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The time‐varying electric and magnetic fields inside a solenoidal coil have recently been calculated by Harmon. He modeled the coil by a cylindrical current sheet, used a physical approach starting from the Maxwell equations, and found results very different from what was believed from circuit theory since the 1920’s. Harmon’s derivation neglects the electric field‐current relation in the wire, i.e., it violates Ohm’s law. In this article, the fields are derived from the Maxwell equations with this omission amended. Instead of the current, the terminal voltage of the coil is taken as the independent input. The relation between the current and the electric field in the coil wire constitutes an additional boundary condition. It allows as additional variable an energy influx from the power source, whereas Harmon considered radiation from the coil outward only. With these modifications, the old circuit theory results are recovered again. Implications are outlined for the physical legitimacy of other models involving a given current sheet.

In the spiral line induction accelerator, an intense electron beam is transported along an open‐ended beam pipe that makes multiple passes through the accelerating cavities. In the straight sections of the beam line, solenoidal focusing is used; in the bends, an l=2 stellarator field is used. At the solenoid/stellarator transition, where the beam equilibrium changes, a mismatch can occur, exciting oscillations of the beam envelope. Numerical simulation is used to show that the frequency, damping rate, and emittance growth associated with these oscillations are sensitive to nonlinear space‐charge forces that depend significantly on the radial profile of the beam. Comparisons between simulation and experimental results illustrate this sensitivity. It is shown that mismatch oscillations can be avoided by using a single thick quadrupole lens at the solenoid/stellarator transition. Simulation and experimental results show excellent agreement.

The Laboratory Microfusion Facility (LMF) has been proposed for the study of high‐gain, high‐yield inertial‐confinement‐fusion targets. The light‐ion LMF approach uses a multimodular system with applied‐B extraction diodes as ion sources. A number of ion‐beam transport and focusing schemes are being considered to deliver the beams from the diodes to the target. These include ballistic transport with solenoidal lens focusing, z‐discharge channel transport, and wire‐guided transport. The energy transport efficiency ηt has been defined and calculated as a function of various system parameters so that point designs can be developed for each scheme. The analysis takes into account target requirements and realistic constraints on diode operation, beam transport, and packing. The effect on ηt of voltage ramping for time‐of‐flight beam bunching during transport is considered here. Although only 5 mrad microdivergence calculations are presented here, results for bunching factors of ≤3 show that transport efficienc...

Sheet electron beams focused by periodically cusped magnetic (PCM) fields are stable against low‐frequency velocity‐shear instabilities (such as diocotron mode). This is in contrast to more familiar unstable behavior in uniform solenoidal magnetic fields. Two rectangular‐cross‐section magnetic configurations capable of focusing in both transverse dimensions are investigated: (i) a closed‐side two‐plane PCM configuration that is topologically equivalent to conventional round‐cross‐section PPM focusing; and (ii) an open‐side configuration that uses ponderomotive PCM focusing in the vertical plane and simple vzBy Lorentz force focusing in the horizontal plane. Both configurations are capable of stable sheet beam confinement. The open‐side configuration appears more practical both for focusing and for realizing matched (cold) beam conditions in which the beam envelope is free from oscillations. For realistic beams with finite emittance, the existence of a matched cold beam solution implies less emittance grow...

Light‐ion inertial confinement fusion requires beam transport over distances of a few meters for isolation of the diode hardware from the target explosion and for power compression by time‐of‐flight bunching. This paper evaluates ballistic transport of light‐ion beams focused by a solenoidal lens. The ion beam is produced by an annular magnetically insulated diode and is extracted parallel to the axis by appropriate shaping of the anode surface. The beam propagates from the diode to the solenoidal lens in a field‐free drift region. The lens alters the ion trajectories such that the beam ballistically focuses onto a target while propagating in a second field‐free region between the lens and the target. Ion orbits are studied to determine the transport efficiency ηt (i.e., the fraction of the beam emitted from the diode which hits the target) under various conditions relevant to light‐ion inertial confinement fusion. Analytic results are given for a sharp boundary, finite thickness solenoidal lens configura...

Dispersion relations are derived to determine the growth rate, dominant wavelength, and group velocity of disturbances caused by the beam breakup instability. Considerations include weak and strong focusing, x‐y coupling in solenoidal transport, the spacing of accelerator cavities, and periodically pulsed beams. Beam breakup growth is minimum when the cavity spacing equals an integral number of half‐betatron wavelengths for quadrupole focusing, and an integral number of betatron wavelengths for solenoidal focusing. Minimum growth is also found for periodic pulses separated by an integral number of half‐periods of the TM110 cavity mode. Expressions for beam breakup growth at the minima are obtained.

