uncertainty
.
NIST
. May 2024
. Retrieved
2024-05-18
.
- ^
Matthew Sadiku (2009).
Elements of electromagnetics
. Oxford University Press. p. 104.
ISBN
9780195387759
.
- ^
a
b
c
Purcell, Edward M. (2013).
Electricity and Magnetism
. Cambridge University Press. pp. 16?18.
ISBN
978-1107014022
.
- ^
"Sur l'attraction des spheroides elliptiques, par M. de La Grange"
.
Mathematics General Collection
.
doi
:
10.1163/9789004460409_mor2-b29447057
. Retrieved
2023-08-11
.
- ^
Gauss, Carl Friedrich (1877),
"Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum, methodo nova tractata"
,
Werke
, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 279?286,
doi
:
10.1007/978-3-642-49319-5_8
,
ISBN
978-3-642-49320-1
, retrieved
2023-08-11
- ^
Poisson, M; sciences (France), Academie royale des (1827).
Memoires de l'Academie (royale) des sciences de l'Institut (imperial) de France
. Vol. 6. Paris.
- ^
Heras, J. A. (2010). "The Galilean limits of Maxwell's equations".
American Journal of Physics
.
78
(10): 1048?1055.
arXiv
:
1012.1068
.
Bibcode
:
2010AmJPh..78.1048H
.
doi
:
10.1119/1.3442798
.
S2CID
118443242
.
Further reading
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]
External links
[
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]
Learning materials related to
Electrostatics
at Wikiversity