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) $r<a$.

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 (ii) $r>a$.

A hollow cylindrical conductor of radii $a$ and $b$ carries a current $I$ uniformly spread over its crosssection. Prove that the magnetic field $B$ for points inside the body of the conductor at a distance $r$ from the axis of the cylinder is given by

$B=\frac{{\mu }_{0}I\left(r^2-a^2\right)}{2\pi \left(b^2-a^2\right)r}$.