A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the fig. 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: 

Magnetic field for infinitely long cylinder at a distance r is given as
 $$\begin{gathered} B_{in}=\frac{{\mu }_{0}Jr}{2} \\ {B}_{out}=\frac{{\mu }_{0}J{a}^2}{2r} \end{gathered}$$
 Assuming bigger cylinder to carry a positive current density and smaller cylinder has a negative current density of magnitude of J then magnetic field P is given as 
  $B_p=B_1+B_2$
Where  $B_1=\frac{{\mu }_{0}Ja}{2}$
And  $B_2=-\frac{{\mu }_{0}J{\left(\frac{a}{2}\right)}^2}{2.\frac{3a}{2}}=-\frac{{\mu }_{0}Ja}{12}$
 $$\begin{gathered} \Rightarrow B_p=\frac{5}{12}{\mu }_{0}Ja \\ \Rightarrow N=5 \end{gathered}$$