Question

Six charges are placed at the vertices of a rectangular 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 (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

The correct option is B $\frac{Qa}{\pi {\epsilon }_0{x}^3}$

Consider any two charges opposite sides of O, they form a dipole where $p=2Qa$.

Electric field due to dipole on it's axis is given by

$E=\frac{kp}{x^3}=\frac{Qa}{2\pi {ϵ}_0{x}^3}$

Now three fields exists as shown in figure, so

$E_{net}=2|E|=\frac{Qa}{\pi {ϵ}_0{x}^3}$