Q$1.$ Inductor and Inductance $(20$ points$)$

As illustrated in Figure Q$1($a$),$ two conducting wires are winded on a hollow cylinder to form inductors. The cross$-$section area of the cylinder is $S=6c{m}^2.$ The green wire has $N_1=62$ turns and a length of $l_1=5cm.$ And red wire has $N_2=24$ turns land a length of $l_2=3cm.$ The winding directions of them are opposite.

$($a$)$ For Figure Q$1($a$),$ find the inductance between point A and $A^{\text{'}}.$

$($b$)$ For Figure Q$1($a$),$ find the inductance between point $B$ and $B\text{'}.$

$($c$)$ As shown in Figure Q$1($b$),$ if point $A^{\text{'}}$ and point $B$ are connected, find the inductance between point A and $B^{\text{'}}.$