A rectangular loop of wire is oriented with the left corner at the origin, one edge along $X$-axis and the other edge along $Y$-axis as shown in the figure. A magnetic field is into the page and has a magnitude that is given by $B=\alpha y$ where $\alpha$ is constant. Find the total magnetic force on the loop if it carries current $i$.

A cylindrical conductor of radius $R$ carries a current along its length. The current density $J$, however, it is not uniform over the cross section of the conductor but is a function of the radius according to $J=br$, where $b$ is a constant. Find an expression for the magnetic field $B$. (i) at $r_1<R$ & (ii) at distance $r_2>R$, measured from the axis

Three infinitely long conductors $R,S$ and $T$ are lying in a horizontal plane as shown in the figure. The currents in the respective conductors are $I_R=I_0\sin \left(\omega t+\frac{2\pi }{3}\right)I_S=I_0\sin \left(\omega t\right)I_T=I_0\sin \left(\omega t-\frac{2\pi }{3}\right)$. Find the amplitude of the vertical component of the magnetic field at a point $P$, distance '$a$' away from the central conductor $S$.

A square cardboard of side $l$ and mass $m$ is suspended from a horizontal axis $XY$ as shown in figure. A single wire is wound along the periphery of board and carrying a clockwise current $I$. At $t=0$, a vertical downward magnetic field of induction $B$ is switched on. Find the minimum value of $B$ so that the board will be able to rotate up to horizontal level.

A $U$-shaped wire of mass $m$ and length $l$ is immersed with its two ends in memory (see figure). The wire is in a homogeneous field of magnetic induction $B$. If a charge, that is, a current pulse $q=\int idt$, is sent through the wire, the wire will jump up. Calculate, from the height $h$ that the wire reaches, the size of the charge or current pulse, assuming that the time of the current pulse is very small in comparison with the time of flight, Make use of the fact that impulse of force equals $\int Fdt=mv$.

Evaluate $q$ for $B=0.1\mathrm{Wb} {\mathrm{m}}^{-2}, m=10 \mathrm{gm}, l=20 \mathrm{cm}$ & $h=3\mathrm{m}$. $\left[g=10 {\mathrm{ms}}^{-2}\right]$

A non-uniform magnetic field $\overrightarrow{B}=B_0\left(1+\frac{y}{d}\right)\widehat{k}$ is present in region of space in between $y=0$ and $y=d$. The lines are shown in the diagram. A particle of mass '$m$' and positive charge '$q$' is moving. Given an initial velocity $\overrightarrow{v}=v_0\widehat{i}$. Find the component of velocity of the particle when leaves the field.