A cylinder cavity of diameter a exists inside a cylinder of diameter 2a as 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 $${N \over {12}}{\mu _0}aJ$$, then the value of N is ______________.

A long circular tube of length 10 m and radius 0.3 m carries a current I along its curved surface as shown. A wire-loop of resistance 0.005 $$\Omega$$ 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 $$I = {I_0}\cos (300t)$$, where I_{0} is constant. If the magnetic moment of the loop is $$N{\mu _0}{I_0}\sin (300t)$$, then N is ___________.

A steady current I goes through a wire loop PQR having shape of a right angle triangle wit6h PQ = 3, PR = 4x and QR = 5x. If the magnitude of the magnetic field at P due to this loop is $$k\left( {{{{\mu _0}I} \over {48\pi x}}} \right)$$, find the value of $$k$$.