Q 12-02-177JEE MainJEE Main 2019 (8 Apr, Shift 2)Medium
A positive point charge is released from rest at a distance $r_0$ from a positive line charge with uniform charge density. The speed ($v$) of the point charge, as a function of instantaneous distance $r$ from line charge, is proportional to
Answer: (B) $v \propto \sqrt{\ln\dfrac{r}{r_0}}$
The field of the line charge is $E = \dfrac{\lambda}{2\pi\varepsilon_0r}$. The work done on the charge as it moves from $r_0$ to $r$ becomes kinetic energy:
$$\frac12mv^2 = \int_{r_0}^{r}\frac{q\lambda}{2\pi\varepsilon_0r}\,dr = \frac{q\lambda}{2\pi\varepsilon_0}\ln\frac{r}{r_0}$$
$$v \propto \sqrt{\ln\frac{r}{r_0}}$$
Solution by Sreeraj P, M.Sc Physics