A particle of charge - q and mass m moves in a circle of radius r...

Attractive electrostatic force due to line charge on the charged particle will provide the required centripetal force. Therefore,

$q\left(\dfrac{2k\lambda}{r}\right) = m{\omega}^{2}r$

^{$⇒ {\omega}^{2} = \dfrac{2k\lambdaq}{m{r}^{2}}$}

^{$⇒ {\left(\dfrac{2\pi}{T}\right)}^{2} = \dfrac{2k\lambdaq}{m{r}^{2}}$}

^{$⇒ T = 2\pir\sqrt{\dfrac{m}{2k\lambdaq}}$}