A charge particle having charge $q=1\mathrm{C}$ and mass $0.25 \mathrm{kg}$ is projected from origin as shown in figure in $x-y$ plane magnetic field of $1\mathrm{T}$ is present perpendicular to plane. Find speed $v$(in $\mathrm{m}/\mathrm{s}$) so that particle can pass through point $A\left(2, 0\right).$

A positive charge particle enters and comes out from a uniform magnetic field which exists with direction inside the paper in a finite space of width $120 \mathrm{cm}$ as shown in the figure. Find the radius of the circular path of charge particle (in meter) when it is inside the magnetic field.

A line charge $\lambda ={10}^{-6} \mathrm{C} {\mathrm{m}}^{-1}$ is fixed on the rim of a wheel of radius $R=1 \mathrm{m}$ which is then suspended horizontally, so that it is free to rotate (the spokes are made of wood). In the central region upto radius $a=\frac{1}{2} \mathrm{m}$, there is uniform magnetic field, $\left|{\overrightarrow{B}}_0\right|=1 \mathrm{T}$ pointing up. Now suddenly the field is turned off. If the moment of inertia $I$ is $=0.25 \mathrm{kg} {\mathrm{m}}^{-2}$, the final angular velocity $\omega$ of the wheel is $n\pi \times {10}^{-6} \mathrm{rad} {\mathrm{s}}^{-1}$. Find $n$.

The long, horizontal pair of rails shown in the figure is connected using resistance $R\mathit{.}$ The distance between the rails is $\ell ,$ the electrical resistance of the rails is negligible. A conducting wire of mass $m$ and length $\ell$ can slide without friction on the pair of rails, in a vertical, homogeneous magnetic field of induction $B\mathit{.}$ A force of magnitude $F_0$ is exerted for sufficiently long time onto the conducting wire, so that the speed of the wire becomes nearly constant. The force $F_0$ is now removed at a certain point $P\mathit{.}$ The distance the conducting wire cover on rails from point $P$ before stopping is $(316+n)$ meters. Find $\text{'}n\text{'}.$ (Given : $F_0=20 \mathrm{N}, m=1.6 \mathrm{g}, R=0.01 \mathrm{\Omega }, \ell =10 \mathrm{cm},\mathit{ }B=0.1 \mathrm{T}$)

Two current carrying wire of infinite length are placed perpendicular to $x$-$y$ plane at $x=0$ and $x=R\sqrt{3}$ as shown in figure. Current in both the wires is same and inwards to the $x$-$y$ plane. Find ratio of $\left|\int \overrightarrow{B}·\mathrm{d}\overrightarrow{l}\right|$ in segment $CDA$ to segment $ABC$ will be :