An iron ring has a circular cross$-$sectional area of $\backslash (\backslash$mathbf$\{5\}\backslash$mathbf$\{$~ c m$\}^\{2\}\backslash )$ and a mean circumference of $\backslash (\backslash$mathbf$\{100\}\backslash$mathbf$\{$~ c m$\}\backslash ).$ The ring is uniformly wound with a coil of $\backslash (\backslash$mathbf$\{1000\}\backslash )$ turns. The relative permeability of iron is $\backslash (\backslash$mathbf$\{800\}\backslash ).$

a$)$ Find the current required to produce a flux of $\backslash (\backslash$mathbf$\{1$ ~ m W b$\}\backslash )$ in the ring.

b$)$ If a saw cut of $\backslash (\backslash$mathbf$\{2\}\backslash$mathbf$\{$~ m m$\}\backslash )$ wide is made in the ring, find the flux produced if the current is the same as that found in part a$.$

c$)$ Find the current required to produce the same flux as in part a for the cut made in the ring in part b$.$