$A$ steady current flows in a long wire. It is bent into a circular loop of one turn and the magnetic field at the centre of the coil is $B$. If the same wire is bent into a circular loop of $n$ turns, the magnetic field at the centre of the coil is

For a circular coil of radius $R$ and $N$ turns carrying current $I$,the magnitude of the magnetic field at a point on its axis at a distance $x$ from its centre is given by,
$B=\frac{\mu_{0} I R^{2} N}{2\left(x^{2}+R^{2}\right)^{3 / 2}}$
$(a)$ Show that this reduces to the familiar result for field at the centre of the coil.
$(b)$ Consider two parallel co-axial circular coils of equal radius $R$ and number of turns $N,$ carrying equal currents in the same direction,and separated by a distance $R$. Show that the field on the axis around the mid-point between the coils is uniform over a distance that is small as compared to $R,$ and is given by,
$B=0.72 \frac{\mu_{0} N I}{R}, \quad \text { approximately }$

$N$ equally spaced charges,each of value $q$,are placed on a circle of radius $R$. The circle rotates about its axis with an angular velocity $\omega$ as shown in the figure. $A$ bigger Amperian loop $B$ encloses the whole circle,whereas a smaller Amperian loop $A$ encloses a small segment. The difference between enclosed currents,$I_A - I_B$,for the given Amperian loops is

In a hydrogen atom,an electron is revolving at $6.6 \times 10^{15} \text{ rev/s}$ around the nucleus in an orbit of radius $0.47 \text{ Å}$. The magnetic field induction produced at the centre of the orbit is nearly: (in $\text{ Wb m}^{-2}$)

$A$ solenoid of $1000$ turns per metre has a core with relative permeability $500$. Insulated windings of the solenoid carry an electric current of $5 \, A$. The magnetic flux density produced by the solenoid is (permeability of free space $= 4 \pi \times 10^{-7} \, H/m$)

Magnetic field intensity at the centre of a coil of $50$ turns,radius $0.5\, m$ and carrying a current of $2\, A$ is: