A cylindrical cavity of diameter a exists inside a cylinder of diameter $2$a shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density $J$ flows along the length. If the magnitude of the magnetic field at the point $P$ is given by $\frac{N}{12} \mu_0$ aJ, then the value of $N$ is :
IIT 2012, Advanced
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$B _1=\frac{\mu_{0 Ja }}{2}-\frac{\mu_{0 Ja }}{12} $
$=\left(\frac{\mu_0 Ja }{2}\right)\left(1-\frac{1}{6}\right)=\frac{5}{6}\left(\frac{\mu_0 Ja }{2}\right)=\frac{5 \mu_0 aJ }{12}=\frac{ N }{12} \mu_0 aJ $
$N =5$
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