A particle of charge $q$ and mass $m$ is subjected to an electric field $E = E _{0}\left(1- ax ^{2}\right)$ in the $x-$direction, where $a$ and $E _{0}$ are constants. Initially the particle was at rest at $x=0 .$ Other than the initial position the kinetic energy of the particle becomes zero when the distance of the particle from the origin is

A simple pendulum with a bob of mass $m = 1\ kg$ , charge $q = 5\mu C$ and string length $l = 1\ m$ is given a horizontal velocity $u$ in a uniform electric field $E = 2 × 10^6\ V/m$ at its bottom most point $A$ , as shown in figure. It is given a speed $u$ such that the particle leave the circular path at its topmost point $C$ . Find the speed $u$ . (Take $g = 10\ m/s^2$ )

When three electric dipoles are near each other, they each experience the electric field of the other two, and the three dipole system has a certain potential energy. Figure below shows three arrangements $(1)$ , $(2)$ and $(3)$ in which three electric dipoles are side by side. All three dipoles have the same magnitude of electric dipole moment, and the spacings between adjacent dipoles are identical. If $U_1$ , $U_2$ and $U_3$ are potential energies of the arrangements $(1)$ , $(2)$ and $(3)$ respectively then

This question contains Statement$-1$ and Statement$-2$. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement$-1$ : For a charged particle moving from point $P$ to point $Q$, the net work done by an electrostatic field on the particle is independent of the path connecting point $P$ to point $Q$.
Statement$-2$ : The net work done by a conservative force on an object moving along a closed loop is zero.

A point charge $q$ is surrounded by eight identical charges at distance $r$ as shown in figure. How much work is done by the forces of electrostatic repulsion when the point charge at the centre is removed to infinity?