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Calculate inductance of finite Solenoid

  1. Dec 8, 2015 #1
    1. The problem statement, all variables and given/known data
    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.
    mvhpwm.png

    2. Relevant 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

    3. 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}##
     
    Last edited: Dec 8, 2015
  2. jcsd
  3. Dec 8, 2015 #2
    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 :wink:.

    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
    11s1soy.png
    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
     
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