$A$ toroid has a non-ferromagnetic core of inner radius $r_{1}$ and outer radius $r_{2}$,around which $N$ turns of wire are wound. If the current in the wire is $I$,then the magnetic field inside the toroid is ($\mu_{0} =$ permeability of free space).

What is the magnetic field at point $P$ in the given figure?

Consider a circular current-carrying loop of radius $R$ in the $x-y$ plane with its centre at the origin. Consider the line integral $\Im(L) = \left| \int_{-L}^{L} \vec{B} \cdot d\vec{l} \right|$ taken along the $z$-axis.
$(a)$ Show that $\Im(L)$ monotonically increases with $L$.
$(b)$ Use an appropriate Amperian loop to show that $\Im(\infty) = \mu_0 I$,where $I$ is the current in the wire.
$(c)$ Verify this result directly.
$(d)$ Suppose we replace the circular coil with a square coil of side $R$ carrying the same current $I$. What can you say about $\Im(L)$ and $\Im(\infty)$?

$A$ rod with a circular cross-section area of $2\,cm^2$ and a length of $40\,cm$ is wound uniformly with $400$ turns of an insulated wire. If a current of $0.4\,A$ flows in the wire windings,the total magnetic flux produced inside the windings is $4\pi \times 10^{-6}\,Wb$. The relative permeability of the rod is (Given: Permeability of vacuum $\mu_0 = 4\pi \times 10^{-7}\,T\cdot m/A$)

$A$ $25\,cm$ long solenoid has radius $2\,cm$ and $500$ total number of turns. It carries a current of $15\,A.$ If it is equivalent to a magnet of the same size and magnetization $\vec M$ (magnetic moment per unit volume),then $\left| {\vec M} \right|$ is

$A$ closely wound solenoid $80\; cm$ long has $5$ layers of windings of $400$ turns each. The diameter of the solenoid is $1.8\; cm$. If the current carried is $8.0\; A$,estimate the magnitude of $B$ inside the solenoid near its centre.