Question

Nine point charges placed at the vertices of an octagon and centre of an octagon as shown in the figure.

Net force experienced by $+q$ charge at the centre of the octagon is zero. If any two charges are removed simultaneously from the vertex, it observed that net force on charge $+q$ is still zero, then the charges will be

• A. $+3Q$ and $+2Q$
• B. $+Q$ and $-Q$
• C. $-4Q$ and $+3Q$
• D. any pair of diagonally opposite charges

Answer 1

The correct option is D any pair of diagonally opposite charges

The force experienced by the charge $+q$ placed at the centre of the octagon is shown in the figure.

Let $a$ be the distance between centre and vertex of the octagon.

So, Force on $+q$ due to $-Q$,
$F_{02}=F_{06}=\frac{kQq}{a^2}$

Similarly, $F_{03}=F_{07}=\frac{k\left(3Q\right)q}{a^2}=\frac{3kQq}{a^2};$

$F_{04}=F_{08}=\frac{4kQq}{a^2}$ and

$F_{01}=F_{05}=\frac{2kQq}{a^2}$

Here, only diagonally opposite charges apply equal and opposite forces on $+q$.

So, if we remove any pair of diagonally opposite charges from vertex of the octagon, net forces on $+q$ will be unaffected.

Hence, option $\left(d\right)$ is the correct answer.