Find the work and power required to move the conductor of length $l$ shown in the figure one full turn in the anticlockwise direction at a rotational frequency of $n$ revolutions per second if the magnetic field is of magnitude $B_0$ every where and points radially outwards from $Z$-axis. The figure shows the surface traced by the wire $AB$.

The figure shows a conductor of weight $1.0 \mathrm{N}$ and length $L=0.5 \mathrm{m}$ placed on a rough inclined plane making an angle $30$ with the horizontal so that conductor is perpendicular to a uniform horizontal magnetic field of induction $B=0.10 \mathrm{T}$. The coefficient of static friction between the conductor and the plane is $0.1$. A current of $I=10$ A flows through the conductor inside the plane of this paper as shown. What is the force needed to be the applied parallel to the inclined plane to sustaining the conductor at rest?

An electron gun $G$ emits electron of energy $2kev$ travelling in the $+ve$ $x$-direction. The electron are required to hit the spot $S$ where $GS=0.1\mathrm{m}$ & the line $GS$ makes an angle of $60°$ with the $x$-axis, as shown in the figure. A uniform magnetic field $\overrightarrow{B}$ parallel to $GS$ exists in the region outsides to electron gun. Find the minimum value of $B$ needed to make the electron hit at $S$.

Two long straight parallel wires are $2 \mathrm{m}$ apart, perpendicular to the plane of the paper. The wire $A$ carries a current of $9.6A$, directed into the plane of the paper. The wire $B$ carries a current such that the magnetic field of induction at the point $P$, at a distance of $\frac{10}{11}\mathrm{m}$ from the wire $B$, is zero. Find

(i) The magnitude and direction of the current in $B$.

(ii) The magnitude of the magnetic field due to induction at the point $S$.

(iii) The force per unit length on the wire $B$.

Two long parallel wires carrying currents $2.5 \mathrm{A}$ and $I$ (amperes) in the same direction (directed into the plane of the paper) are held at $P$ and $Q$ respectively such that they are perpendicular to the plane of paper. The points $P$ and $Q$ are located at the distance of $5 \mathrm{m}$ and $2 \mathrm{m}$ repetitively from a collinear point $R$ (see figure).

(i) An electron moving with a velocity of $4\times {10}^5 \mathrm{m} {\mathrm{s}}^{-1}$ along the positive $x$-direction experiences a force of magnitude  $3.2\times {10}^{-20} \mathrm{N}$ at the point $R$. Find the value of $I$.

(ii) Find all the positions at which a third long parallel wire carrying a current of magnitude $2.5$ A may be placed, so that the magnetic induction at $R$ is zero.

Figure shows a small magnetised needle $A$ placed at a point, $O$, the arrow shows the direction of its magnetic moment. The other arrows show different positions (and orientation of the magnetic moment) of another identical magnetised needle $B$.

(i) In which configurations is the system not in equilibrium?

(ii) In which configuration is the system in stable, and unstable equilibrium?

(iii) Which configuration corresponding to the lowest potential energy among all the configurations shown?