Abstract:
In the summer of 1995, in a magnetic particle inspection experiment, the author was surprised by the magnetic particle pattern around a square hole on the work piece surface, because its profile wasn't a square as estimated according to the traditional concept, but four long straight lines starting from the four corners of the square hole and spreading out along the extension lines of the two diagonals of the square hole. The cause of this knotty phenomenon is explained by the calculated result of the author's magnetic dipole theory in this paper.
According to the traditional concept of magnetic particle inspection, the profile of magnetic particle pattern on the discontinuity of a work piece surface is similar to this defect. So the length, width and character, etc. of a flaw on the work piece surface can be estimated by the magnetic particle pattern on it [13].
In the summer of 1995, however, in a magnetic particle inspection experiment of square hole (0.6×0.6mm) on a sample surface, the author surprisingly discovered that the magnetic particle pattern didn't pile along the two edges perpendicular to the magnetizing field as predicted by the classical conception (Fig.1.a), but four long straight lines starting from the four corners of the square hole and spreading out along the extension lines of the two diagonals of the hole (Fig.1.b).
Why does this strange magnetic particle pattern appear? The author made a research as follows.

a. Estimated by traditional concept  b. Experimental result

Fig 1: The profile of magnetic particle pattern around a square holr on the work piece surface

Keywords: Magnetic particle pattern, Square hole, Magnetic leakage field, Linear magnetic dipole,Magnetic dipole theory.
1. Theoretical fundamentals
According to the view point of magnetic charge in electromagnetism, at the initial transient moment of magnetization, the uniform positive and negative charges ±Q_{m}(W_{b}) will be excited out over the two lateral faces of the square hole on the work piece surface perpendicular to the magnetizing field H_{a}(A/m)[4].
 (1) 
in which, X_{m} is the magnetic susceptibility of the sample material, and is a pure value.
m_{o} is the magnetic permeability for vacuum, and equal to 4px10^{7}.
2w and d is respectively the width and the depth of the square hole (m).
Due to the repulsion between the same kind magnetic charges, these excited magnetic charges will be squeezed upon the surface edges of the square hole to form uniform linear magnetic charge density s_{ml}(W_{b}/m)[5]:
 (2) 
here, C is a constant.
As mentioned above it isn't difficult to find out that in the stable condition of magnetization the square hole on the work piece surface corresponds to a linear magnetic dipole of finite length [6].
On the basis of reference [6], the magnetic leakage field components , caused by the square hole on the work piece surface are:
 (3) 
 (4) 
2. Calculation and analysis
For the four corners of the square hole there will be x = w,y = w; x = w; y = w;x = w, y = w; x = w, y = w. Substitute these four groups of values into expressions (3) and (4), the results are respectively:
 (5) 
 (6) 
So the composite magnetic field strength at the four corners of the square hole must also approach to infinity and along the two extension lines of the two diagonals of the square hole as shown in Fig.2.
 (7) 
If the magnetic charge excited out by the magnetic leakage field of the square hole at the two ends of a magnetic particle is ±q_{m}(W_{b}), and one end a of the magnetic particle just conforms with one corner of the square hole, the magnetic force F_{ma} acting upon the magnetic particle at the end a is:

a. The force of magnetic field
 b. The balance of forces

Fig 3: The force acting upon a magentic particle, whose one end is at a corner of the square hole

Fig 2: The magnetic field strength at the corners of the square hole on the sample surface

 (8) 
And F_{ma} is along the extension line of one diagonal of the square hole.
From expressions (3) and (4), it is known that the force acting upon another end b of the same magnetic particle F_{mxb}, F_{myb} are finite respectively. For the balance of this magnetic particle, there must be another force acting upon the magnetic particle (No.1) in order that the following vectorial equation exists.
 (9) 
here, F_{r} is the resistance to magnetic particle No.1 of the medium(N).
F_{21} is the force acting upon magnetic particle No.1 by another magnetic particle No.2(N).
In other words, according to Newton's third law, the magnetic particle No.1 transfers the magnetic force F_{12} to the magnetic particle No.2. The rest may be inferred, the magnetic force must be transferred to magnetic particle No.3,
No.4,......F_{1b},F_{2b},...... and F_{r} are all finite, so there will be:
 (10) 
Thus, there must appeared magnetic particle pattern in four long straight lines starting from the four corners of the square hole and spreading out along the extension lines of the two diagonals of the square hole as shown in experiment mentioned above (Fig.1.b).
3. Conclusion
3.1 The magnetic particle pattern around a square hole on the work piece surface ought to be four long straight lines along the extension lines of the two diagonals of the square hole. The very strange profile of the magnetic particle pattern, which is hard to understand initially is in fact a normal phenomenon.
3.2 The successful explanation of the cause of this knotty magnetic particle pattern by the theory of linear magnetic dipole with finite length proves that the magnetic dipole theory [6,7,8] is correct.
Acknowledgment
The author gratefully acknowledges Senior Engineer Jia  Han Du, who supplied the test block with square hole on its surface formed by electric spark machining, Engineer Lian  Hua Huang, who carried out the magnetic particle inspection testing, Senior Engineer Hua  Zhang Shi, who accurately measured the square hole and the financial support of this research by The National Natural Science Foundation of China.
References
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