Correct Answer - Option 3 :

\(\sqrt {\frac{{4\pi {\varepsilon _0}F{d^2}}}{{{e^2}}}} \)
__Concept__:

According to Coulomb’s law, the force of repulsion between the two positive ions each of charge q, separated by a distance d is given by:

\(F = \frac{1}{{4\pi {\varepsilon _0}}}\frac{{\left( q \right)\left( q \right)}}{{{d^2}}}\)

\(F = \frac{{{q^2}}}{{4\pi {\varepsilon _0}{d^2}}}\) ---(1)

q = ne

n = number of electrons missing from each ion

e = magnitude of charge on electron

__Analysis__:

Rearranging Equation (1), we get:

\({q^2} = 4\pi {\varepsilon _0}F{d^2}\)

\(q = \sqrt {4\pi {\varepsilon _0}F{d^2}} \)

With q = ne

\( n = \frac{q}{e} = \sqrt {\frac{{4\pi {\varepsilon _0}F{d^2}}}{{{e^2}}}} \)