$(a)$ Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (i.e., where $E =0$ ) of the configuration. Show that the equilibrium of the test charge is necessarily unstable.

$(b)$ Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.

Four charges are placed on corners of a square as shown in figure having side of $5\,cm$. If $Q$ is one microcoulomb, then electric field intensity at centre will be

Five charges, $\mathrm{q}$ each are placed at the corners of a regular pentagon of side $\mathrm{'a'}$ as in figure.

$(a)$ $(i)$ What will be the electric field at $O$, the centre of the pentagon ?

$(ii)$ What will be the electric field at $O$ if the charge from one of the corners (say $A$ $)$ is removed ?

$(iii)$ What will be the electric field at $O $ if the charge $q$ at $A$ is replaced by$ -q$ ?

$(b) $ How would your answer to $(a)$ be affected if pentagon is replaced by $n\,-$ sided regular polygon with charge $q$ at each of its corners ?

The intensity of the electric field required to keep a water drop of radius ${10^{ – 5}}\, cm$ just suspended in air when charged with one electron is approximately

A total charge $q$ is divided as $q_1$ and $q_2$ which are kept at two of the vertices of an equilateral triangle of side a. The magnitude of the electric field $E$ at the third vertex of the triangle is to be depicted schematically as a function of $x=q_1 / q$. Choose the correct figure.