A point charge \(q\) is situated at a distance \(r\) on axis from one end of a thin conducting rod of length \(L\) having a charge \(Q\) (uniformly distributed along its length). The magnitude of electric force between the two is __________.

(D)

\(\Rightarrow\)As shown in figure, consider on elementary charge \(d q\) having length \(d x\) at a distance \(x\) from the charge \(q\), then
\(
\begin{array}{l}
d q=\lambda d x \\
d q=\frac{Q}{L} d x \quad\left(\because \lambda=\frac{Q}{L}\right)
\end{array}
\)
By Coloumb's law, force act between \(d q\) and \(q\).
\(
\begin{aligned}
d F & =\frac{k q d q}{x^2} \\
d F & =\frac{k q Q d x}{L x^2}
\end{aligned}
\)
\(\begin{array}{l} F =\int d F \\ F =\int_r^{r+ L } \frac{k Q q}{L x^2} d x \\ F=\frac{k Q q}{L} \int_r^r x^2 d x \\ F=\frac{k Q q}{L}\left|-\frac{1}{x}\right|_r^{+ L } \\ F =\frac{k Q q}{L}\left|-\frac{1}{r+ L }+\frac{1}{r}\right| \\ F =\frac{k Q q}{L}\left|\frac{1}{r}-\frac{1}{r+ L }\right| \\ F =\frac{k Q q}{L}\left|\frac{r+ L -r}{r(r+ L )}\right| \\ F =\frac{k Q q}{r(r+ L )}\end{array}\)