A standard matrix method for tracing rays through an ion optical system is extended to include a recently discovered [W. Z. Liu and F. D. Bechetti, Rev. Sci. Instrum. 60, 1228 (1989)] defocusing solenoidal lens. The results of the matrix approach are shown to correspond with results obtained by numerical integration of particle trajectories through real coil configurations. First‐order radial miss distances due to radial divergence of the beam, chromatic aberration of the lens system, and physical extent of the source are compared for one and two lens systems using the matrix formulation. It is demonstrated that a two lens system in a focusing‐defocusing arrangement can simultaneously correct to first order for chromatic aberration and beam divergence, producing a better focus for thin annular beams with significant energy spread and divergence. A new calculation of the focal length for the negative solenoidal lens is also included.

The magnetization reversal process in an array micron sized NiFe patterns was studied using magnetic force microscopy in the presence of external fields. The behavior of the magnetization distribution was correlated with the aspect ratio and the direction of the applied fields. Magnetizing along the hard axis was found to produce solenoidal magnetization at remanence while applying the field along the easy axis tend to form nonsolenoidal configurations. The micromagnetic evolution, which involved domain wall, crosstie, and vortex displacements, was studied and the correlations were consistent with previously reported M–H loop observations and theoretical predictions.

The Boltzmann transport equation for phonons is recast directly in terms of the heat-flux by means of iteration followed by truncation at the second order in the spherical harmonic expansion of the distribution function. This procedure displays the heat-flux in an explicitly coordinate-invariant form, and leads to a natural decomposition into two components, namely the solenoidal component in addition to the usual irrotational component. The solenoidal heat-flux is explicitly shown to arise by applying the heat-flux equation to a right-circular cylinder. These findings are important in the context of phonon resonators that utilize the strong quasi-ballistic thermal transport reported recently in silicon membranes at room temperature.

In this work, we show how the complexity of magnetic sensors for scanning applications can be reduced about one dimension by utilizing the redundancy of the magnetic field. This means a three-dimensional magnetic field can be scanned with a two-dimensional sensor, and a two-dimensional magnetic field with a one-dimensional sensor. For this purpose, the fact is used that the magnetic field is purely solenoidal and that the curl of the magnetic induction in air can be assumed to be zero. After an explanation of the basic principal of the reconstruction of one field component, the performance of the method is analyzed with a two- and three-dimensional example where the magnetic field is computed analytically. Finally, results of a real measurement example are presented which are in excellent agreement with the theoretical considerations and which demonstrate the practical applicability of the method.

We investigated the evolution of the electron energy distribution function (EEDF) in a solenoidal inductively coupled plasma surrounded by an axial dc magnetic field. The increase in the dc magnetic field caused the EEDF to evolve from a bi-Maxwellian to a Maxwellian distribution. At the discharge center, the number of low energy electrons was significantly reduced while the high energy electron population showed little change when a weak dc magnetic field was present. However, at the discharge radial boundary, the high energy electron population decreased significantly with the magnetic field while the change in low energy population was not prominent compared to the discharge boundary. These changes in EEDFs at the boundary and center of the discharge are due to the radial confinement and the restriction of radial transport of electrons by dc magnetic field.

The electromagnetic field (both E and B fields) is calculated for a solenoidal inductively coupled plasma (ICP) discharge. The model is based on two-dimensional cylindrical coordinates, and the finite difference method is used for solving Maxwell equations in both the radial and axial directions. Through one-turn coil measurements, assuming that the electrical conductivity has a constant value in each cross section of the discharge tube, the calculated E and B fields rise sharply near the tube wall. The nonuniform radial distributions imply that the skin effect plays a significant role in the energy balance of the stable ICP. Damped distributions in the axial direction show that the magnetic flux gradually dissipates into the surrounding space. A finite difference calculation allows prediction of the electrical conductivity and plasma permeability, and the induction coil voltage and plasma current can be calculated, which are verified for correctness.

A class of LC resonators for micromagnetic sensor devices is reported, which is fabricated by means of the microelectromechanical system (MEMS) technique. The micro-LC resonator consists of a solenoidal microinductor with a bundle of soft magnetic microwire cores and a capacitor connected in parallel to the microinductor. The core magnetic material is a tiny glass-coated Co83.2B3.3Si5.9Mn7.6 microwire fabricated by a glass-coated melt-spinning technique. The solenoidal microinductors fabricated by the MEMS technique were 500–1000μm in length with 10–20 turns. The changes of inductance as a function of external magnetic field in microinductors with appropriately annealed microwire cores were varied as much as 370%. Because the permeability of ultrasoft magnetic microwire is changing rapidly as a function of external magnetic field, the inductance ratio as well as magnetoimpedance ratio (MIR) in a LC resonator is varied accordingly as a function of external magnetic field. The MIR curves can be tuned very precisely to obtain maximum sensitivity.

