The following figures show regular hexagons with charges placed at their vertices. In which of the following cases is the electric field at the center zero?

Six infinitely large and thin non-conducting sheets are fixed in configurations $I$ and $II$. As shown in the figure,the sheets carry uniform surface charge densities which are indicated in terms of $\sigma_0$. The separation between any two consecutive sheets is $d = 1 \mu m$. The various regions between the sheets are denoted as $1, 2, 3, 4$ and $5$. If $\sigma_0 = 9 \mu C / m^2$,then which of the following statements is/are correct: (Take permittivity of free space $\epsilon_0 = 9 \times 10^{-12} F / m$):

Two identical balls having like charges $Q_1$ and $Q_2$ are placed at a certain distance $r$ apart and repel each other with a force $F_1$. They are brought into contact and then moved apart to a distance equal to half their initial separation $(r/2)$. The force of repulsion between them increases $4.5$ times in comparison with the initial value. The ratio of the initial charges of the balls is

Which of the following statement$(s)$ is/are correct?
$(A)$ If the electric field due to a point charge varies as $r^{-2.5}$ instead of $r^{-2}$,then the Gauss law will still be valid.
$(B)$ The Gauss law can be used to calculate the field distribution around an electric dipole.
$(C)$ If the electric field between two point charges is zero somewhere,then the sign of the two charges is the same.
$(D)$ The work done by the external force in moving a unit positive charge from point $A$ at potential $V_A$ to point $B$ at potential $V_B$ is $(V_B - V_A)$.

$A$ point charge '$Q$' is placed at the centre of the line joining two equal charges '$+q$' and '$+q$'. The value of '$Q$' when the system is in equilibrium is

Two identical charges $Q$ are separated by a distance $r$. $A$ third charge $q$ is placed at the midpoint of the line joining the two charges such that all three charges are in equilibrium. In this state,$q = $ . . . . . . .