A steady current $I$ flows down a long rod of radius $a$in such a way that its current density $\overrightarrow{\mathrm{J}}=k{r}^2\hat{r},$ where $k$ is a positive constant of appropriate dimensions, $r$ is the radial distance from the axis and  $\hat{r}$ is the unit vector along the radial direction. Then the magnitude of the magnetic field at any point  $r\left(r<a\right)$ inside the rod is

A very long straight wire of radius $r$ carries current $I$. Intensity of magnetic field $B$ at a point lying at a perpendicular distance $a$ from the axis is $\propto$ _____ $(\text{where} a<r)$.

Two very long straight wires are set parallel to each other. Each carries a current $I$ in the same direction and the separation between them is $2r.$ The intensity of the magnetic field at point $P$ as shown in the figure is _________.

A toroidal coil has $3000$ turns. The inner and outer radii are $10 \mathrm{cm}$ and $12 \mathrm{cm}$, respectively. If current flowing is $5 \mathrm{A}$, then the magnetic field inside the toroid will be

A solenoid of $1000$ turns per metre has a core with relative permeability $500.$ Insulated windings of the solenoid carry an electric current of $5 \mathrm{A}.$ The magnetic flux density produced by the solenoid is:

(Permeability of free space$=4\pi \times {10}^{-7} \mathrm{H} {\mathrm{m}}^{-1}$)