Abstract:
According to the distribution of the magnetic field caused by the threading  bar technique and the magnetization law of magnetic medium, the analytic expressions for the linear magnetic charge density along the edge of slot on the cylindrical surface of a disc and the magnetic leakage field of the slot have been given in this paper. Their digital magnitudes and spatial distribution have also been calculated and plotted. These theoretical results approximately conform to the classical experiment of magnetic particle inspection.
For theoretical explanation of the experimental result about the magnetic leakage field inside and outside a slot on cylindrical surface of a disc at circumferential magnetization (Fig.1~Fig.3)[1], the author preliminarily analyzed this problem several years ago with a model, in which the direct electric current flows through the disc, and the obtained theoretical curves very well conform to the experimental result tendency [2]. Now this classical experiment will be investigated from the viewpoint of alternating magnetization by the threading  bar technique.
Keywords: Magnetic leakage field, Circumferential magnetization, Threading  bar technique,Exponential function, Triangular distribution
1. Theoretical bases
 1.1 The magnetic field strength excited by the threading  bar technique at a point in the work piece is inversely proportional to the distance between the point and the center of the threading  bar [3].
 1.2 At circumferential magnetization the distribution law for magnetic field strength of a cylinder is similar to that for a plane [4]. And the latter is a minus exponential function of the depth from the surface (Fig.4) [5,6].
 1.3 According to the magnetization law of magnetic medium [3], the excited magnetic charge areal density s_{ms}(W_{b}/m^{2})
over the two lateral faces of a slot on the sample surface is:
 (1) 
here, H_{a} is the magnetizing field strength (A/m).
X_{m} is the magnetic susceptibility of the sample material, and is a pure number.
m_{o} is the magnetic permeability for vacuum, and equal to 4px10^{7}
2. Basic hypotheses
 2.1 The magnetization through an alternating electric current can be approximately replaced by that through a direct electric current.
 2.2 From 1.2 it may be known, that at circunferential magnetization, the excited magnetic charge Q_{m}(W_{b}) over the two lateral faces of a slot on the cylindrical surface of a disc is also a minus exponential function of the depth from the cylindrical surface (Fig.5, curve):
 (2) 
in which, S is the area of the slot lateral face (m^{2}).
For the convenience of analysis, the minus exponential distribution of magnetic charges may first be replaced approximately by a triangular distribution (the straight line in Fig.5).
 2.3 The magnetic charges of the same kind excited over the lateral surfaces of a slot at the initial transient moment of magnetization must be squeezed by the Coulomb's repulsion [3]underneath the slot and onto the edges of the slot. So uniform linear magnetic charge density s_{ml}(W_{b}/m)
will appear along these edges [7].
3. Analyses and derivation
The hypothesis 2.2, in fact, has the same model of the reference [2], so all its results can be utilized.
In Fig.5, o is the center of the disc, R_{o} is its radius, d is the depth of the slot, u is the depth of the watershed, D is a small distance between magnetic charges, r is the radius of a field point. According to reference [7], at the section of watershed the centrifugal magnetic force is balanced by the centripetal magnetic force:
that is,
 (3) 
If D = mR_{o}, u = nR_{o} and d =DR_{o}, so equation (3) can be rewritten as:
 (3') 
In addition, for the linear magnetic charge density s_{ml} squeezed onto the outward edges of the slot, there will be [7]:
 (4) 
And the magnetic leakage field strength H caused by the magnetic charges along the slot edges of the sample is proportional to s_{ml}[7]
 (5) 
4. Checking computation and plotting
Suppose there are two slots: D,n and hD,N, from expression (3') and (5), there will be
 (6) 
 (7) 
According to reference [1]: R_{o} = 72mm, d=6，12，18，24mm, if h =3, k =23/8, from expressions (6) and (7), the value of n can be calculated, and from expression (4), s_{ml} can be gained as in the Table.1.
D (mm)
 D = d/R_{o}(%)
 n
 s_{ml}a(2  n)n
 s_{ml}/s_{ml}(D = 1 %)

0.70
 1.00
 1/192.8
 0.010346542
 1.00

6.00
 8.33
 1/21.1
 0.092540599
 8.9441091

12.00
 16.66
 1/10.3
 0.18474880
 17.856092

18.00
 25.00
 1/7
 0.26530612
 25.642009

24.00
 33.33
 1/5.5
 0.33057851
 31.950628

Table 1: The watershed and linear magnetic charge density of different slot depth 
The author already gave the analytic expression for the magnetic leakage field inside and outside a slot on the cylindrical surface of a disc at the symmetrical axis of the slot as [2]:
 (8) 
in which, 2b is the width of the slot (m), 2w is the length of the slot (m).
And the magnetic leakage field H_{sum}, measured by the classical experiment [1] is the sum of' and the applied magnetizing field H_{a}
 (9) 
or
 (10) 
here, I_{o} is the magnetizing electric current through the threading  bar (A).
According to reference [1], w = 7.5mm, if b = 6mm and H_{a} = 0.2861084 A/m, from expressions (8) and (10) the result listed in the Table.2 and curves plotted in Fig.6 can be obtained.
No.
 d
 6
 12
 18
 24

1

 1.4722377
 3.2305838
 4.8085795
 6.0875369

2
 H_{sum}
 1.7583461
 3.5166932
 5.0946879
 6.3736453

3
 H_{sum}/H_{sum}(6)
 1
 2
 2.8974318
 3.6247957

4
 Experimental result
 1
 2.0196078
 2.9019608
 3.3921569

5
 Absolute error
 0
 0.0196078
 4.52899×10^{3}
 0.23263880

6
 Relative error (%)
 0
 0.9708716
 0.156066
 +6.8581380

Table 2: The magnetic leakage field strength at the center of slots with different depth

5. Conclusion
 5.1 From the contrast of Row 3 and Row 4 in Table 2 and the comparison of Fig.2 and Fig.6, it is known that the theoretical calculation approximately conform to the experimental results. This clearly shows that the simple model and some hypotheses used in this paper basically reflects the practical situation of magnetization by the threading  bar technique.
 5.2 Some difference between theoretical computation and experimental measure is due to the far too simple hypotheses mentioned above. So the research on this difficult theoretical problem must be furthered on.
Acknowledgment
The author gratefully acknowledges the financial support of this research by The National Natural Science Foundation of China under Grant No. 59571064.
References
 The Japanese Society for Non  Destructive Inspection edited: Magnetic Particle Inspection Testing A. 2nd Ed. Tokyo, Japan, 1973: 36  37
 W. C. Zhong: Analytic expression for the magnetic leakage field inside and outside of a slot on the cylindrical surface of a disc. <Proceedings of The 14th World Conference on NDT> New Delhe, India. December 8  13, 1996: 1581  1584
 K. H. Zao & X. M. Chen: Electromagnetism. People's Education Press, Beijing, China. 1978
 W. B. Tao: Eddy effect in magnetic particle inspection. <ChSNDT 7th Conference on NDT and International Research Symposium> Santou, China. October 26  30, 1999: 486  488
 L. A. Pipes; H. S. Cao & L. J. Zhang translated: Applied mathematics for engineers and physicists. Long  Men Integrated Press. Shanghai, China. 1952
 B. N. Domashevsky & A. I. Greiser: Polarization of crack magnetized by a longitudinal a. c. field. Defectoskopiya. 1976. (2): 89  95
 W. C. Zhong: Effect of slot depth in work piece surface on magnetic leakage field. (Chinese Journal of ) NDT. 1997. 19 (11): 304  307