Two charged particles each having charge ' \(q\) ' and mass ' \(m\) ' are held at rest while their separation is ' \(r\) '. The speed of the particles when their separation is ' \(\frac{\mathrm{r}}{2}\), will be ( \(\varepsilon_0=\) permittivity of the medium)

By conservation of energy for the system of two charges,
\((\text { K.E. + P.E. })_{\text {initial }}=(\text { K.E. }+ \text { P.E. })_{\text {final }}\)
Initially the charges are at rest, \(\mathrm{KE}_{\text {initial }}=0\),
Final K.E. \(=-\frac{1}{2}(2 \mathrm{~m}) \mathrm{v}^2\)
...(Like charges repel each other)
\(\left(0+\left(\frac{1}{4 \pi \varepsilon_0} \frac{q^2}{r}\right)\right)=\left(-\frac{1}{2}(2 m) v^2+\left(\frac{1}{4 \pi \varepsilon_0} \frac{q^2}{r / 2}\right)\right) \)
\( \left(\frac{1}{4 \pi \varepsilon_0} \frac{q^2}{r}\right)=\left(-m v^2+\left(\frac{1}{4 \pi \varepsilon_0} \frac{q^2}{r / 2}\right)\right) \)
\( m v^2=\frac{q^2}{4 \pi \varepsilon_0}\left(\frac{2}{r}-\frac{1}{r}\right) \)
\( \therefore v=\frac{q}{\sqrt{4 \pi \varepsilon m r}}\)