Coulomb

The coulomb, symbol C, is the coherent derived unit of electric charge in the International System of Units (SI). One coulomb is the charge transferred through a cross-section by a steady current of one ampere in one second. It is named after the French physicist Charles-Augustin de Coulomb (1736–1806), whose experimental work established the inverse-square relationship between electrostatic force and distance. The unit measures an amount of charge, rather than its rate of flow or the energy associated with it. (nist.gov)

Definition and SI status

The defining unit relation is

The coulomb is therefore a derived unit, while the ampere and second are SI base units. Its physical dimension is electric current multiplied by time. “Coherent” means that no additional numerical conversion factor is required when it is expressed through the relevant SI base units. (bipm.org)

Since May 20, 2019, the SI has fixed the value of the elementary charge $e$ exactly:

Consequently,

The reciprocal expression is exact; its displayed decimal approximation is rounded. This definition establishes the size of the coulomb through a constant of nature rather than through a particular measuring apparatus. (bipm.org)

The electron carries charge $-e$, whereas the proton carries $+e$. Approximately $6.24\times10^{18}$ excess electrons therefore correspond to a net charge of $-1$ C. The coulomb itself has no polarity: positive and negative numerical values describe the sign of the measured charge. (openstax.org)

Charge and current

Electric current is the rate at which charge passes through a specified surface. If $Q$ denotes transferred charge and $t$ denotes time,

For a varying current, the transferred charge is calculated using an integral:

For constant current, this reduces to $\Delta Q=I\Delta t$. Thus, a current of 2 A maintained for 3 s transfers 6 C. Current in amperes and charge in coulombs are related quantities, but they are not interchangeable. (openstax.org)

The direction assigned to current is conventionally the direction of positive-charge motion. In a metallic conductor, the mobile electrons drift in the opposite direction. In an electrolyte, positively and negatively charged ions can both contribute to current. A measured charge transfer therefore need not represent electrons alone moving through the material. (openstax.org)

Relation to electrostatic quantities

The coulomb appears throughout electrostatics. In Coulomb’s law, the magnitude of the force between two stationary point charges in vacuum is

where the charges are expressed in coulombs, their separation $r$ in metres, and the force in newtons. Two point charges of 1 C each, separated by 1 m in vacuum, would exert a force of approximately $8.99\times10^9$ N. This illustrates the large scale of one coulomb relative to charges commonly encountered in elementary static-electricity experiments. (openstax.org)

An electric field is defined through the force it exerts per unit charge, so its SI unit can be written as newtons per coulomb, N/C. Electric potential and potential difference instead concern energy per unit charge: one volt equals one joule per coulomb. Thus, moving 1 C through a potential difference of 1 V corresponds to an energy difference of 1 J. (openstax.org)

Charge must therefore be distinguished from energy. The same amount of charge can be associated with different energy transfers, depending on the potential difference involved. (bipm.org)

Capacitors and practical scales

For a capacitor, capacitance relates the magnitude of charge on either conductor to the potential difference between them:

The subscript distinguishes capacitance from the unit symbol C. Capacitance is measured in farads, with $1\ \mathrm{F}=1\ \mathrm{C/V}$. A 1 F capacitor at 1 V consequently has charges of $+1$ C and $-1$ C on its two conductors. Its total net charge can still be zero: the quoted stored charge refers to the magnitude on either conductor. (openstax.org)

SI prefixes provide convenient smaller or larger units:

• 1 millicoulomb (mC) = $10^{-3}$ C;
• 1 microcoulomb (μC) = $10^{-6}$ C;
• 1 nanocoulomb (nC) = $10^{-9}$ C;
• 1 kilocoulomb (kC) = $10^3$ C.

As with other SI units named after people, the written name coulomb begins with a lowercase letter, while its symbol is uppercase. Symbols do not take plural endings; the standard form is “5 C,” not “5 Cs.” (bipm.org)

Related units and chemical quantities

The ampere-hour expresses charge on a scale useful for describing battery capacity. Because an hour contains 3,600 seconds,

A capacity of 2,000 mAh therefore corresponds to 7,200 C. This is a charge specification, not an energy specification; conversion to energy also requires information about voltage. (nist.gov)

In electrochemistry, the Faraday constant connects charge with amount of substance:

where $N_{\mathrm A}$ is the Avogadro constant. Multiplying the exact SI values gives approximately $96,485.33212\ \mathrm{C\,mol^{-1}}$. One mole of elementary charges thus has this charge magnitude. The constant’s unit, coulombs per mole, distinguishes it from both the coulomb and the farad. (bipm.org)