There is another useful system of units, besides the SI/mks A system, called the cgs (centimeter-gram-second) system. In this system Coloumb’s law is given by
`F=(Qq)/r^2 hatr` 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 1es unit of charge =`1/[3] ×10^(-9)`
The number [3] actually arises from the speed of light in vaccum which is now taken to be exactly given by c = 2.99792458 × 108 m/s. An approximate value of c then is c = [3] × 108 m/s.
(i) Show that the coloumb law in cgs units yields 1 esu of charge = 1 (dyne)1/2 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
`1/4pi epsilon_0 =10^(-9)/x^2(Nm^2)/C^2`
with `x=1/[[3]] xx10^(-9)`
we have
`1/(4pi epsilon_o)= [3]^2 xx 10^9 Nm^2/C^2`
`1/(4pi epsilon_o)=(2.99792458)^2xx10^9 Nm^2/C^2 (exactly)`