Coulomb constant

Coulomb constant
Coulomb constant
Common symbols	ke
Dimension

The Coulomb constant,^[1] the electric force constant,^[2] or the electrostatic constant^[3] (denoted $k e$ , $k$ or $K$ ) is a proportionality constant in electrostatics equations.^[4] In SI base units it is equal to 8.9875517923(14)×10⁹ kg⋅m³⋅s⁻⁴⋅A⁻².^[5] It was named after the French physicist Charles-Augustin de Coulomb (1736–1806) who introduced Coulomb's law.^[6]

Value of the constant[edit]

Value of $k$	Units
8.9875517923(14)×10⁹	N·m²/C²
14.3996	eV·Å·e⁻²
1.00000000059(15)×10⁻⁷	(N·s²/C²)c²

The Coulomb constant is the constant of proportionality in Coulomb's law,

\mathbf {F} =k_{\text{e}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}

where $ê r$ is a unit vector in the $r$ -direction. In SI, ^[7]

k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}},

where $\varepsilon _{0}$ is the vacuum permittivity. This formula can be derived from Gauss' law,

$\oiint$

{\scriptstyle S}

\mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}

Taking this integral for a sphere, radius $r$ , centered on a point charge, the electric field points radially outwards and is normal to a differential surface element on the sphere with constant magnitude for all points on the sphere.

$\oiint$

{\scriptstyle S}

\mathbf {E} \cdot {\rm {d}}\mathbf {A} =|\mathbf {E} |\int _{S}dA=|\mathbf {E} |\times 4\pi r^{2}

Noting that $E = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}F/q$ for some test charge $q$ ,

{\begin{aligned}\mathbf {F} &={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}=k_{\text{e}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}\\[8pt]\therefore k_{\text{e}}&={\frac {1}{4\pi \varepsilon _{0}}}\end{aligned}}

Coulomb's law is an inverse-square law, and thereby similar to many other scientific laws ranging from gravitational pull to light attenuation. This law states that a specified physical quantity is inversely proportional to the square of the distance.

In some modern systems of units, the Coulomb constant $k e$ has an exact numeric value; in Gaussian units $k e = 1$ , in Heaviside–Lorentz units (also called rationalized) $k e = 1 / 4π$ . This was previously true in SI when the vacuum permeability was defined as $μ 0 = 4 π \times 10 -7 H \cdotm -1$ . Together with the speed of light in vacuum $c$ , defined as 299792458 m/s, the vacuum permittivity $ε 0$ can be written as 1/μ₀c², which gave an exact value of^[8]

{\begin{aligned}k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}={\frac {c^{2}\mu _{0}}{4\pi }}&=c^{2}\times (10^{-7}\ \mathrm {H{\cdot }m} ^{-1}).\end{aligned}}

Since the redefinition of SI base units, the Coulomb constant is no longer has an exactly defined value and is subject to the measurement error in the fine structure constant, as calculated from CODATA 2018 recommended values being^[5]

k_{\text{e}}=8.987\,551\,7923\,(14)\times 10^{9}\;\mathrm {kg{\cdot }m^{3}{\cdot }s^{-4}{\cdot }A^{-2}} .

Use[edit]

The Coulomb constant is used in many electric equations, although it is frequently expressed as the following product of the vacuum permittivity constant:

k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}.

The Coulomb constant appears in many expressions including the following:

Coulomb's law: $\mathbf {F} =k_{\text{e}}{Qq \over r^{2}}\mathbf {\hat {e}} _{r}.$
Electric potential energy: $U_{\text{E}}(r)=k_{\text{e}}{\frac {Qq}{r}}.$
Electric field: $\mathbf {E} =k_{\text{e}}\sum _{i=1}^{N}{\frac {Q_{i}}{r_{i}^{2}}}\mathbf {\hat {r}} _{i}.$

References[edit]

