The magnetic field a distance \(z\) above the centre of a circular loop of radius \(r\) carrying current \(I\) in the \(x y\) plane is \(\vec{B}=\frac{\mu_{0} I r^{2}}{2\left(r^{2}+z^{2}\right)^{3 / 2}} \hat{k}\). 

(a) Use this result to re-derive the formula for the magnetic field inside a very long solenoid of radius \(r\) with \(n\) windings per unit length, carrying a current \(I\) per winding.

(b) Now replace the very long solenoid with one of arbitrary length \(L\), and find the magnetic field on its central axis, a distance \(D\) from its centre.