Two electrons are moving along parallel lines in the same direction with the same velocity. They will:

$A$ charged particle of mass $m$ and charge $q$ moving under the influence of a uniform electric field $E\hat{i}$ and a uniform magnetic field $B\hat{k}$ follows a trajectory from point $P$ to $Q$ as shown in the figure. The velocities at $P$ and $Q$ are respectively $v\hat{i}$ and $-2v\hat{j}$. Which of the following statements $(A, B, C, D)$ are correct? (Trajectory shown is schematic and not to scale)
$(A)$ $E = \frac{3}{4}\left(\frac{mv^{2}}{qa}\right)$
$(B)$ Rate of work done by the electric field at $P$ is $\frac{3}{4}\left(\frac{mv^{3}}{a}\right)$
$(C)$ Rate of work done by both the fields at $Q$ is zero
$(D)$ The difference between the magnitude of angular momentum of the particle at $P$ and $Q$ is $2mav$.

Two insulated rings,one of slightly smaller diameter than the other,are suspended along their common diameter as shown. Initially,the planes of the rings are mutually perpendicular. When a steady current is set up in each of them:

Two very long straight parallel wires,parallel to the $y-$axis,carry currents $4I$ and $I$ along the $+y$ direction and $-y$ direction,respectively. The wires pass through the $x-$axis at the points $(d, 0, 0)$ and $(-d, 0, 0)$ respectively. The graph of the magnetic field $z-$component as one moves along the $x-$axis from $x=-d$ to $x=+d$ is best given by:

The dimensional formula of $\frac{1}{2} \mu_0 H^2$ (where $\mu_0$ is the permeability of free space and $H$ is the magnetic field intensity) is:

The dimensions of $\left(\frac{B^{2}}{\mu_{0}}\right)$ will be. (where $\mu_{0}$ is the permeability of free space and $B$ is the magnetic field)