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## Strip Magnetic Dipole of Finite Length and Magnetic Particle Inspection

Wei-Chang Zhong
Nanjing Gas Turbine Research Institute, 47 Northern Central Road,
Nanjing, 210037, People's Republic of China.
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### Abstract

For simulating the rectangular crack on a work piece surface by a strip magnetic dipole of finite length, this paper derives the expression of the magnetic field strength yielded by the positive and negative magnetic charges respectively distributing uniformly over two parallel rectangular areas, and calculates their spacial distribution . From the comparison of this theoretical plotting and the classical experiment of magnetic particle inspection, a conclusion can be reached : the traditional hypothesis, that the magnetic charges distribute uniformly over the area can not be tenable . So at the first the only one real distribution of magnetic charges over the rectangular area must be searched .

Keywords Strip magnetic dipole, Finite length, Margnetic charge .

The nearest simulation of a rectangular crack on the work piece surface seems to be the strip magnetic dipole of finite length - that is, two parallel rectangular plane, over which respectively uniformly distribute positive and negative magnetic charges ( Fig.1 )

 Fig 1: A strip magnetic dipole with finite length in a Cartesian coordinates system.

In October, 1986, three months after the author derived the analytic expression of the magnetic field strength components yielded by this model, according to basic theory of electromagnetics[1,2], it is known that scholars in England [3] and Japan [4] have studied in this region . Another five months passed, the author finally read the Soviet article [5], which first investigated this problem . Although the analytic expressions obtained by various authors are very different due to distinct coordinates systems and integral variables used by them, the author already proves that they are all equvalent ( it is ignored, because this paper bas limited space ) . For the conveniece of calculation now the most succinct expression is adopted as follows :

 (1)

 (2)

 (3)

The selection of the coordinates is shown in Fig.1, in which Hx, Hy and Hz respectively is the x, y and z component of the magnetic field strength yielded by a strip magnetic dipole with finite length at an arbitrary point p ( x, y, z ) .

ss is the areal density of uniformly distributing magnetic charges on the strip magnetic dipole ( unit is Wb /m2 ) .
msis the magnetic permeability for the vacuum, and ms= 4p´10-7 H/m, in the MKSA International Standard System .
2b, 2l and h is respectively the width, length and depth of the strip magnetic dipole ( units are all metre ) .

For verifying this theory the Soviet experiment made by V. E. Shcherbinin and A. I. Pashagin can be referred to ( the litter circles in Fig.2, the width of the cracks is 1mm, the depth of the cracks is 10mm, and the length of the cracks isn't given ) [6] . But from reference [5] it is known that the artificial cracks length on their testing block 2l = 1~30 mm . Substitute these values into expression ( 1 ), and then calculated, the theoretical curves of magnetic field strength yielded by the cracks in this scope mentioned above can be plotted ( as shown by the solid line and dotted line in Fig.2 ) .

 Fig 2: The comparison between this theory and the Soviet's experiment [6].

Evidently, from Fig.2 though the shape and the tendency of theoretical curve and experimental curve are quite identical, the two curves in the end can not all coincide with each other . So, this theory only in the approximation of first degree explains the principle of magnetic particle inspection for the defects of deep cracks type .

Because the geometrical model of this theory is identical with the testing block, the only reason, which can lead to the discrepancy between theory and experiment may be the physical model - the hypothesis that the magnetic charges distribute uniformly over the area don't conform to the practical condition . Up to now scholars of the magnetic particle inspection theory, electromagnetic NDT the world over all develop themselves theories based on the uniformly distributing magnetic charges over the area, as N. N. Zatsepin, V. E. Shcherbinin [7], A. I. Pashagin [5,6] from Soviet Union, F. Förster [8] from Germany, C. N. Owston [9], C. R. Edwards, S. B. Palmer [3] from England, W. L. Ko, P. H. Francis [10] from U. S. A. Toshio Shiraiwa, Tatsuo Hiroshima [11], Shizuo Mukae, Mitsuaki Katoh, Kazumasa Nishio [4] from Japan, Jia-Zhen Xiu[12], Chen-Ling Gou and Xiu-Tang He[13] from China ...... Now it looks as if we have to give up this traditional physical model . The state of affairs remains as V. V. Kleuev, a corresponding member of Russian Academy of Sciences said "This problem hasn't been solved" [14] in 1978. Today we must start from the very beginning, to seek the same as one correct distribution of magnetic charges from the infinite probable distributions of magnetic charges[15]. Other wise the basic principle of the magnetic particle inspection for deep cracks can not be explained truly, exactly and quantitatively .

