J 1 4 π r p φ 1 1 4 σ π r p φ 1 4 πσ δ δ lim

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J = ∇∇ 1 4 π R p Φ = 1 1 4 0 σ π R p Φ = × ′ → ′ + ( ) 1 4 0 0 πσ δ δ lim r I I r r r r r Φ = 1 4 0 σ π I R J R = 1 4 2 π I R ˆ 77 Modeling of Electromagnetic Testing
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Small Spherical Inclusion The spherical inclusion problem, usually found in textbooks as a problem in electrostatics involving a dielectric sphere, has a solution that satisfies the laplace equation inside and outside the sphere. Interface conditions on its surface ensure that the normal current and tangential electric field are continuous. Given a uniform field E 0 in the Z direction, which is also the polar direction of a spherical coordinate system ( z = R cos θ ) and defining the parameter s as the conductivity ratio s = σ · σ 0 –1 , the internal potential (volt) is: 65 (38) whereas outside the sphere the potential is: (39) where θ is the polar angle (radian). The external potential can also be written: (40) where the dipole intensity and direction are given by: (41) Perhaps of greater interest here is the fact that the external electric field can be written: (42) where E 0 = E 0 ^ z . This goes beyond the basic textbook account by expressing the field of the dipole in terms of a dyadic Green’s function, σ 0 –1 ∇∇ (4 π R ) –1 . Equation 42 can apply to a dipole of arbitrary orientation. Figure 11 shows the current associated with the perturbed field that when added to the unperturbed current σ 0 E 0 ^ z gives the total current density. An additional point of interest is that the dipole intensity can be related to a uniform current dipole density P distributed in the spherical region. By putting p = 4·3 –1 × π a 3 P , it is found that: (43) where E is the electric field in the sphere given by taking the negative gradient of Eq. 38. Dynamic Current Dipole In eddy current testing, the fields are dynamic rather than static. Therefore, the dynamic current dipole has a more significant elemental discontinuity field than does the field of static current dipole. The dynamic current dipole for a time harmonic field is described by essentially the same equations as those used for the textbook treatment of the hertzian dipole. 78 The difference arises from the fact that in eddy current applications the host medium is a conductor, not air. In a good conductor such as a metal, the charge current is much larger than the displacement current. Consequently, the latter can be neglected. This means that Ampère’s law (Eq. 2), ∇ × H = J , is adequate and Maxwell’s addition of the displacement current j ω D to the right hand side of this relationship is not needed. Here, the field is expressed in terms of complex phasors, which means, for example, that the magnetic field varies in time as the real part of H e j ω t , ω being the angular frequency (radian per second) of the excitation. The neglect of displacement current means that solutions are sought in the quasistatic limit. As a short cut from the description of waves in air to fields in
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  • Fall '19
  • Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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