Two identical point charges are placed at a separation of $l$. $P$ is a point on the line joining the charges,at a distance $x$ from any one charge. The field at $P$ is $E$. $E$ is plotted against $x$ for values of $x$ from close to zero to slightly less than $l$. Which of the following best represents the resulting curve?

Two positively charged particles $X$ and $Y$ are initially far away from each other and at rest. $X$ begins to move towards $Y$ with some initial velocity. The total momentum and energy of the system are $p$ and $E$.

In $1959$,Lyttleton and Bondi suggested that the expansion of the Universe could be explained if matter carried a net charge. Suppose that the Universe is made up of hydrogen atoms with a number density $N$,which is maintained constant. Let the charge on the proton be: $e_p = -(1 + y)e$,where $e$ is the electronic charge.
$(a)$ Find the critical value of $y$ such that expansion may start.
$(b)$ Show that the velocity of expansion is proportional to the distance from the centre.

An empty thick conducting shell of inner radius $a$ and outer radius $b$ is shown in the figure. If it is observed that the inner face of the shell carries a uniform charge density $-\sigma$ and the outer surface carries a uniform charge density $\sigma'$,and a point charge $q_A$ is placed at the center of the shell,then choose the correct statement$(s)$.

Consider two point charges of equal magnitude and opposite sign separated by a certain distance. The neutral point due to them

As shown in the figure,a point charge $q_{1} = +1 \times 10^{-8} \ C$ is placed at the origin in the $x-y$ plane and another point charge $q_{2} = +3 \times 10^{-6} \ C$ is placed at the coordinate $(10, 0)$. In that case,which of the following graph$(s)$ shows most correctly the electric field vector $E_{x}$ in the $x$-direction?