An E$-$core with three air$-$gaps is given in the figure with dimensions of left mean core length $10$cm$,$right mean core length $10$cm$,$middle leg mean core length of $4$cm and core cross section of $1$cm$^2$and uniform through the geometry except $2$cm$^2$at middle leg.

i$)$Core is made out of a low$-$quality material $$A$$with relative permeability of $1000.$The coil with N $=300$turns is carrying a current of $1$Amps$.$

a$)$Calculate the coil inductance when there are no air$-$gaps. Left air gap is equal to the right air gap and when gleft$=$gright$=$x cm$,$gmiddle$=$x $+0\mathrm{.}1$cm$.$While left and right air gaps are simultaneously changed between $0$cm $$x $0\mathrm{.}1$cm $($use at least

$100$data points$)$:

b$)$Calculate and plot inductance $($L$)$of the coil with respect to middle gap length. What is the reason behind this change? Give your comments.

c$)$Calculate and plot the middle and left air gap flux density with respect to respective air gap lengths. Comment on your graphs.

ii$)$Now assume the core is made out of a high$-$quality material $$B$$with a relative

permeability of $15000.$Repeat the parts above. Comment on your findings. What is the reason of change, or lack thereof? How does adding an air$-$gap to a ferrous core affect the inductance?

Neglect leakage fluxes and fringing. Assume a uniform distribution of magnetic field inside the core. $($$0=4$ $10-7$H$/$m$)$