Euler's totient function: Difference between revisions

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The '''statcoulomb''' ('''statC''') or '''franklin''' ('''Fr''') or '''electrostatic unit of charge''' ('''esu''') is the [[units of measurement|physical unit]] for [[electrical charge]] used in the [[centimetre gram second system of units|centimetre–gram–second system of units]] (cgs) and [[Gaussian units]]. It is a derived unit given by
: 1 statC = 1 g<sup>1/2</sup> cm<sup>3/2</sup> s<sup>−1</sup> = 1 erg<sup>1/2</sup> cm<sup>1/2</sup>.
 
The [[SI]] system of units uses the [[coulomb]] (C) instead. The conversion between C and statC is different in different contexts. The most common contexts are:
:1 C ↔ {{Gaps|2|997|924|580}} statC ≈ {{Val|3.00|e=9|u=statC}} (for [[electric charge]])
:1 statC ↔ ≈{{Val|3.33564|e=-10|u=C}} (for [[electric charge]]).
 
:1 C ↔ 4π×{{Gaps|2|997|924|580}} statC ≈{{Val|3.77|e=10|u=statC}} (for [[electric flux]] Φ<sub>D</sub>)
:1 statC ↔ ≈{{Val|2.65|e=-11|u=C}} (for [[electric flux]] Φ<sub>D</sub>).
 
:1 C/m<sup>2</sup> ↔ 4π×{{Gaps|2|997|924|580}}×10<sup>−4</sup> statC/cm<sup>2</sup> ≈ {{Val|3.77|e=6|u=statC/cm<sup>2</sup>}} (for [[electric displacement field]] D)
:1 statC/cm<sup>2</sup> ↔ ≈{{Val|2.65|e=-7|u=C/m<sup>2</sup>}} (for [[electric displacement field]] D).
 
The symbol "↔" is used instead of "=" because the two sides are not necessarily interchangeable, as discussed [[#Dimensional relation between Statcoulomb and Coulomb|below]]. The number {{Gaps|2|997|924|580}} is 10 times the value of the [[speed of light]] expressed in meters/second, and the conversions are ''exact'' except where indicated.
 
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 1&nbsp;cm apart, they will electrically repel each other with a force of 1 [[dyne]]. This repulsion is governed by [[Coulomb's law]], which in the [[Gaussian units|Gaussian-cgs system]] states:
:<math>F=\frac{q_1q_2}{r^2}</math>
where ''F'' is the force, ''q''<sub>1</sub> and ''q''<sub>2</sub> are the two charges, and ''r'' is the distance between the charges. Performing [[dimensional analysis]] on Coulomb's law, the dimension of [[electrical charge]] in cgs must be [mass]<sup>1/2</sup> [length]<sup>3/2</sup> [time]<sup>−1</sup>. (This statement is ''not'' true in [[SI]] units; see below.) We can be more specific in light of the definition above: Plugging in ''F'' = 1 dyne, ''q''<sub>1</sub> = ''q''<sub>2</sub> = 1 statC, and ''r'' = 1&nbsp;cm, we get:
: 1 statC = g<sup>1/2</sup> cm<sup>3/2</sup> s<sup>−1</sup>
as expected.
 
==Dimensional relation between Statcoulomb and Coulomb==
{{or section|date=February 2013}}
{{offtopic|Gaussian units #Major differences between Gaussian and SI units|date=February 2013}}
===General incompatibility===
[[Coulomb's law]] in [[Gaussian units|cgs-Gaussian unit system]] and [[SI]] are respectively:
:<math>F=\frac{q_1q_2}{r^2}</math> (cgs-Gaussian)
:<math>F=\frac{q_1q_2}{4\pi\epsilon_0 r^2}</math> (SI)
Since ''ε''<sub>0</sub>, the [[vacuum permittivity]], is ''not'' dimensionless, the [[coulomb]] (the SI unit of charge) is '''not''' dimensionally equivalent to [mass]<sup>1/2</sup> [length]<sup>3/2</sup> [time]<sup>−1</sup>, 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" can be 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'' {{Val|3.00|e=9|u=statC}}" 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 {{Val|3.00|e=9|u=statC}}. Likewise, "1 C ''corresponds to'' {{Val|3.77|e=10|u=statC}}" when describing an [[electric displacement field]] 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&nbsp;cm apart, they will electrically repel each other with a force of 1 [[dyne]]. From this definition, it is straightforward to find an equivalent charge in [[SI]] [[coulomb]]s. Using the [[SI]] equation
:<math>F=\frac{q_1q_2}{4\pi\epsilon_0 r^2}</math> (SI),
and plugging in F = 1 dyne = 10<sup>−5</sup> N, and r = 1 cm = 10<sup>−2</sup> m, and then solving for ''q'' = ''q''<sub>1</sub> = ''q''<sub>2</sub>, the result is ''q'' = (1/2997924580)C ≈ {{Val|3.34|e=-10|u=C}}. Therefore an object with a charge of 1 statC has a charge of {{Val|3.34|e=-10|u=C}}.
 
This can also be expressed by the following conversion, which is fully dimensionally consistent, and often useful for switching between SI and cgs formulae:
:<math>1 \; \mathrm{C}/\sqrt{4 \pi \epsilon_0} = 2997924580 \; \mathrm{statC}</math>
 
===As a unit of electric displacement field or flux===
An [[electric flux]] (specifically, a flux of the [[electric displacement field]] '''D''') has units of charge: statC in cgs and coulombs in SI. The conversion factor can be derived from [[Gauss's law]]:
:<math>\Phi_D = 4\pi Q</math> (cgs)
:<math>\Phi_D = Q</math> (SI)
where
:<math>\Phi_D \equiv \int_S \mathbf{D}\cdot \mathrm{d}\mathbf{A}</math>
Therefore the conversion factor for flux is 4π different than the conversion factor for charge:
:<math>1 \; \mathrm{C} \text{ corresponds to } 3.7673 \times 10^{10} \; \mathrm{statC}</math> (as unit of Φ<sub>D</sub>).
The dimensionally consistent version is:
:<math>1 \; \mathrm{C}\sqrt{4 \pi /\epsilon_0} = 3.7673 \times 10^{10} \; \mathrm{statC}</math> (as unit of Φ<sub>D</sub>).
 
[[Category:Units of electrical charge]]
[[Category:Centimetre–gram–second system of units]]

Revision as of 00:14, 11 February 2014

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