Question

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 $\sqrt{\ln \left(\frac{ r }{ r _{0}}\right)^{ N }}:$ The value of $\frac{1}{2 N }$ is ________.

Answer 1

At any point $P$, the force $\left(f_{P}\right)$ experienced by charge $=\frac{2 k \lambda q }{ x }$
Using Newton's second law of motion,
$mv \frac{ dv }{ dx }=\frac{2 k \lambda q }{ x } $
$\ldots .\left(\because \frac{ dv }{ dt }=\frac{ dv }{ dx } \cdot \frac{ dx }{ dt }= v \frac{ dv }{ dx }\right)$
$\therefore \int\limits_{0}^{v} mvdv=2 k \lambda q \int\limits_{r_{0}}^{r} \frac{d x}{x}$
$\left.\therefore \frac{m v^{2}}{2}\right|_{0} ^{v}=\left.2 k \lambda q \ln x\right|_{r_{0}} ^{r}$
$\therefore \frac{1}{2} m v^{2}=2 k \lambda q \ln \left(\frac{r}{r_{0}}\right)$
$\therefore v=\sqrt{\frac{4 k \lambda q}{m}} \sqrt{\ln \left(\frac{r}{r_{0}}\right)}$
$\therefore v \propto \sqrt{\ln \left(\frac{r}{r_{0}}\right)}$
$\therefore N =1 $
$ \Rightarrow \frac{1}{2 N }=0.5$