$(a)$ 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.

Two charged particles, each with a charge of $+q$, are located along the $x$ -axis at $x = 2$ and $x = 4$, as shown below. Which of the following shows the graph of the magnitude of the electric field along the $x$ -axis from the origin to $x = 6$?

Obtain the equation of electric field at a point by system of $\mathrm{'n'}$ point charges.

Two uniform spherical charge regions $S_1$ and $S_2$ having positive and negative charges overlap each other as shown in the figure. Point $O_1$ and $O_2$ are their centres and points $A, B, C$ and $D$ are on the line joining centres $O_1$ and $O_2$. Electric field from $C$ to $D$

For given arrangement, where four charge fixed at ends of as quare as given, find value of additional charge $Q$ to be put on one of the vertices so that component of net electric field along the vertical symmetric axis is zero at every point on the vertical