Answer:
L = N^2Aμ/L
Explanation:
To find the inductance you use the following general formula:
[tex]L=N\frac{\Phi_B}{I}\\\\\Phi_B=BA[/tex]
N: turns of the solenoid
I: current in the solenoid
ФB: magnetic flux
B: magnitude of the magnetic field
A: cross area of the solenoid
Next, you take into account the magnitude of magnetic field inside a solenoid:
[tex]B=\frac{\mu NI}{L}\\\\\mu=k\mu_o[/tex]
where k is the relative permeability of the material inside the solenoid. the last expression for B also takes into account the long solenoid a approximation, which makes possible that magnetic fields effect in the borders of the solenoid is negligible.
Next, you replace the equation for B in the equation for L:
[tex]L=N\frac{A\mu_oNI}{IL}=N^2\frac{A\mu_o}{L}[/tex]
where you have used k=1 (for air).
If a ferromagnetic material is inserted in the solenoid the value of k is different of 1 and you have for L:
[tex]L=N^2\frac{Ak\mu_o}{L}[/tex]
