Calculate inductance of finite Solenoid

[Cata]

Homework Statement

A finite solenoid with "N" turns of wire, "L" length , "R" is the radius of the solenoid and passes through it a current "I".
The objective is to calculate "L" of a finite solenoid. Not the basic formula $L=\frac{\mu_0·N^2·S}{Length}$ which is for a infinite solenoid.
See picture.

Homework Equations

Magnetic field produced by 1 coil at a point far from the coil a distance "x"
$B=\frac{\mu_0·I·R^2}{2(R^2+x^2)^\frac{3}{2}}$
x=distance from the center of the coil to a point in it's axes

The total magnetic flux into a solenoid is proportional to the current : $\phi_m=L·I$ where L=inductance of the solenoid

The Attempt at a Solution

First of all I calculate the magnetic field produced by the solenoid in a point out of the solenoid as follows:
The elementary magnetic field by a proportion of conductors in the region dx is:
$dB=\frac{\mu_0·I·R^2}{2(R^2+x^2)^\frac{3}{2}}·\frac{N}{L}dx$

And from the figure I find out that: $x=R·ctg\beta \Rightarrow dx=-R·(cosec\beta)^2·d\beta$ and $R^2+x^2=R^2(cosec\beta)^2$
So substituing the elementary magnetic field is: $dB=\frac{\mu_0·N·I}{2L} (-sin\beta d\beta)$

The total magnetic field in that point is:
$B=\frac{\mu_0·N·I}{2L}\int_(\beta_1)^(\beta_2) -sin\beta d\beta=\frac{\mu_0·N·I}{2L}(cos\beta_2 - cos\beta_1)$

And if the point is placed in the center of the first coil --> $cos\beta_1=0 ; cos\beta_2=\frac{L}{(L^2+R^2)^\frac{1}{2}}$

So the magnetic field in the first coil is : $B=\frac{\mu_0·N·I}{2L}\frac{L}{(L^2+R^2)^\frac{1}{2}}$

And now to calculate the magnetic flux through the first coil --> $\phi=\int_S^· BdS$

Before I keep doing my calculations my questions are:
1) It is correct what I have done until now ?
2) How do I calculate the magnetic flux $\phi_m$ through all the solenoid so then I can calculate the inductance $L=\frac{\phi_m}{I}$

 

[Cata]

Trying to find out the inductance like that is a difficult task. I thought it would be easy to find it but I am wrong... I have just found out by searching properly (with the accurate words) that there are expressions for most used forms of solenoids so I am going to share them with you... If somebody knows more about this field would be great to post them .

Cylindrical air core coil : $L=\frac{\mu_0 ·N^2 ·A·K}{L}$ where K=Nagaoka coefficient , A=area of cross section, N=number of turns, L=length of the solenoid

Here is a curve to detremine the Nagoaka coefficient

On the x axis: length = length of the solenoid; diameter=diameter of the solenoid (do not confuse it with the diameter of the wire you make the solenoid)

If the length >>>> diameter => Nagoaka coefficient is 1 so it is the case of the infinite solenoid