Two parallel beams of protons and electrons,carrying equal currents,are fixed at a separation $d$. The protons and electrons move in opposite directions. $P$ is a point on a line joining the beams,at a distance $x$ from the proton beam. The magnetic field at $P$ is $B$. If $B$ is plotted against $x$,which of the following best represents the resulting curve?

The adjoining figure shows a very long semi-cylindrical conducting shell of radius $R$ carrying a current $i$. An infinitely long straight current-carrying conductor is lying along the axis of the semi-cylinder. If the current flowing through the straight wire is $i_0$,then the force per unit length on the conducting wire is

$A$ loop of irregular shape made of flexible conducting wire carrying a clockwise current is placed in a uniform inward magnetic field,such that its plane is perpendicular to the field. Then the loop:

Six point charges,each of magnitude $q$,are arranged in different manners as shown in the image. In each case,a point $M$ and a line $PQ$ passing through $M$ are shown. Let $E$ be the electric field and $V$ be the electric potential at $M$ (potential at infinity is zero) due to the given charge distribution when it is at rest. Now,the whole system is set into rotation with a constant angular velocity about the line $PQ$. Let $B$ be the magnetic field at $M$ and $\mu$ be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current. Match the conditions in Column $I$ with the configurations in Column $II$.

Column $I$	Column $II$
$(A)$ $E=0$	$(p)$ Charges at corners of a regular hexagon. $M$ is the centre. $PQ$ is perpendicular to the plane.
$(B)$ $V \neq 0$	$(q)$ Charges on a line perpendicular to $PQ$ at equal intervals. $M$ is the mid-point.
$(C)$ $B=0$	$(r)$ Charges on two coplanar concentric rings. $M$ is the common centre. $PQ$ is perpendicular to the plane.
$(D)$ $\mu \neq 0$	$(s)$ Charges at corners and mid-points of a rectangle. $M$ is the centre. $PQ$ is parallel to the longer sides.
	$(t)$ Charges on two coplanar,identical rings. $M$ is the mid-point between centres. $PQ$ is perpendicular to the line joining centres.

Two parallel long wires carry currents $i_1$ and $i_2$ with $i_1 > i_2$. When the currents are in the same direction,the magnetic field midway between the wires is $10 \, \mu T$. When the direction of $i_2$ is reversed,it becomes $40 \, \mu T$. The ratio $i_1/i_2$ is

$A$ 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.