In a co-axial straight cable,the central conductor and the outer conductor carry equal currents in opposite directions. The magnetic field is zero:

The magnetic field intensity inside a current-carrying solenoid is $H = 2.4 \times 10^3 \ A/m$. If the length and the number of turns of the solenoid are $15 \ cm$ and $60$ turns respectively,the current flowing in the solenoid is: (in $A$)

$A$ long solenoid with $20$ turns per $cm$ is made. To produce a magnetic field of $20$ $mT$ inside the solenoid, the necessary current will be approximately......$A$.

Assertion : If the current in a solenoid is reversed in direction while keeping the same magnitude,the magnetic field energy stored in the solenoid decreases.
Reason : Magnetic field energy density is proportional to the square of the current.

Rank the value of $\oint \vec{B} \cdot d\vec{l}$ for the closed paths shown in the figure from the smallest to largest.

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 $\Im(\infty)$?