A circular coil has a $7\mathrm{.}52$ cm radius and consists of $32\mathrm{.}0$ closely wound turns of wire. An externally produced magnetic eld of magnitude $2\mathrm{.}97$ mT is perpendicular to the coil. $($a$)$ If no current is in the coil, what magnetic ux links its turns?$($b$)$ When the current in the coil is $4\mathrm{.}27$ A in a certain direction, the net ux through the coil is found to vanish. What is the inductance of the coil?