In gravity free space, a bead of charge $1\mu \mathrm{C}$ and mass $3mg$ is threaded on a rough rod of friction coefficient $\mu =0.3.$A magnetic field of magnitude $0.2T$ exists perpendicular to the rod. The bead is projected along the rod with a speed of $4 \mathrm{m}/\mathrm{s}$. How much distance (in $m$) will the bead cover before coming to rest?

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 :

A constant current source as shown in figure is connected across a galvanometer. If galvanometer is shunted by $4 \mathrm{\Omega }$ resistance then current in it is reduced to ${\left(\frac{1}{4}\right)}^{th}$ of initial value. If it is further shunted by $2 \mathrm{\Omega }$ resistance, current is further reduced by $\frac{k}{100}i$ while original current being same. Find value of $k$.