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 $x=2 q$, the potential at $O$ is $\frac{7 q}{4 \sqrt{3} \pi \in_0 a}$.
• D. When $x=-3 q$, the potential at $O$ is $\frac{3 q}{4 \sqrt{3} \pi \epsilon_0 a}$.

Answer 1

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

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

$ E _{\text {net }}=\frac{ kq }{(2 d )^2} \times 2=\frac{2 q \times 4}{4 \pi \varepsilon_0 \cdot 4 \cdot 3 a ^2} $
$=\frac{ q }{6 \pi \varepsilon_0 a ^2}$
(C) $v =\frac{7 kq }{2 d }=\frac{7 q }{4 \pi \varepsilon_0 \cdot \sqrt{3} a }=\frac{7 q }{4 \sqrt{3} \pi \varepsilon_0 q }$
(D) $v =\frac{2 kq }{2 d }=\frac{2 q }{4 \pi \varepsilon_0 \cdot \sqrt{3} a }=\frac{ q }{2 \sqrt{3} \pi \varepsilon_0 q }$