Question

Six charges are placed at the vertices of a regular hexagon as shown in the figure. The electric field on the line passing through point $O$ and perpendicular to the plane of the figure as a function of distance $x$ from point $O$ is (assume $x>>a$ )

• A. 0
• B. $\frac{Qa}{\pi {\epsilon }_0{x}^3}$
• C. $\frac{2Qa}{\pi {\epsilon }_0{x}^3}$
• D. $\frac{\sqrt{3}Qa}{\pi {\epsilon }_0{x}^3}$

Answer 1

Answer: (B)

Two opposite charges situated at opposite vertices forms a dipole and point lies on bisector line of dipole for one dipole.

$\vec{E}=\frac{-K\vec{p}}{x^3}$
Direction of ficld due to these three dipoles is in a horizontal plane at distance $x$ from plane of hexagon and parallel to it, mutually having angle $60^{\circ }$ with each other.
$E_0=E+E=2E=2\frac{Q(2a)}{4\pi {\epsilon }_0{x}^3}=\frac{Qa}{\pi {\epsilon }_0{x}^3}$