The dimension of $\frac{1}{2} \varepsilon_0 E^2$ (where $\varepsilon_0$ is the permittivity of free space and $E$ is the electric field) is:

The dimension of $\frac{1}{2} \varepsilon_0 E^2$,where $\varepsilon_0$ is the permittivity of free space and $E$ is the electric field,is:

$A$ charged cork of mass $m$ suspended by a light string is placed in a uniform electric field of strength $E = (\hat{i} + \hat{j}) \times 10^5 \ NC^{-1}$ as shown in the figure. If in the equilibrium position,the tension in the string is $\frac{2mg}{(1 + \sqrt{3})}$,then the angle $\alpha$ with the vertical is:

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	$4. -y$

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.

$A$ negatively charged plate has a surface charge density of $2 \times 10^{-6} \, C/m^2$. Find the minimum initial distance of an electron moving toward the plate such that it does not strike the plate,given that its initial kinetic energy is $200 \, eV$.