Quick Calculation 3.7

A First Course in String Theory

The force $\displaystyle{\vec F}$ on a test charge $\displaystyle{q}$ in an electric field $\displaystyle{\vec E}$ is $\displaystyle{\vec F = q \vec E}$ . What are the units of charge in various dimensions?

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Eq. (3.74):

$\displaystyle{E(r) = \frac{\Gamma\left( \frac{d}{2} \right)}{2 \pi^{\frac{d}{2}}} \frac{Q}{r^{d-1}}}$

$\displaystyle{ \begin{aligned} \vec F &= q \vec E \\ &= \frac{\Gamma\left( \frac{d}{2} \right)}{2 \pi^{\frac{d}{2}}} \frac{qQ}{r^{d-1}} \\ \\ \\ [\vec F] &= \left[ \frac{\Gamma\left( \frac{d}{2} \right)}{2 \pi^{\frac{d}{2}}} \frac{qQ}{r^{d-1}} \right] \\ &= \left[ \frac{1}{1} \frac{q^2}{r^{d-1}} \right] \\ &= \left[ \frac{q^2}{r^{d-1}} \right] \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} [r^{d-1} \vec F] &= \left[ q^2 \right] \\ \\ \left[ q \right] &= [\sqrt{r^{d-1} \vec F}] \\ \\ &= \sqrt{m^{d-1} N} \\ \\ \end{aligned}}$

— Me@2022-06-08 11:09:27 AM

Lorentz–Heaviside units (or Heaviside–Lorentz units) constitute a system of units (particularly electromagnetic units) within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant $\displaystyle{\epsilon_0}$ and magnetic constant $\displaystyle{\mu_0}$ do not appear, having been incorporated implicitly into the electromagnetic quantities by the way they are defined. Heaviside-Lorentz units may be regarded as normalizing $\displaystyle{\epsilon_0 = 1}$ and $\displaystyle{\mu_0 = 1}$ , while at the same time revising Maxwell’s equations to use the speed of light $\displaystyle{c}$ instead.

Heaviside–Lorentz units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of $\displaystyle{4 \pi}$ appearing explicitly in Maxwell’s equations. That these units are rationalized partly explains their appeal in quantum field theory: the Lagrangian underlying the theory does not have any factors of $\displaystyle{4 \pi}$ in these units. Consequently, Heaviside-Lorentz units differ by factors of $\displaystyle{\sqrt{4\pi}}$ in the definitions of the electric and magnetic fields and of electric charge. They are often used in relativistic calculations, and are used in particle physics. They are particularly convenient when performing calculations in spatial dimensions greater than three such as in string theory.