A variable reluctance sensor consists of a core, a variable air gap and an armature. The core is a steel rod of diameter $1$ cm and relative permeability $100,$ bent to form a semi$-$circle of diameter $4$ cm$.$ A coil of $500$ turns is wound onto the core. The armature is a steel plate of thickness $0\mathrm{.}5$ cm and relative permeability $100.$ Assuming the relative permeability of air $1\mathrm{.}0$ and the permeability of free space $=4$pi $*10^-7*$ H $*$ m $^-1.$ calculate the inductance of the sensor for air gaps of $1$ mm and $3$ mm$.$