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 `vecF =(Qq)/(r^(2)).hatr` where the distance is measured in cm (=`10^(-2)`m), F in dynes (=`10^(-5)` N) and the charges in electrostatic units (es units), where 1 es unit of charge `1/(3) xx 10^(-9) C`.
The number [3] actually arises from the speed of light in vacuum which is now taken to be exactly given by `c = 2.99792458 xx 10^8` m/s. An approximate value of c then is `c = [3] xx 10^8` 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/(4piepsilon_(0)) = 10^(-9)/x^(2) (Nm^(2))/C^(2)` with `x=1/[3] xx 10^(-9)`, we have `1/(4pi epsilon_(0)) = [3]^(2) xx 10^(9) (Nm^(2))/C^(2)` or `1/(4pi epsilon_(0)) = (2.99792458)^(2) xx 10^(9) (Nm^(2))/C^(2)` (exactly).

Explanation

(i) `F = (Qq)/r^(2)`
`therefore` 1 dyne `=("1 esu charge")^(2)/(1 cm)^(2)`
`therefore 1 esu = (1 "dyne")^(1//2) xx 1 cm`
`=F^(1//2) L`
`therefore` Dimensional formula of 1 esu,
`=[M^(1//2)L^(3//2)T^(-1)]`
Hence, in dimensional formula of esu charge, power of M is `1/2` and of L is `3/2`, which is non-integer.
(ii) Suppose 1 esu = xC, where x is a dimensionless number. The force between two charges of 1 esu magnitude is `10^(-5)` N ( = 1 dyne) when they are at distance `10^(-2)` m (= 1 cm).
`therefore F = 1/(4pi epsilon_(0)).x^(2)/(10^(-2))^(2)`
`= 10^(-9)/x^(2).(Nm^(2))/C^(2)`
but `x = 1/(|3| xx 10^(9))`
`1/(4pi epsilon_(0)) = 10^(-9) xx |3|^(2) xx 10^(18) (Nm^(2))/C^(2)`
`=9 xx 10^(9) (Nm^(2))/C^(2)`
If `|3| = 2.99792458`, then
`=1/(4pi epsilon_(0)) = 10^(-9) xx (2.99792548)^(2) xx 10^(18)`
`=8.98755 xx 10^(9)`