$A$ 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 $\mu_0 = 4\pi \times 10^{-7}\,T\cdot m/A$)

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 $\Im(\infty)$?

$A$ thick uniformly charged hollow cylinder of inner radius $a$ and outer radius $b$ rotates with constant angular speed $\omega$ about its axis $APB$,with charge density $\rho$. Given that $L \gg a$ and $L \gg b$,and $P$ is the midpoint of $AB$. Choose the incorrect option.

$A$ current $i$ ampere flows along the inner conductor of a coaxial cable and returns along the outer conductor of the cable. The magnetic induction at any point outside the conductor at a distance $r$ metre from the axis is

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 $i$?

For a long current-carrying solenoid,the magnetic field inside is $0.6 \ T$. The magnetic energy per unit volume is . . . . . . .