In the figure shown,a coil of single turn is wound on a sphere of radius $R$ and mass $m$. The plane of the coil is parallel to the inclined plane and lies in the equatorial plane of the sphere. The current in the coil is $i$. If the sphere is in equilibrium,the value of the magnetic field $B$ is:

$A$ uniform magnetic field of $0.3 \text{ T}$ is established along the $+z$-direction. $A$ rectangular loop of sides $10 \text{ cm}$ and $5 \text{ cm}$ carries a current of $12 \text{ A}$ and is placed in the $xy$-plane as shown in the figure. The torque acting on the loop is $....... \text{ Nm}$.

$(a)$ $A$ current-carrying circular loop lies on a smooth horizontal plane. Can a uniform magnetic field be set up in such a manner that the loop turns around itself (i.e.,turns about the vertical axis)?
$(b)$ $A$ current-carrying circular loop is located in a uniform external magnetic field. If the loop is free to turn,what is its orientation of stable equilibrium? Show that in this orientation,the flux of the total field (external field $+$ field produced by the loop) is maximum.
$(c)$ $A$ loop of irregular shape carrying current is located in an external magnetic field. If the wire is flexible,why does it change to a circular shape?

Two similar wires of equal lengths are bent in the form of a square and a circular loop. They are suspended in a uniform magnetic field and the same current is passed through them. The torque experienced by:

$A$ square loop of side $2a$ carrying current $I$ is kept in the $XZ$ plane with its centre at the origin. $A$ long wire carrying the same current $I$ is placed parallel to the $z$-axis and passes through the point $(0, b, 0)$,where $b >> a$. The magnitude of the torque on the loop about the $z$-axis is given by:

$A$ square loop of side $2a$ carrying current $I$ is kept in the $xz$-plane with its centre at the origin. $A$ long wire carrying the same current $I$ is placed parallel to the $z$-axis and passes through the point $(0, b, 0)$,where $b \gg a$. The magnitude of the torque on the loop about the $z$-axis is: