A rectangular loop of metallic wire is of length $a$ and breadth $b$ and carries current $I$. Find the magnitude of the magnetic field at the centre $O$ of the loop.

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.

An infinitely long conductor is bent into the shape as shown in Figure. The semicircular part lies in the $YZ$-plane ; while of the two straight parts, one is along $X$-axis and the other parallel to $Y$-axis at a distance $r$ from it. If the conductor carries a current $I$, as shown into the figure, find the magnetic field at the centre $C$ of the semicircular loop.

The wire loop $PQRSP$ formed by joining two semicircular wires of radii $R_1$ and $R_2$ carries a current I, as shown in Figure. Find the magnitude and direction of the magnetic field at the centre $O$.

A current $I$ is flowing in an infinitely long conductor bent into the shape shown in Figure. If the radius of the curved part is $R$, find the magnetic field at the centre $O$.

If the current density in a linear conductor of radius a varies with $r$ according to the relation : $J=k{r}^2$, where $k$ is a constant and $r$ is the distance of a point from the axis of the conductor. Find the magnetic field at a point distance $r$ from the axis when (i) $r<a$.