# Coulomb

Short description: SI derived unit of electric charge
coulomb Charles-Augustin de Coulomb, the unit's namesake
General information
Unit systemInternational System of Units
Unit ofElectric charge
SymbolC
Named afterCharles-Augustin de Coulomb
Conversions
1 C in ...... is equal to ...
SI base units   As
CGS units   2997924580 statC
Atomic units   6.241509074e×1018

The coulomb (symbol: C) is the SI derived unit of electric charge. Historically, it was defined as the charge delivered by an electric current of one ampere in one second, with the ampere being defined as the current needed to generate a magnetic force of 0.2 micronewtons per metre between 2 parallel wires 1 metre apart. Under the 2019 redefinition of the SI base units, which took effect on 20 May 2019, the elementary charge was assigned the exact value of 1.602176634×10−19 C, making the coulomb exactly 1/(1.602176634×10−19) = 1019/1.602176634 elementary charges. The same number of electrons has the same magnitude but opposite sign of charge, that is, a charge of −1 C. This makes the coulomb one of the few SI units to depend on the value of only one of the SI defining constants. Despite this, the coulomb is, somewhat incongruously, still classified as a derived unit, and the ampere is still classified as an SI base unit.

## Name and notation

The coulomb is named after Charles-Augustin de Coulomb. As with every SI unit named for a person, its symbol starts with an upper case letter (C), but when written in full it follows the rules for capitalisation of a common noun; i.e., "coulomb" becomes capitalised at the beginning of a sentence and in titles, but is otherwise in lower case.

## Definition

The SI defines the coulomb in terms of the ampere and second: 1 C = 1 A × 1 s. The 2019 redefinition of the ampere and other SI base units fixed the numerical value of the elementary charge when expressed in coulombs, and therefore fixed the value of the coulomb when expressed as a multiple of the fundamental charge (the numerical values of those quantities are the multiplicative inverses of each other). The ampere is defined by taking the fixed numerical value of the elementary charge e to be 1.602176634×10−19 coulombs. The ampere was previously defined in terms of two wires of infinite extent.

Thus 1 coulomb is exactly \displaystyle{ \begin{alignat}{2} 1 ~ \mathrm{C} &= \frac{1}{1.602\,176\,634 \times 10^{-19}} ~ e \\[0.9ex] &= \frac{1}{1.602\,176\,634} \times 10^{19} ~ e \\[0.9ex] &= \frac{10^{28}}{1\,602\,176\,634} ~ e \\[0.9ex] &= \frac{5 \times 10^{27}}{801\,088\,317} ~ e \qquad \text{ (written in lowest terms) } \\[0.9ex] &= 6\,241\,509\,074\,460\,762\,607 + \tfrac{621\,837\,581}{801\,088\,317} ~ e \\ \end{alignat} } elementary charges where $\displaystyle{ \tfrac{621\,837\,581}{801\,088\,317} = \tfrac{621\,837\,581}{3^2 \times 19 \times 389 \times 12\,043} \approx 0.776\,24\ldots }$ and the numerator $\displaystyle{ 621\,837\,581 }$ is a prime number. Thus one coulomb is the charge of approximately 6241509074460762607.776 elementary charges, where the number is the reciprocal of 1.602176634×10−19 C. It is impossible to realize exactly 1 C of charge, since the number of elementary charges is not an integer.

By 1878, the British Association for the Advancement of Science had defined the volt, ohm, and farad, but not the coulomb. In 1881, the International Electrical Congress, now the International Electrotechnical Commission (IEC), approved the volt as the unit for electromotive force, the ampere as the unit for electric current, and the coulomb as the unit of electric charge. At that time, the volt was defined as the potential difference [i.e., what is nowadays called the "voltage (difference)"] across a conductor when a current of one ampere dissipates one watt of power. The coulomb (later "absolute coulomb" or "abcoulomb" for disambiguation) was part of the EMU system of units. The "international coulomb" based on laboratory specifications for its measurement was introduced by the IEC in 1908. The entire set of "reproducible units" was abandoned in 1948 and the "international coulomb" became the modern coulomb.

## SI prefixes

Main page: Orders of magnitude (charge)

Like other SI units, the coulomb can be modified by adding a prefix that multiplies it by a power of 10.

Submultiples Multiples Value SI symbol Name Value 10−1 C dC decicoulomb 101 C daC decacoulomb 10−2 C cC centicoulomb 102 C hC hectocoulomb 10−3 C mC millicoulomb 103 C kC kilocoulomb 10−6 C µC microcoulomb 106 C MC megacoulomb 10−9 C nC nanocoulomb 109 C GC gigacoulomb 10−12 C pC picocoulomb 1012 C TC teracoulomb 10−15 C fC femtocoulomb 1015 C PC petacoulomb 10−18 C aC attocoulomb 1018 C EC exacoulomb 10−21 C zC zeptocoulomb 1021 C ZC zettacoulomb 10−24 C yC yoctocoulomb 1024 C YC yottacoulomb Common multiples are in bold face.

## Conversions

• The magnitude of the electrical charge of one mole of elementary charges (approximately 6.022×1023, the Avogadro number) is known as a faraday unit of charge (closely related to the Faraday constant). One faraday equals 96485.33212... coulombs. In terms of the Avogadro constant (NA), one coulomb is equal to approximately 1.036×10−5 mol × NA elementary charges.
• A capacitor of one farad can hold one coulomb at a drop of one volt.
• One ampere hour equals 3600 C, hence 1 mA⋅h = 3.6 C.
• One statcoulomb (statC), the obsolete CGS electrostatic unit of charge (esu), is approximately 3.3356×1010 C or about one-third of a nanocoulomb.

## In everyday terms

• The charges in static electricity from rubbing materials together are typically a few microcoulombs.
• The amount of charge that travels through a lightning bolt is typically around 15 C, although for large bolts this can be up to 350 C.
• The amount of charge that travels through a typical alkaline AA battery from being fully charged to discharged is about 5 kC = 5000 C ≈ 1400 mA⋅h.
• A typical smartphone battery can hold 10,800 C ≈ 3000 mA⋅h.

## Notes and references

1. 6.241509126(38)×1018 is the reciprocal of the 2014 CODATA recommended value 1.6021766208(98)×10−19 for the elementary charge in coulomb.
2. "SI Brochure (2019)". SI Brochure. BIPM. p. 127.
3. "SI brochure (2019)". SI Brochure. BIPM. p. 130.
4. "SI brochure (2019)". SI Brochure. BIPM. p. 132.
5. "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
6. W. Thomson, et al. (1873) "First report of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units," Report of the 43rd Meeting of the British Association for the Advancement of Science (Bradford, September 1873), pp. 222–225. From p. 223: "The "ohm," as represented by the original standard coil, is approximately 109 C.G.S. units of resistance; the "volt" is approximately 108 C.G.S. units of electromotive force; and the "farad" is approximately 1/109 of the C.G.S. unit of capacity."
7. (Anon.) (September 24, 1881) "The Electrical Congress," The Electrician, 7 .
8. Donald Fenna, A Dictionary of Weights, Measures, and Units, OUP (2002), 51f.
9. "2018 CODATA Value: Faraday constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
10.
11. Hasbrouck, Richard. Mitigating Lightning Hazards , Science & Technology Review May 1996. Retrieved on 2009-04-26.
12. How to do everything with digital photography – David Huss, p. 23, at Google Books, "The capacity range of an AA battery is typically from 1100–2200 mAh."