Quest - a variant of laboratory work "Investigation of the axial magnetic induction of a solenoid"

. A variant of the laboratory work "Investigation of the axial magnetic induction of the solenoid" is given for the case when the solenoid is located in a separate block and its parameters are unknown. The results of calculations of axial induction of a solenoid with a different ratio of the length of the solenoid to the diameter are given. It is shown that the maximum of the derivative of axial induction in relative length corresponds to the physical end of the solenoid with sufficient accuracy for practice.An algorithm for determining the parameters of a solenoid based on experimental data on measuring the axial induction of the solenoid is described. Laboratory work has a character that orients students to independent research. Processing of digital data is carried out using the Excel program.


Introduction
Laboratory work "Investigation of axial magnetic induction of solenoid" is included in the set of laboratory works, accompanying the study of physics course in technical universities [1,2,3].During the work students get acquainted with a technical device consisting of solenoid, special rod (probe) with a Hall sensor [4,5].
The methodical literature [1,2,6] and the Internet [7, 8] contain an analysis of the dependence of the magnetic field induction modulus on the solenoid axis on the distance from the solenoid center [9].In most practical implementations, the solenoid is open and its geometrical parameters are not the subject of research.

Materials and Methods
At the Department of Physics of MSCU students are offered a variant, where the solenoid is placed in a closed unit ("black box") and is inaccessible for observation.Lack of visibility of the device can be easily eliminated by taking a picture of a sample solenoid in the text of the laboratory work.It can be recommended to install an additional solenoid on a common bench.An essential advantage of this option is the possibility in the course of the experiment to determine the parameters of the solenoid enclosed in the "black box".A picture of the setup is shown in Fig. 1.To solve the problem, we consider some theoretical points that allow us to solve the problem [10].As a rule, in laboratory works we study the magnetic field of solenoid along its axis rather than along the radius [11].In this case, the ratio of the axial induction at distance   from the middle of the solenoid to the induction in the center is equal:    We find the extremum of the derivative by repeatedly differentiating

Results
Analyzing the resulting equation, we expect (based on experimental data) that the extremum is at the edge of the solenoid.Therefore, it is approximately equal to 2 and the first term will be a value close to 2 -4 .For the extremum (0-value of second derivative) to appear, the second term should be negative.
The results of the exact numerical solution of the equation are shown in Fig. 4.

Fig.4. Distribution of the relative induction derivative
As follows from Fig. 4, the position of the extremum at the length of the solenoid equal to the diameter for the ratio equal to 2 is 0.25%.Consequently, in laboratory practice [12], the position of the extremum with sufficient accuracy determines the position of the solenoidal edge.
The value of the extremum of the derivative at the edge of the solenoid is .
For small values , it asymptotically approaches the value of , (see Figure 5).5 Set the probe with the measuring bar 3 to position 0. 6 Gradually pushing the probe inside the solenoid, fixing its position and the voltage value corresponding to it, find the derivative of the relative induction from the distance.Figure 6 shows a sample of processing the experimental data of measuring the axial induction of the solenoid and calculating the derivative in Excel.
7 Using the data obtained, find the length of the solenoid as the position of the edge of the solenoid corresponding to the minimum (maximum) of the derivative, (see Fig. 6).
8 Using the dependence of the relative induction on the geometric characteristics of the solenoid Fig. 2 and Fig. 3. Find the diameter of the solenoid.

Fig. 6. Results of the study
To increase the accuracy of determining the parameters, the averaging algorithm "sliding sum" is applied.
As can be seen from fig. 6, the maximum value of the derivative of the magnetic field induction has pronounced changes at the initial and final positions of the probe.In the middle part, this value is significantly less.When considering the distribution of the induction of the magnetic field of the solenoid [13] and its derivative, Fig. 6.The following areas can be noted.Jump-like changes in the derivative correspond to the boundaries of the solenoid.Further, the values of the magnetic field induction and its derivative change monotonously, gradually taking on a "practically" constant value in the middle part of the solenoid.

Conclusions
1.The description of the variant of laboratory work "Investigation of axial magnetic induction of solenoid" for the case when the solenoid is located in a separate block and its parameters are unknown.
2. An algorithm of solenoid parameters calculation on the basis of experimental data of solenoid axial induction measurement is defined.
3. An example of processing the experimental digital data with the help of Excel program is given.

𝐶𝐶 2 E3SFigure 2
Figure2shows the dependence of the relative axial induction on the ratio of or

Fig. 2 3 E3S
Fig. 2 Distribution of induction in the solenoid.The numbers in the figure correspond to the L/d ratio of the different solenoid designs.

Figure 3 .
Figure 3. Relative induction derivative distribution The numbers in the figure correspond to those in Fig. 2.

Fig. 5 .
Fig. 5. Minimum of the derivative of the relative induction