A non-polar molecule is located on the axis of a thin uniformly charged ring of radius $R$. The mole

Question

A non-polar molecule is located on the axis of a thin uniformly charged ring of radius $R$. The molecule is at a distance $x$ from the centre of the ring. Let $F$ be the magnitude of the electric force acting on the molecule. Which of the following statements is/are correct?

Professor PC Thomas Classes · Answer

Field $E = \frac{1}{4\pi\varepsilon_0}\frac{qx}{(R^2+x^2)^{3/2}}$. Induced dipole $p \propto E$. Force $F = p \frac{dE}{dx} \propto \frac{1}{2} \frac{d(E^2)}{dx}$. $E$ is max at $x = R/\sqrt{2}$, where $F=0$ and $p$ is max. Also $F=0$ at $x=0$. $F \propto \frac{xR^2-2x^3}{(R^2+x^2)^4}$. Setting $\frac{dF}{dx}=0$ gives $10x^4 - 13R^2x^2 + R^4 = 0 \Rightarrow x = R\sqrt{\frac{13 \pm \sqrt{129}}{20}}$. Force is repulsive for $x < R/\sqrt{2}$ and attractive for $x > R/\sqrt{2}$.