The surface of a planet is found to be uniformly charged. When a particle of mass $m$ and no charge is thrown at an angle from the surface of the planet,it has a parabolic trajectory as in projectile motion with horizontal range $L$. $A$ particle of mass $m$ and charge $q$,with the same initial conditions,has a range $L/2$. The range of a particle of mass $m$ and charge $2q$,with the same initial conditions,is:

The figure represents a crystal unit of cesium chloride, $CsCl$. The cesium atoms, represented by open circles, are situated at the corners of a cube of side $0.40 \, nm$, whereas a $Cl$ atom is situated at the centre of the cube. The $Cs$ atoms are deficient in one electron while the $Cl$ atom carries an excess electron.
$(i)$ What is the net electric field on the $Cl$ atom due to eight $Cs$ atoms?
$(ii)$ Suppose that the $Cs$ atom at the corner $A$ is missing. What is the net force now on the $Cl$ atom due to the seven remaining $Cs$ atoms?

Four point charges,each of $+q$,are rigidly fixed at the four corners of a square planar soap film of side $a$. The surface tension of the soap film is $\gamma$. The system of charges and the planar film are in equilibrium,and $a = k \left[ \frac{q^2}{\gamma} \right]^{1/N}$,where $k$ is a constant. Then $N$ is:

Two particles each of mass $m$ and charge $q$ are separated by distance $r_1$ and the system is left free to move at $t = 0$. At time $t$,both the particles are found to be separated by distance $r_2$. The speed of each particle is

Given below are two statements:
Statement $I$: An electric dipole is placed at the centre of a hollow sphere. The flux of electric field through the sphere is zero,but the electric field is not zero anywhere in the sphere.
Statement $II$: If $R$ is the radius of a solid metallic sphere and $Q$ is the total charge on it,the electric field at any point on a spherical surface of radius $r$ ( < R ) is zero,and the electric flux passing through this closed spherical surface of radius $r$ is also zero.
In the light of the above statements,choose the correct answer from the options given below:

In a regular polygon of $n$ sides,each corner is at a distance $r$ from the centre. Identical charges are placed at $(n - 1)$ corners. At the centre,the intensity is $E$ and the potential is $V$. The ratio $V/E$ has magnitude.