Preform a Dimensional analysis of Electric permittivity to show that the units of epsilon zero are C^2/(Nm^2).
by Coulomb's law the electric force is F = (1/4pi eo) q1q1 / d^2 so that electric permittivity of free space eo = q1 q2 / 4 pi F d^2 since q1,q2 are measured in coulomb C, F is measure in newton N and 'd' is measure in meter 'm' the units for ' eo ' will be C^2 / N m^2. that's it.
Since you didn't say what we're allowed to assume, I'm going to present two different approaches. The first is to assume Coulomb's Law, which say that the force between two charged particles is proportional to the magnitude of each charge, and inversely proportional to the distance between them. Thus, Coulomb said: F = [some constant] times Q1 Q2 / r^2 We are used to writing the constant of proportionality as 1 / (4 pi epsilon-nought), and the epsilon-nought is called the permittivity. Evidently epsilon-nought = Q1 Q2 / (4 pi F r^2), which would be in coulomb*coulomb/(newton*meter*meter). The second approach would be to start with some other definition of the permittivity: it can be defined as 1 / (mu-nought c^2), where mu-nought is vacuum permeability and c is the speed of light. c = 299 792 458 m/s (exact definition), and mu-nought = 4 pi x 10^(-7) H/m. But what is a henry in the definition of mu-nought? Well, if the current in a circuit is changing, it produces an electromotive force that is proportional to the rate of change of the current. The constant of proportionality is called inductance and is measured in henrys. Thus, (henrys) times (amps/seconds) = volts 1 henry = 1 volt-second/amp = 1 joule-sec/amp-coulomb = 1 joule-sec^2/coulomb^2 Going back to epsilon-nought, it is then 1 / (mu-nought times c^2), and its units are 1 / [ (joule-sec^2/coulomb^2) (m/s)^2 ] = coulomb^2 / (newton meter^2), the same result we got from assuming Coulomb's Law.
