What is the magnetic field outside an ideal solenoid?

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 $\Im(\infty)$?

What is the magnetic field at point $P$ located at a distance of $\frac{3R}{2}$ from the center of the cylindrical conductor with inner radius $R$ and outer radius $2R$ carrying a current $i$?

$A$ toroid core has an inner radius of $0.24 \ m$ and an outer radius of $0.26 \ m$. $A$ current of $10 \ A$ flows through the wire having $2500$ turns around it. Find the magnetic field inside the core of the toroid.

$A$ current $i$ is uniformly distributed over the cross section of a long hollow cylindrical wire of inner radius $R_1$ and outer radius $R_2$. The magnetic field $B$ varies with distance $r$ from the axis of the cylinder as:

Ampere's circuital law is associated with which law?