Sketch of the hollow cylinder rotating inside a magnetic field . The Cartesian frame and cylindrical coordinates used in the calculation are also shown.
Comparison of our results (red curves) with those of Ref. 8 (black curves) for a plain cylinder (left) and a hollow one (right). The figure from Ref. 8 was scanned and used as the background on the top of which we plotted our data on the same scale. The coincidence is almost perfect. The power P was expressed as with λ = a 1/a 2, μ = b/a 2, and f(λ, μ) being plotted. Reprinted with permission from R. Schäfer and C. Heiden, Appl. Phys. 9, 121–125 (1976). Copyright 1976 Springer Science and Business Media.
Variation of the power (P) with side lengths for a small rectangular cross-section toroid as a function of ɛ (dashed-dotted curve) or β (plain curve); ɛ and β are related to b and (a2 − a1) by Eq. (12). P is normalised with respect to its value P = P0 for ɛ0/β0 = 2, the value corresponding to a square cross section. The upper and lower limits of P, corresponding to ɛ/β ≫ 1 and ɛ/β ≪ 1, respectively, are shown as red and green curves.
Current trajectories (presented in the laboratory frame) in a plain cylinder with length/diameter = 2 seen from two directions. Only half of the trajectories are shown, the other half being obtained from mirror symmetry along the yOz plane. Data obtained using the 3D numerical electromagnetics software Flux3D. The values of the parameters used for the calculation are: a = 1 mm, b = 2 mm, B0 = 9.6 T, νr = 1 kHz, and σ = 5.8 107 S/m (copper). The magnetic field is applied perpendicular to the axis of the cylinder, which coincides with the rotation axis. This would correspond to a cylinder spun at the magic angle inside a magnetic field of B0 = 11.7 T. The cylinder and surrounding volume was meshed with 360 000 volumic elements.
Current trajectories calculated analytically using Eqs. (8a)–(8c). (a) a1 = 0.75, a2 = 1.25, b = 1 (a.u.); (b) a1 = 0.75, a2 = 1.25, b = 0.25 (a.u).
Map of Jz in the xy half-plane (x > 0) for the two cases displayed in Fig. 5.
Plot showing the domains of existence of the two topologies of current lines: on the right hand side of the plot the current lines are similar to those of a plain cylinder. On the left hand side a new type of trajectory appears as shown in the inset.
Field coefficients h1, h2, and g3 plotted as a function of ρ for a copper loop of dimensions a1 = 0.36 mm, a2 = 0.41 mm, b = 0.025 mm spun at magic angle, at frequency νr = 10 kHz in a field B0 = 11.7 T. Symbols used: z = 0 (blue -), z = 0.5b (red -.), z = b (magenta - -), z = 1.5b (green ..).
Contour lines of the central peak intensity I0 of the NMR line inside a 5 turn coil, modeled as a five-ring stack. Only the first quadrant of the ρ-z plane has been represented; the other ones are obtained by symmetry about the ρ and z-axis. The internal diameter of the coil is φ = 0.72 mm and its overall length 0.75 mm. The 9 levels of the contour lines going toward the axis of the coil are equally spaced between 0.1 and 0.9. The wire cross sections are shown as square brown blocks.
Angular variation of the 3 components Bfx, Bfy, and Bfz expressed in ppm of B0 (1st, 2nd, and 3rd row) of the field induced by eddy currents in a cylinder of diameter = 2.9 mm, length = 2b = 14 mm. Conductivity is 83 S/m and rotation frequency νr = 10 kHz. Three altitudes z = 0, z = 0.35b, and z = 0.7b have been considered (1st, 2nd, and 3rd columns). On each plot the data for 3 radii (a/4, a/2, and 3a/4) are superimposed. The highest amplitude modulation corresponds to the largest radii.
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