A long circular tube of length 10 m and radius 0.3 carries a current   along its curved surface as shown. A wire- loop of resistance $0.005\mathrm{\Omega }$  and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube

 

The current varies as $\mathrm{I}={\mathrm{I}}_0\cos 300\mathrm{t}$  where  I₀ is constant. If the magnetic moment of the loop is ${\mathrm{N\mu }}_0{\mathrm{I}}_0\sin \left(300\mathrm{t}\right),$  then  N is

Take the circular tube as a long solenoid. The wires are closely wound . Magnetic filed inside the  solenoid is
$\mathrm{B}={\mathrm{\mu }}_0\mathrm{ni}$
Here , n = number of turns per unit length
 $\therefore \mathrm{ni}$  current per unit length
In the given problem  $\mathrm{ni}=\frac{\mathrm{I}}{\mathrm{L}}$
$\therefore \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{I}}{\mathrm{L}}$
Flux passing through the circular coil is
$\mathrm{\varphi }=\mathrm{BS}=\left(\frac{{\mathrm{\mu }}_0\mathrm{I}}{\mathrm{L}}\right)\left({\mathrm{\pi r}}^2\right)$
Induced emf  $\mathrm{e}=-\frac{\mathrm{d\varphi }}{\mathrm{dt}}=-\left(\frac{{\mathrm{\mu }}_0{\mathrm{\pi r}}^2}{\mathrm{L}}\right).\frac{\mathrm{dI}}{\mathrm{dt}}$
Induced current,  $\mathrm{i}=\frac{\mathrm{e}}{\mathrm{R}}=-\left(\frac{{\mathrm{\mu }}_0{\mathrm{\pi r}}^2}{\mathrm{LR}}\right).\frac{\mathrm{dI}}{\mathrm{dt}}$
Magnetic moment,  $\mathrm{M}=\mathrm{iA}={\mathrm{i\pi r}}^2$
Or     $\mathrm{M}=-\left(\frac{{\mathrm{\mu }}_0{\mathrm{\pi }}^2{\mathrm{r}}^4}{\mathrm{LR}}\right).\frac{\mathrm{dI}}{\mathrm{dt}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}...\left(\mathrm{i}\right)$            
Given ,  $\mathrm{I}={\mathrm{I}}_0\cos \left(300\mathrm{t}\right)$
$\therefore$  $\frac{\mathrm{dI}}{\mathrm{dt}}=-300{\mathrm{I}}_0\sin \left(300\mathrm{t}\right)$
Substituting in Eq.(i)   we get
    $\mathrm{M}=\left(\frac{300{\mathrm{\pi }}^2{\mathrm{r}}^4}{\mathrm{LR}}\right){\mathrm{\mu }}_0{\mathrm{I}}_0\sin \left(300\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{t}\right)$
$\therefore$ $\mathrm{N}=\frac{300{\mathrm{\pi }}^2{\mathrm{r}}^4}{\mathrm{LR}}$
Substituting  the values , we get
$\mathrm{N}=\frac{300{\left(22/7\right)}^2{\left(0.1\right)}^4}{\left(10\right)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(0.005\right)}\text{}=5.926\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{or}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{N}\simeq 6$