$A$ particle of mass $2.2 \times 10^{-30} \,kg$ and charge $1.6 \times 10^{-19} \,C$ is moving at a speed of $10 \,km/s$ in a circular path of radius $2.8 \,cm$ inside a solenoid. The solenoid has $25 \,turns/cm$ and its magnetic field is perpendicular to the plane of the particle's path. The current in the solenoid is (Take $\mu_0 = 4\pi \times 10^{-7} \,H/m$) (in $\,mA$)

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 $\Im(\infty)$?

The magnetic field inside a $200$ turns solenoid of radius $10 \ cm$ is $2.9 \times 10^{-4} \ T$. If the solenoid carries a current of $0.29 \ A$,then the length of the solenoid is . . . . . . $\pi \ cm$.

$A$ coaxial cable consists of an inner wire of radius $a$ surrounded by an outer shell of inner and outer radii $b$ and $c$ respectively. The inner wire carries an electric current $i_o$,which is distributed uniformly across its cross-sectional area. The outer shell carries an equal current in the opposite direction,also distributed uniformly. What will be the ratio of the magnetic field at a distance $x$ from the axis when $(i)$ $x < a$ and $(ii)$ $a < x < b$?

$A$ solenoid is $1 \ m$ long and $4 \ cm$ in diameter. It has five layers of windings of $1000$ turns each and carries a current of $7 \ A$. The magnetic field at the centre of the solenoid is

$A$ long solenoid has a radius $a$ and the number of turns per unit length is $n$. If it carries a current $i$,then the magnetic field on its axis is directly proportional to