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Coulomb's constant, the electric force constant, or the electrostatic constant (denoted k_e, k or K) is a proportionality constant in electrodynamics equations. In SI units, it is exactly equal to 7009898755178736817♠8987551787.3681764 N·m²·C⁻², or roughly equaling 7009899000000000000♠8.99×10⁹ N·m²·C⁻². It was named after the French physicist Charles-Augustin de Coulomb (1736–1806) who introduced Coulomb's law.

Value of the constant

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

F=keQqr2e^rdisplaystyle mathbf F =k_textefrac Qqr^2mathbf hat e _r

where ê_r is a unit vector in the r-direction and

ke=αℏce2displaystyle k_texte=alpha frac hbar ce^2,

where α is the fine-structure constant, c is the speed of light, ħ is the reduced Planck constant, and e is elementary charge.^[1] In SI:

ke=14πε0displaystyle k_texte=frac 14pi varepsilon _0,

where ε0displaystyle varepsilon _0 is the vacuum permittivity.
This formula can be derived from Gauss' law,

Sdisplaystyle scriptstyle S E⋅dA=Qε0displaystyle mathbf E cdot rm dmathbf A =frac Qvarepsilon _0

Taking this integral for a sphere, radius r, around a point charge, we note that the electric field points radially outwards at all times and is normal to a differential surface element on the sphere, and is constant for all points equidistant from the point charge.

Sdisplaystyle scriptstyle S E⋅dA=|E|e^r∫SdA=|E|e^r×4πr2mathbf E

Noting that E = F/Q for some test charge q,

F=14πε0Qqr2e^r=keQqr2e^r∴ke=14πε0displaystyle beginalignedmathbf F &=frac 14pi varepsilon _0frac Qqr^2mathbf hat e _r=k_textefrac Qqr^2mathbf hat e _r\[8pt]therefore k_texte&=frac 14pi varepsilon _0endaligned

In modern systems of units Coulomb's constant k_e is an exact constant, in Gaussian units k_e = 1, in Lorentz–Heaviside units (also called rationalized) k_e = 1/4π and in SI k_e = 1/4πε₀, where the vacuum permittivity ε₀ = 1/μ₀c² ≈ 6988885418782000000♠8.85418782×10⁻¹² F m⁻¹, the speed of light in vacuum c is 7008299792458000000♠299792458 m/s, the vacuum permeability μ₀ is 4π×10⁻⁷H m⁻¹,^[2]
so that^[3]

ke=14πε0=c2μ04π=c2×(10−7 H m−1)=8.9875517873681764×109 N m2 C−2.displaystyle beginalignedk_texte=frac 14pi varepsilon _0=frac c^2mu _04pi &=c^2times (10^-7 mathrm H m ^-1)\&=8.987,551,787,368,1764times 10^9~mathrm N m^2 C^-2 .endaligned

Use of Coulomb's constant

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

ke=14πε0.displaystyle k_texte=frac 14pi varepsilon _0.

Coulomb's constant appears in many expressions including the following:

Coulomb's law:

F=keQqr2e^r.displaystyle mathbf F =k_texteQq over r^2mathbf hat e _r.

Electric potential energy:

UE(r)=keQqr.displaystyle U_textE(r)=k_textefrac Qqr.

Electric field:

E=ke∑i=1NQiri2r^i.displaystyle mathbf E =k_textesum _i=1^Nfrac Q_ir_i^2mathbf hat r _i.

See also

• Gravitational constant


• Vacuum permittivity


• Vacuum permeability

References