$A$ Van de Graaff generator has a spherical metal shell as an electrode which is at a potential $15 \times 10^6 \,V$. If the dielectric strength of the surrounding medium is $5 \times 10^7 \,V/m$, then the diameter of the shell is (in $\,cm$)

$A$ charge $+q$ is fixed at each of the points $x = x_0, x = 3x_0, x = 5x_0, \dots$ up to $\infty$ on the $X$-axis and a charge $-q$ is fixed at each of the points $x = 2x_0, x = 4x_0, x = 6x_0, \dots$ up to $\infty$. Here $x_0$ is a positive constant. Taking the potential at a point due to a charge $Q$ at a distance $r$ from it to be $\frac{Q}{4\pi\varepsilon_0 r}$,find the potential at the origin due to the above system of charges.

There are four concentric shells $A, B, C$ and $D$ of radii $a, 2a, 3a$ and $4a$ respectively. Shells $B$ and $D$ are given charges $+q$ and $-q$ respectively. Shell $C$ is now earthed. The potential difference $V_A - V_C$ is:

Charges of $+ \frac{10}{3} \times 10^{-9} \, C$ are placed at each of the four corners of a square of side $8 \, cm$. The potential at the intersection of the diagonals is

The diagram shows a small bead of mass $m$ carrying charge $q$. The bead can freely move on a smooth fixed ring placed on a smooth horizontal plane. In the same plane,a charge $+Q$ has also been fixed as shown. The potential at point $P$ due to $+Q$ is $V$. The velocity with which the bead should be projected from point $P$ so that it can complete a circle should be greater than:

Two point charges $4q$ and $-q$ are fixed on the $x$-axis at $x = -d/2$ and $x = d/2$,respectively. If a third point charge $q$ is moved from the origin to $x = d$ along the semicircle as shown in the figure,the potential energy of the charge will: