Question

Four charges $+q,+q,-q\text{and}-q$ are placed respectively at the corners $A,B,C\text{and}D$ of a square of side $a$ arranged in the given order. $E\text{and}F$ are midpoints of side $BC$ and $CD$ respectively. $O$ is the center of the square.

$\left(i\right)$ The electric field at $O$ is

​​​​​​​$[$1 Mark$]$

• A. $\frac{q}{\sqrt{2}\pi {\epsilon }_0{a}^2}$
• B. $\frac{q}{\sqrt{3}\pi {\epsilon }_0{a}^2}$
• C. $\frac{\sqrt{3}q}{\pi {\epsilon }_0{a}^2}$
• D. $\frac{\sqrt{2}q}{\pi {\epsilon }_0{a}^2}$

Answer 1

The correct option is D $\frac{\sqrt{2}q}{\pi {\epsilon }_0{a}^2}$

The magnitude of electric field at $O$ due to charges at $A,B,C\text{and}D$ will be equal as they have same magnitude of charge and also all the points are equidistant from $O$. $(AO=BO=CO=DO=\frac{a}{\sqrt{2}})$
$E_A=E_B=E_C=E_D=\frac{q}{4\pi {\epsilon }_0{\left(\frac{a}{\sqrt{2}}\right)}^2}$
Net electric field along $OD,E_1=E_B+E_D$
$E_1=2\left(\frac{q}{4\pi {\epsilon }_0{\left(\frac{a}{\sqrt{2}}\right)}^2}\right)$
Net electric field along $OC$
$E_2=E_A+E_C$
$E_2=2\left(\frac{q}{4\pi {\epsilon }_0{\left(\frac{a}{\sqrt{2}}\right)}^2}\right)$

Resultant of $E_1$ and $E_2$
$E_{net}=\sqrt{E_1^2+E_2^2}$
$E_{net}=\sqrt{2}\times \frac{2q}{4\pi {\epsilon }_0{\left(\frac{a}{\sqrt{2}}\right)}^2}$
$E_{net}=\frac{\sqrt{2}q}{\pi {\epsilon }_0{a}^2}$
$\therefore$ option $\left(d\right)$ is correct.