Calculate the net force acting on the charge present at the origin.

  

• A

  $\frac{q{q}_1}{4\pi {\epsilon }_0{a}^2}\times \sqrt{2}$
• B

  $\frac{q{q}_1}{4\pi {\epsilon }_0\sqrt{2}{a}^2}+\frac{q{q}_1}{8\pi {\epsilon }_0{a}^2}$
• Correct answer
  C

  $\frac{q{q}_1}{4\pi {\epsilon }_0{a}^2}\left(\sqrt{2}+\frac{1}{2}\right)$
• D

  $\frac{q{q}_1}{4\pi {\epsilon }_0{a}^2}\left(\frac{1}{2}-\sqrt{2}\right)$

Solution

$F_1=k\frac{q{q}_1}{a^2};F_2=k\frac{q{q}_1}{{\left(\sqrt{2}a\right)}^2}$

Resultant force $F=\sqrt{2}{F}_1+F_2$

$=\sqrt{2}\frac{kq{q}_1}{a^2}+\frac{kq{q}_1}{2a^2}$

$=\frac{kq{q}_1}{a^2}\left(\sqrt{2}+\frac{1}{2}\right)$

$=\frac{q{q}_1}{4\pi {\epsilon }_0{a}^2}\left(\sqrt{2}+\frac{1}{2}\right)$

Hence, the correct answer is option: 'C'.