$A$ long thin hollow metallic cylinder of radius $R$ carries a current $i$ ampere. The magnetic induction $B$ at a distance $r$ from the axis varies as shown in:

$A$ long solenoid of radius $1\,mm$ has $100$ turns per $mm$. If $1\,A$ current flows in the solenoid,the magnetic field strength at the centre of the solenoid is:

$A$ solenoid has a core of a material with relative permeability $400$. The windings of the solenoid are insulated from the core and carry a current of $1 \text{ A}$. If the number of turns is $1000 \text{ per metre}$,the magnetic field $(B)$ is . . . . . . $\text{T}$. (Given: $\mu_0 = 4\pi \times 10^{-7} \text{ SI units}$)

What is a solenoid? Provide information about the magnetic field through a qualitative discussion.

Derive the expression for the magnetic field inside a long straight solenoid.

Using Ampere's circuital law,derive the expression for the magnetic field at a perpendicular distance $r$ from an infinitely long current-carrying wire. Explain it.