Statcoulomb

Coulomb's law in the Gaussian unit system and the SI are respectively:

${\displaystyle F={\frac {q_{1}^{\text{G}}q_{2}^{\text{G}}}{r^{2}}}}$ (Gaussian)

${\displaystyle F={\frac {q_{1}^{\text{SI}}q_{2}^{\text{SI}}}{4\pi \epsilon _{0}r^{2}}}}$ (SI)

Since ε₀, the vacuum permittivity, is not dimensionless, the coulomb is not dimensionally equivalent to [mass]^1/2 [length]^3/2 [time]⁻¹, 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" is 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 centimetres and metres. One can, however, find a correspondence between coulombs and statcoulombs in different contexts. As described below, "1 C corresponds to 3.00×10⁹ 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 3.00×10⁹ statC. Likewise, "1 C corresponds to 3.77×10¹⁰ statC" when describing an electric displacement field flux.

The statcoulomb is defined as follows: If two stationary objects each carry a charge of 1 statC and are 1 cm apart in vacuum, they will electrically repel each other with a force of 1 dyne. From this definition, it is straightforward to find an equivalent charge in coulombs. Using the SI equation

${\displaystyle F={\frac {q_{1}^{\text{SI}}q_{2}^{\text{SI}}}{4\pi \epsilon _{0}r^{2}}}}$ (SI),

and plugging in F = 1 dyn = 10⁻⁵ N, and r = 1 cm = 10⁻² m, and then solving for q = q^SI
₁ = q^SI
₂, the result is q = (1/2997924580) C ≈ 3.34×10⁻¹⁰ C. Therefore, an object with a charge of 1 statC has a charge of 3.34×10⁻¹⁰ 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:[lower-alpha 1]

${\displaystyle \mathrm {1~C} \times {\sqrt {\frac {10^{9}}{4\pi \epsilon _{0}}}}=\mathrm {1~C} \times 2997924580=\mathrm {1~statC} }$

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:

$${\displaystyle \Phi _{\mathbf {D} }^{\text{G}}=4\pi Q^{\text{G}}}$$

$${\displaystyle \Phi _{\mathbf {D} }^{\text{SI}}=Q^{\text{SI}}}$$

where

$${\displaystyle \Phi _{\mathbf {D} }\equiv \int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} }$$

Therefore, the conversion factor for flux is 4π different from the conversion factor for charge:

$${\displaystyle \mathrm {1~C} ~{\overset {\frown }{=}}~\mathrm {3.7673\times 10^{10}~statC} \qquad {\text{(as unit of }}\Phi _{\mathbf {D} }{\text{).}}}$$

The dimensionally consistent version is:[lower-alpha 1]

$${\displaystyle \mathrm {1~C} \times {\sqrt {\frac {4\pi \times 10^{9}}{\epsilon _{0}}}}=\mathrm {3.7673\times 10^{10}~statC} \qquad {\text{(as unit of }}\Phi _{\mathbf {D} }{\text{).}}}$$