A particle of charge $$-q$$ and mass $$m$$ moves in a circle of radius $$r$$ around an infinitely long line charge of linear density $$+\lambda$$. Then time period will be given as: (Consider $$k$$ as Coulomb's constant)

Electric field due to infinite line charge at distance $$r$$: $$E = \frac{2k\lambda}{r}$$.

For circular motion: $$qE = \frac{mv^2}{r}$$.

$$q \cdot \frac{2k\lambda}{r} = \frac{mv^2}{r} \Rightarrow v^2 = \frac{2k\lambda q}{m}$$

$$v = \sqrt{\frac{2k\lambda q}{m}}$$

Time period: $$T = \frac{2\pi r}{v} = 2\pi r\sqrt{\frac{m}{2k\lambda q}}$$

The answer is Option (2): $$\boxed{T = 2\pi r\sqrt{\frac{m}{2k\lambda q}}}$$.