A thin insulated tightly wound wire forms a plane spiral of $N$ turns and carries current $I.$ The radii of inside and outside turns are a and $3a$ respectively. Then

$ABCD$ is a rigid square wire frame carrying a steady clockwise current $i$ with its centre at the origin. At time $t=0,$ the frame is at rest in the position shown in the figure. Each side of the frame is of mass $m$ and length $l.$ A uniform, constant magnetic field $B$ is directed at an angle of $45°$ to the $x$-axis in the $x-y$ plane. If the initial angular acceleration is $\frac{xBi}{12\mathrm{m}},$ the value of $x$ is

Two circular rings each of radius $\text{'}a\text{'}$ are joined together such that their planes are perpendicular to each other as shown in figure. (circle with solid line representation in vertical plane and the other circle with dotted line representation is horizontal plane respectively) A very small loop of mass $m$ and radius $r$ carrying a current $I_0$ is placed in the plane of the paper at common centre of each ring. The loop can freely rotate about any of its diametric axis. Find the time period (in s) of loop for small oscillations $\left(ma=\frac{2\pi {\mu }_0{I}_0}{{\pi }^2}\right)·$ Resistances of each half ring is shown in figure.