### Acknowledgment

The author gratefully acknowledges Professor Zhi-Di Song, Senior Engineer Hong-Sheng Li, at Wu-Han Research Institute for Materials Protection, Wu-Han, China, who gave the author enthusiastic advice and a series of references, and the finacial support of this work by The National Natural Science Foundation of China .

### References

1. Kei-Hua Zhao, Xi-Mo Chen : Electromagnetism. Higher Education Publisher, Beijing. July, 1978
2. Wei-Chang Zhong : Magnetic dipole and magnetic particle inspection-Principle of magnetic particle inspection, Part I. ( Chinese Journal of )ááNon-Destructive Testingññ1990,12 ( 3 ) : 66-70
3. C. R. Edwards, S. B. Palmer : The magnetic leakage field of surface-breaking cracks. áá Journal of Physics D. Applied Physics ññ 1986, ( 4 ) : 657-673
4. Shizuo Mukae, Mitsuaki Katoh, Kazumasa Nishio : Investigation on quantitization of defect and effect of factors affecting leakage flux density in magnetic leakage flux testing method. ááJapanese Journal of Non-Destructive Inspectionññ1988, 37 ( 10/11 ) : 885-894
5. V. E. Shcherbinin, A. I. Pashagin : The influence of the length of defect on the value of its magnetic field. ááDefectoskopiyaññ1972, ( 4 ) : 74-82
6. V. E. Shcherbinin, A. I. Pashagin : On the volume polarization of cracks. ááDefectoskopiyaññ 1974, ( 4 ) : 106-110
7. N. N. Zatsepin, V. E. Shcherbinin : Calculation of the magnetostatic field of surface defects. ááDefectoskopiyaññ1966, ( 5 ) : 50-65
8. Friedriech Förster : Nondestructive inspection by means of magnetic leakage field. Theoretical and experimental foundations of detection of surface defects of finite and infinite depth. ááDefectoskopiyaññ 1982, ( 11 ) : 1-14
9. C. N. Owston : The magnetic flux leakage technique of non-destructive testing. ááBritish Journal of Non-Destructive Testingññ1974, 16 ( 6 ) : 162-168
10. W. L. Ko, P. H. Francis : Magnetic field leakage due to a surface crack. ááBritish Journal of NDTññ1975, 17 ( 5 ) : 141-144
11. Toshio Shiraiwa, Tatsuo Hiroshima : Automatic magnetic inspection method for steel bars and tubings ( I ). ááJapanese Journal of Non-Descructive Inspectionññ1972, 12 ( 3 ) : 166-176
12. Jia-Zhen Xiu : Detection of the magnetic particle patterns of defects with uniformly-distributed strip magnetic dipole. ( Chinese Journal of )ááNDTññ1984, 5 ( 6 ) : 1-3
13. Jia-Zheng Xiu, Chen-Ling Gou, Xiu-Tang He : A study on defect detectability of magnetic particle technique. ( Chinese Journal of )ááNDTññ1985, 6 ( 2 ) : 1-5
14. V. V. Kleuev : Problems of physical methods of quality testing. ááDefectoskopiyaññ1978, ( 9 ) : 5-12
15. The teaching group of electrodynamics, teaching and research section of theoretical physics, department of physics, Beijing University compiled : Electrodynamics. People's Education Publisher, Beijing. July, 1961

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