Question

a] two equal point charges are fixed at x = -a ,and x=+a on the x axis .another point charge Q is placed at origin .the charge in the electrical potential energy of Q,when it is displaced by a small distance x along the x axis is approximately proportional to

Answer 1

Potential Energy of a charge $Q$ after it is displaced by a distance $x$ from the origin,
$$\begin{gathered} U_E=\frac{1}{4\pi {\epsilon }_0}Qq\left[\frac{1}{a+x}+\frac{1}{a-x}\right]=\frac{Qq}{4\pi {\epsilon }_0}\frac{2a}{\left(a^2-x^2\right)} \\ {U}_E=\frac{Qq}{4\pi {\epsilon }_0}\frac{2a}{a^2}{\left(1-\frac{x^2}{a^2}\right)}^{-1} \\ \approx \frac{Qq}{2\pi {\epsilon }_{0}a}\left(1+\frac{x^2}{a^2}\right) \end{gathered}$$