The total flux linked with the solenoid is
(
)(
)( )
0
NB
nl
n I
A
Φ
µ
=
0n2AlI

where nl is the total number of turns Thus, the self-inductance is,
L
I
ΝΦΒ
=
   
2
=µ0n Al
(6 15)
If we fill the inside of the solenoid with a material of relative permeability
mr (for example soft iron, which has a high value of relative permeability),
then,
2
0
r
L
=µ µn Al
(6 16)
The self-inductance of the coil depends on its geometry and on the
permeability of the medium