Two charges each of charge  $+10\mu \mathrm{c}$  are kept on Y-axis at y = -a and y = +a espectively. Another point charge $-20\mu \mathrm{c}$ is placed at the origin and given a small displacement x(x<<a) along X-axis. The force acting on the point charge is (x and a are metres ,$\left.\frac{1}{4\pi {\in }_0}=9\times {10}^9{\mathrm{Nm}}^2{\mathrm{C}}^{-2}\right)$

Given data: at y=-a and y=+a, two charges each of charge   $+10\mu \mathrm{C}$  are maintained on the y-axis. A second point charge, $-20\mu \mathrm{C}$  , is positioned at the origin with a tiny displacement along the x-axis , X(X a).

Concept used: electric field and electric field lines

Detailed solution:

 $F=k\frac{q_1{q}_2}{r^2}$

 $k=\frac{1}{4\pi {\epsilon }_0}=9\times {10}^9{\mathrm{Nm}}^2/C^2$ 

 $F=9\times {10}^9\times \frac{10\times {10}^{-6}\times 20\times {10}^{-6}}{{\left(\sqrt{a^2+x^2}\right)}^2}$ 

 $=9\times {10}^9\times \frac{200\times {10}^{-12}}{\left(a^2+x^2\right)}$

 $Net \text{force, }$ 

 $F_{\text{net }}=2\times 9\times {10}^9\times \frac{200\times {10}^{-12}}{\left(a^2+x^2\right)}·\cos \theta$

$\therefore F_{\text{net }}=2\times 9\times \frac{2\times {10}^{-1}}{\left(a^2+x^2\right)}\left(\frac{x}{\sqrt{a^2+x^2}}\right)$ 

 $x<<a\text{ Neglecting }{x}^2$ 

$F_{\text{net }}=3.6\frac{x}{a^3} \left(\because {\left(\sqrt{a^2+x^2}\right)}^3\approx a^3\right)$