State Biot-Savart law.
A current $I$ flows in a conductor placed perpendicular to the plane of the paper. Indicate the direction of the magnetic field due to a small element $d\overrightarrow{l}$ at point $P$ situated at a distance $\overrightarrow{r}$ from the element as shown in Figure.

A charge $\text{'}q\text{'}$ moving along the $X$-axis with a velocity $\overrightarrow{v}$ is subjected to a uniform magnetic field $B$ acting along the $Z$-axis as it crosses the origin $O$.

(i) Trace its trajectory.

A charge $\text{'}q\text{'}$ moving along the $X$-axis with a velocity $\overrightarrow{v}$ is subjected to a uniform magnetic field $B$ acting along the $Z$-axis as it crosses the origin $O$.

(ii) Does the charge gain kinetic energy as it enters the magnetic field ? Justify your answer.

A long wire is bent into a circular coil of one turn and then into a circular coil of smaller radius having n turns. If the same current passes in both the cases, find the ratio of the magnetic fields produced at the centres in the two cases.

Two semi-infinitely long straight current carrying conductors are held at right angles to each other so that their common end lies at the origin, as shown in Figure. If both the conductors carry the same current I as shown, find the magnetic field at the point $P (a, b)$.

Two insulating infinitely long conductors carrying currents $I_1$ and $I_2$ lie mutually perpendicular to each other in the same plane, as shown in Figure. Find the locus of the point at which the net magnetic field is zero.

As shown in Figure, a cell is connected across two points $A$ and $B$ of a uniform circular conductor. Prove that the magnetic field at its centre $O$ will be zero.