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. $\frac{Q a}{\pi \varepsilon_{0} x^{3}}$
• C. $\frac{2 Q a}{\pi \varepsilon_{0} x^{3}}$
• D. $\frac{\sqrt{3} Q a}{\pi \varepsilon_{0} x^{3}}$

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.
$E_{0}=E+E=2 E=2 \frac{Q(2 a)}{4 \pi \varepsilon_{0} x^{3}}=\frac{Q a}{\pi \varepsilon_{0} x^{3}}$