Two point charges each of charge +q are fixed at (+a, 0) and (–a, 0). Another positive point charge q placed at the origin is free to move along x- axis. The charge q at origin in equilibrium will have

minimum force & minimum potential energy

The net force on q at origin is

$\vec{F}=\vec{F}_1+\vec{F}_2=\frac{1}{4 \pi \varepsilon_0} . \frac{q^2}{r^2} \hat{i}+\frac{1}{4 \pi \varepsilon_0} . \frac{q^2}{r^2}(-\hat{i})=0$

The P.E. of the charge q in between the extreme charges at a distance x from the origin along +ve x axis is

$U=\frac{1}{4 \pi \varepsilon_0} . \frac{q^2}{(a-x)}+\frac{1}{4 \pi \varepsilon_0} . \frac{q^2}{(a+x)}=\frac{1}{4 \pi \varepsilon_0} . q^2\left[\frac{1}{a-x}+\frac{1}{a+x}\right]$

$\frac{dU}{dx}=\frac{q^2}{4 \pi \varepsilon_0}\left[-\frac{1}{(a-x)^2}+\frac{1}{(a+x)^2}\right]$

For U to be minimum, $\frac{d U}{d x}=0$, and $\frac{d^2 U}{d x^2}>0$,

$\Rightarrow(a-x)^2=(a+x)^2$

⇒ a + x = ±(a – x)

⇒ x = 0, because other solution is relevant.

Thus, the charged particle at the origin will have minimum force and minimum P.E.

∴ (D)