question_answer

Equal charge q are placed at the four comers A, B, C, D of a square of length a. The magnitude of the force on the charge at B will be

A) \[\frac{3{{q}^{2}}}{4\pi {{\varepsilon }_{0}}{{a}^{2}}}\]                 

B) \[\frac{4{{q}^{2}}}{4\pi {{\varepsilon }_{0}}{{a}^{2}}}\]

C) \[\left( \frac{1+2\sqrt{2}}{2} \right)\frac{{{q}^{2}}}{4\pi {{\varepsilon }_{0}}{{a}^{2}}}\]      

D) \[\left( 2+\frac{1}{2} \right)\frac{{{q}^{2}}}{4\pi {{\varepsilon }_{0}}{{a}^{2}}}\]

Correct Answer: C

Solution :

   

\[{{F}_{net}}={{F}_{AC}}+{{F}_{D}}=\sqrt{F_{A}^{2}+F_{C}^{2}}+{{F}_{D}}\]

Since \[{{F}_{A}}={{F}_{C}}=\frac{k{{q}^{2}}}{{{a}^{2}}}\]and \[{{F}_{D}}=\frac{k{{q}^{2}}}{{{\left( a\sqrt{2} \right)}^{2}}}\]

\[{{F}_{net}}=\frac{\sqrt{2}k{{q}^{2}}}{{{a}^{2}}}+\frac{k{{q}^{2}}}{2{{a}^{2}}}=\frac{k{{q}^{2}}}{{{a}^{2}}}\left( \sqrt{2}+\frac{1}{2} \right)\]

\[=\frac{{{q}^{2}}}{4\pi {{\varepsilon }_{0}}{{a}^{2}}}\,\left( \frac{1+\sqrt{2}}{2} \right)\]