Two small conducting spheres of equal radius have charges of $10\ \mu C$ and $-20\ \mu C$ respectively and are placed at a distance $R$ from each other,experiencing a force $F_1$. If they are brought in contact and then separated to the same distance,the new force between them is $F_2$. Find the ratio $F_1 : F_2$.

Two point charges $+q$ and $-q$ are held fixed at $(-d, 0)$ and $(d, 0)$ respectively of a $(X, Y)$ coordinate system. Then:

Identify the correct statements:
$A$. Electrostatic field lines form closed loops.
$B$. The electric field lines point radially outward when charge is greater than zero.
$C$. The Gauss-Law is valid only for inverse-square force.
$D$. The work done in moving a charged particle in a static electric field around a closed path is zero.
$E$. The motion of a particle under Coulomb's force must take place in a plane.
Choose the correct answer from the options given below:

Four charges $Q_1, Q_2, Q_3$, and $Q_4$ of the same magnitude are fixed along the $x$-axis at $x = -2a, -a, +a$, and $+2a$, respectively. A positive charge $q$ is placed on the positive $y$-axis at a distance $b > 0$. Four options regarding the signs of these charges are given in List-$I$. The direction of the net force on the charge $q$ is given in List-$II$. Match List-$I$ with List-$II$ and select the correct answer using the codes given below the lists.

List-$I$	List-$II$
$P. Q_1, Q_2, Q_3, Q_4$ all positive	$1. +x$
$Q. Q_1, Q_2$ positive; $Q_3, Q_4$ negative	$2. -x$
$R. Q_1, Q_4$ positive; $Q_2, Q_3$ negative	$3. +y$
$S. Q_1, Q_3$ positive; $Q_2, Q_4$ negative	$4. -y$

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.

$A$ charge $Q$ is fixed at a distance $d$ in front of an infinite metal plate. The lines of force are represented by