$A$ and $B$ are two parallel conductors carrying a current $i$ in the same direction. $x$ and $y$ are two electron beams moving in the same direction as shown in the figure.

$A$ compass needle free to turn in a horizontal plane is placed at the centre of a circular coil of $30$ turns and radius $12 \;cm$. The coil is in a vertical plane making an angle of $45^{\circ}$ with the magnetic meridian. When the current in the coil is $0.35 \;A$,the needle points west to east.
$(a)$ Determine the horizontal component of the earth's magnetic field at the location.
$(b)$ The current in the coil is reversed,and the coil is rotated about its vertical axis by an angle of $90^{\circ}$ in the anticlockwise sense looking from above. Predict the direction of the needle. Take the magnetic declination at the place to be zero.

Two long current-carrying thin wires,both with current $I$,are held by insulating threads of length $L$ and are in equilibrium as shown in the figure,with threads making an angle $\theta$ with the vertical. If the wires have mass $\lambda$ per unit length,then the value of current $I$ is ($g =$ gravitational acceleration).

$A$ very long wire $ABDMNDC$ is shown in the figure carrying current $I$. $AB$ and $BC$ parts are straight,long,and at a right angle. At $D$,the wire forms a circular turn $DMND$ of radius $R$. $AB$ and $BC$ parts are tangential to the circular turn at $N$ and $D$. The magnetic field at the centre of the circle is

$A$ metal sample carrying a current along $X-$ axis with density $J_x$ is subjected to a magnetic field $B_z$ (along $z-$ axis). The electric field $E_y$ developed along $Y-$ axis is directly proportional to $J_x$ as well as $B_z$. The constant of proportionality has $SI$ unit:

$A$ long thin-walled pipe of radius $R$ carries a current $I$ along its length. The current density is uniform over the circumference of the pipe. The magnetic field at the center of the pipe due to the quarter portion of the pipe shown is: