There is another useful system of units, besides the SI/MKS. A system called the CGS (Centimeter-Gram-Second) system. In this system, Coulomb's law is given by $\mathrm{F}=\frac{\mathrm{Qq}}{{\mathrm{r}}^2}\hat{\mathrm{r}}.$

Where the distance r is measured in cm (=10^-2 m), F in dynes (=10^-5 N) and the charges in electrostatic units (es units), where 1 es units of charge$=\frac{1}{\left[3\right]}\times {10}^{-9}<mspace\; width="1em"></mspace>\mathrm{C}$. The number [3] actually arises from the speed of light in the vacuum which is now taken to be exactly given by $\mathrm{c}=2.99792458\times {10}^8<mspace\; width="1em"></mspace>\mathrm{m}/\mathrm{s}$. An approximate value of c then is $\mathrm{c}=3\times {10}^8<mspace\; width="1em"></mspace>\mathrm{m}/\mathrm{s}$.

(i) Show that  Coulomb's law in CGS units yields 1 esu of charge$=1{\left(\mathrm{dyne}\right)}^{1/2}<mspace\; width="1em"></mspace>\mathrm{cm}.$ Obtain the dimensions of units of charge in terms of mass M, length L and time T. Show that it is given in terms of fractional powers of M and L.

(ii) Write 1 esu of charge = x C, where x is a dimensionless number. Show that this gives $\frac{1}{4{\mathrm{\pi \epsilon }}_0}=\frac{{10}^{-9}}{{\mathrm{x}}^2}\frac{{\mathrm{Nm}}^2}{{\mathrm{C}}^2}$ with $\mathrm{x}=\frac{1}{\left[3\right]}\times {10}^{-9}$, we have $\frac{1}{4{\mathrm{\pi \epsilon }}_0}={\left[3\right]}^2\times {10}^9<mspace\; width="1em"></mspace>\frac{{\mathrm{Nm}}^2}{{\mathrm{C}}^2},<mspace\; width="1em"></mspace>\frac{1}{4{\mathrm{\pi \epsilon }}_0}={(2.99792458)}^2\times {10}^9\frac{{\mathrm{Nm}}^2}{{\mathrm{C}}^2}<mspace\; width="1em"></mspace>\left(\mathrm{exactly}\right)$