Question

Six charges are placed around a regular hexagon of side length a as shown in the figure. Five of them have charge $q$, and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the center $O$ of the hexagon and is bisected by the side.

Which of the following statement(s) is(are) correct in SI units?

• A. When $x=q$, the magnitude of the electric field at $O$ is zero.
• B. When $x=-q$, the magnitude of the electric field at $O$ is $\frac{q}{6\pi {\in }_0{a}^2}$.
• C. When $x=2q$, the potential at $O$ is $\frac{7q}{4\sqrt{3}\pi {\in }_{0}a}$.
• D. When $x=-3q$, the potential at $O$ is $\frac{3q}{4\sqrt{3}\pi {ϵ}_{0}a}$.

Answer 1

Answer: (A), (B), (C)

(A) Due to symmetry ${\overrightarrow{E}}_0=0$

$E_{\text{net }}=\frac{kq}{(2d{)}^2}\times 2=\frac{2q\times 4}{4\pi {\epsilon }_0\cdot 4\cdot 3a^2}$
$=\frac{q}{6\pi {\epsilon }_0{a}^2}$
(C) $v=\frac{7kq}{2d}=\frac{7q}{4\pi {\epsilon }_0\cdot \sqrt{3}a}=\frac{7q}{4\sqrt{3}\pi {\epsilon }_{0}q}$
(D) $v=\frac{2kq}{2d}=\frac{2q}{4\pi {\epsilon }_0\cdot \sqrt{3}a}=\frac{q}{2\sqrt{3}\pi {\epsilon }_{0}q}$