Statcoulomb
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The franklin (Fr) or statcoulomb (statC) electrostatic unit of charge (esu) is the
physical unit A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multip ...
for
electrical charge Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described ...
used in the cgs-esu and
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
. It is a derived unit given by : 1 statC = 1 dyn1/2⋅cm = 1 cm3/2⋅g1/2⋅s−1. That is, it is defined so that the Coulomb constant becomes a dimensionless quantity equal to 1. It can be converted using : 1 newton = 105 dyne : 1 cm = 10−2 m The SI system of units uses the
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
(C) instead. The conversion between C and statC is different in different contexts. The most common contexts are: * For
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
: *: 1 C ≘ ≈ *: ⇒ 1 statC ≘ ~. * For
electric flux In electromagnetism, electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow. The electric field E can exert a force on an electric charge at any point in space. The electric fi ...
D): *: 1 C ≘ 4π × ≈ *: ⇒ 1 statC ≘ ~. The symbol "≘" ('corresponds to') is used instead of "=" because the two sides are not interchangeable, as discussed below. The number is 10 times the numeric value of the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
expressed in meters/second, and the conversions are ''exact'' except where indicated. The second context implies that the SI and cgs units for an
electric displacement field In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
(D) are related by: : 1 C/m2 ≘ 4π × ≈ : ⇒ 1 statC/cm2 ≘ ~ due to the relation between the
metre The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
and the
centimetre 330px, Different lengths as in respect to the Electromagnetic spectrum, measured by the Metre and its deriveds scales. The Microwave are in-between 1 meter to 1 millimeter. A centimetre (international spelling) or centimeter (American spellin ...
. The coulomb is an extremely large charge rarely encountered in electrostatics, while the statcoulomb is closer to everyday charges.


Definition and relation to cgs base units

The statcoulomb is defined as follows: if two stationary objects each carry a charge of 1 statC and are apart, they will electrically repel each other with a force of 1 
dyne The dyne (symbol: dyn; ) is a derived unit of force specified in the centimetre–gram–second (CGS) system of units, a predecessor of the modern SI. History The name dyne was first proposed as a CGS unit of force in 1873 by a Committee of ...
. This repulsion is governed by
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's 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 conventiona ...
, which in the Gaussian-cgs system states: :F=\frac , where ''F'' is the force, ''q'' and ''q'' are the two charges, and ''r'' is the distance between the charges. Performing
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
on Coulomb's law, the dimension of
electrical charge Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described ...
in cgs must be asssup>1/2 engthsup>3/2 imesup>−1. (This statement is ''not'' true in SI units; see below.) We can be more specific in light of the definition above: Substituting ''F'' = 1 dyn, ''q'' = ''q'' = 1 statC, and ''r'' = 1 cm, we get: : 1 statC = g1/2⋅cm3/2⋅s−1 as expected.


Dimensional relation between statcoulomb and coulomb


General incompatibility

Coulomb's law Coulomb's inverse-square law, or simply Coulomb's 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 conventiona ...
in the Gaussian unit system and the SI are respectively: :F=\frac (Gaussian) :F=\frac (SI) Since ''ε''0, the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
, is ''not'' dimensionless, the
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
is not dimensionally equivalent to asssup>1/2 engthsup>3/2 imesup>−1, unlike the statcoulomb. In fact, it is impossible to express the coulomb in terms of mass, length, and time alone. Consequently, a conversion equation like "1 C = ''n'' statC" is misleading: the units on the two sides are not consistent. One ''cannot'' freely switch between coulombs and statcoulombs within a formula or equation, as one would freely switch between centimeters and meters. One can, however, find a ''correspondence'' between coulombs and statcoulombs in different contexts. As described below, "1 C ''corresponds to'' " when describing the charge of objects. In other words, if a physical object has a charge of 1 C, it also has a charge of . Likewise, "1 C ''corresponds to'' " when describing an
electric displacement field In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
flux.


As a unit of charge

The statcoulomb is defined as follows: If two stationary objects each carry a charge of 1 statC and are 1 cm apart in vacuum, they will electrically repel each other with a force of 1 
dyne The dyne (symbol: dyn; ) is a derived unit of force specified in the centimetre–gram–second (CGS) system of units, a predecessor of the modern SI. History The name dyne was first proposed as a CGS unit of force in 1873 by a Committee of ...
. From this definition, it is straightforward to find an equivalent charge in
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
s. Using the SI equation :F=\frac (SI), and plugging in = 1 dyn = 10−5 N, and = 1 cm = 10−2 m, and then solving for , the result is = (1/2997924580) C ≈ . Therefore, an object with a charge of 1 statC has a charge of . This can also be expressed by the following conversion, which is fully dimensionally consistent, and often useful for switching between SI and cgs formulae: :1 \; \mathrm \times = 2997924580 \; \mathrm


As a unit of electric displacement field or flux

An
electric flux In electromagnetism, electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow. The electric field E can exert a force on an electric charge at any point in space. The electric fi ...
(specifically, a flux of the
electric displacement field In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
) has units of charge: statC in cgs and coulombs in SI. The conversion factor can be derived from
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
: :\Phi_\mathbf^\text = 4\pi Q^\text :\Phi_\mathbf^\text = Q^\text where :\Phi_\mathbf \equiv \int_S \mathbf\cdot \mathrm\mathbf Therefore, the conversion factor for flux is 4π different from the conversion factor for charge: :1 \; \mathrm ~\overset\frown=~ 3.7673 \times 10^ \; \mathrm (as unit of ). The dimensionally consistent version is: :1 \; \mathrm \times = 3.7673 \times 10^ \; \mathrm (as unit of {{math, ΦD) Units of electrical charge Centimetre–gram–second system of units