^ Cleri, Fabrizio (2016). The Physics of Living Systems. Springer International Publishing. p. 611. ISBN 9783319306476.
^ Oxford, Cambridge and RSA Examinations (2010). Data, Formulae and Relationships Booklet (Revised Version 2.2) – GCE Advanced Level and Advanced Subsidiary – Physics B (Advancing Physics) (PDF). p. 2.
^ Milne, Edward Arthur (6 March 1937). "Letters to Editor: The Constant of Gravitation" (PDF). Nature. 139 (3514): 409. Retrieved 15 March 2023. Note added in proof, Feb. 26.⠀Complete agreement between Dirac's results and kinematic cosmology can be obtained, and his proportionalities M ∝ t², γ ∝ t⁻¹, replaced by M = const. = M₀, γ ∝ t, if we take the 'electrostatic' constant γ′ in the inverse square law to be proportional to t; γ′ is, of course, usually taken as unity. This is formally the same as taking e² ∝ t. The t-dynamics then gives the radius of the Bohr atom as proportional to t, and a material rod is 'rigid' on the τ-scale. Electrostatic forces then become to some extent analogous to gravitational forces, and might be discussed kinematically by examining the properties of an expanding sphere of mingled positive and negative particles.
^ "Coulomb force". Encyclopedia Britannica. Retrieved 13 March 2023. In the metre–kilogram–second and the SI systems, the unit of force (newton), the unit of charge (coulomb), and the unit of distance (metre), are all defined independently of Coulomb's law, so the proportionality factor k is constrained to take a value consistent with these definitions, namely, k in a vacuum equals 8.98 × 10⁹ newton square metre per square coulomb. This choice of value for k permits the practical electrical units, such as ampere and volt, to be included with the common metric mechanical units, such as metre and kilogram, in the same system.
^ ^a ^b Derived from k_e = 1/(4πε₀) – "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
^ "Charles-Augustin de Coulomb". Encyclopedia Britannica. Retrieved 3 March 2021.
^ Tomilin, K. (1999). "Fine-structure constant and dimension analysis". European Journal of Physics. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404. S2CID 250841835.
^ Coulomb's constant, HyperPhysics [bad link]

[1] Cleri, Fabrizio (2016). The Physics of Living Systems. Springer International Publishing. p. 611. ISBN 9783319306476.

[2] Oxford, Cambridge and RSA Examinations (2010). Data, Formulae and Relationships Booklet (Revised Version 2.2) – GCE Advanced Level and Advanced Subsidiary – Physics B (Advancing Physics) (PDF). p. 2.

[3] Milne, Edward Arthur (6 March 1937). "Letters to Editor: The Constant of Gravitation" (PDF). Nature. 139 (3514): 409. Retrieved 15 March 2023. Note added in proof, Feb. 26.⠀Complete agreement between Dirac's results and kinematic cosmology can be obtained, and his proportionalities M ∝ t², γ ∝ t⁻¹, replaced by M = const. = M₀, γ ∝ t, if we take the 'electrostatic' constant γ′ in the inverse square law to be proportional to t; γ′ is, of course, usually taken as unity. This is formally the same as taking e² ∝ t. The t-dynamics then gives the radius of the Bohr atom as proportional to t, and a material rod is 'rigid' on the τ-scale. Electrostatic forces then become to some extent analogous to gravitational forces, and might be discussed kinematically by examining the properties of an expanding sphere of mingled positive and negative particles.

[4] "Coulomb force". Encyclopedia Britannica. Retrieved 13 March 2023. In the metre–kilogram–second and the SI systems, the unit of force (newton), the unit of charge (coulomb), and the unit of distance (metre), are all defined independently of Coulomb's law, so the proportionality factor k is constrained to take a value consistent with these definitions, namely, k in a vacuum equals 8.98 × 10⁹ newton square metre per square coulomb. This choice of value for k permits the practical electrical units, such as ampere and volt, to be included with the common metric mechanical units, such as metre and kilogram, in the same system.

[ke-5] Derived from k_e = 1/(4πε₀) – "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.

[6] "Charles-Augustin de Coulomb". Encyclopedia Britannica. Retrieved 3 March 2021.

[Tomilin1999-7] Tomilin, K. (1999). "Fine-structure constant and dimension analysis". European Journal of Physics. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404. S2CID 250841835.

[8] Coulomb's constant, HyperPhysics [bad link]

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Coulomb constant
Common symbols	$k e$
Dimension	${\mathsf {L}}^{3}{\mathsf {M}}{\mathsf {T}}^{-2}{\mathsf {Q}}^{-2}$

Value of the constant[edit]

Use[edit]

See also[edit]

References[edit]