A numerical micromagnetic technique is used to simulate magnetization processes in two‐dimensional thin metallic platelets. The platelets are modeled as an array of interacting polycrystalline grains. The technique assumes a triangular discretization at the subgrain level with the magnetization varying linearly over each triangle. The coupling between the grains has a profound effect on the magnetic structure of the platelets as does the grain size. For systems with strongly exchange coupled grains, approximately solenoidal magnetization structures exist. A single domain behavior exists for systems with weakly coupled grains. The magnetization pattern of the platelets has been characterized by the vorticity of the magnetization vector field.

Experiments were performed on high‐vacuum transport of a relativistic electron beam (130 A, 300 keV) through a 180° section of a curved transport system. The system was constructed for high‐current betatron experiments. The electron beam was directed by a sector‐type bending field and focused by an array of 18 solenoidal lenses. The lenses could be operated with the same field polarity (modulated toroidal field) or with alternating polarity (cusp array). The reversing field of the cusp array canceled beam drift motions. Consequently, the beam was quite stable and could tolerate vertical field errors of ±100%. In contrast, beam motion in the toroidal field was dominated by drifts. After propagation through the 3.2‐m‐long transport tube, significant beam loss was observed with only an 8% vertical field error. Measured displacements of the beam centroid at the end of the transport system were well described by theory in both field geometries.

A trapped distribution of low‐energy electrons was used to alleviate spherical aberration in a solenoidal magnetic lens. The space‐charge‐correction method has potential application to the focusing of heavy, negatively charged particles such as H− ions. In the geometry investigated, the field contributions from trapped electrons cancel net electric fields along magnetic flux surfaces. The lens magnetic fields therefore define equipotential surfaces. Experimental results were obtained using a 10‐keV electron beam as a nondestructive probe in a solenoid lens with 65‐G peak field. Annular distributions of trapped electrons were achieved that were stable against the diochotron instability. Deflections of the probe beam were consistent with theoretical predictions. The peak focused current density of the 10‐keV beam was improved twentyfold with the addition of trapped electrons in the lens.

A coherent and systematic multipole formulation of the magnetic field generated, both in the near and remote regions, by axially symmetric field and gradient systems is presented. These include an open spherical winding, an arbitrary number of coaxial coils, and a combination of coils and spherical or solenoidal windings. Optimum configurations for maximum uniformity of the field or field gradient are discussed and illustrated by numerical results. Legendre series resulting from the multipole expansion are analyzed at the system’s boundary where convergence is very slow, using a summation method particularly suitable for numerical computation.

A self‐consistent domain theory, based on micromagnetic principles, is unfolded for two‐dimensional solenoidal M distributions present in ideally soft‐magnetic plane‐parallel thin‐film objects at zero external field and in the absence of conduction currents. Two types of domain configurations are distinguished: the basic structures in simply connected regions and the parallel configurations in special types of multiply connected regions—the parallel regions. A decomposition of the objects’s area into disjunct subregions, either simply connected or of the parallel type, whose union completely covers the object, is put forward. A procedure for constructing all feasible parallel regions is presented. In each region, the appropriate solenoidal M distribution is specified with which M is taken parallel to the subregion’s boundary. Thus, all the domain structures possible in thin‐film objects with arbitrary lateral shapes can be constructed. Experimental examples are provided.

The basic concepts of a self‐consistent domain theory for the class of ideal soft ferromagnetic media is unfolded. The treatment is based on micromagnetic equilibrium and stability equations which are simplified by removing the contributions due to the intrinsic anisotropy and the spatial variation term in the exchange energy density. Only the solenoidal magnetization distributions in thin film objects are investigated. This limitation makes confinement to two‐dimensional dipole distributions possible. Cauchy’s method of characteristics is employed to derive dipole distributions that satisfy the relation ∇⋅M=0, where M is the magnetization, and the constitutive equation. It is proved that the characteristic base curves are straight lines perpendicular to the magnetization vector. In an ellipse, four different characteristic base curves that do not coincide intersect at one single point in certain regions, thus generating ambiguities in the magnetization direction. It is demonstrated that these ambiguities originate in the incompatibility of the magnetization distribution that is imposed by the various segments of the edge of the ellipse. Based on differential geometrical principles, a partitioning of the edge into segments is introduced, and along with it two adjoining regions in which the magnetization direction is uniquely determined are defined. It is proved that a domain wall is required in the cross section of both regions to accomplish a dipole configuration in stable equilibrium. Experimental confirmation is given by means of the ferrofluid pattern of a 3500‐A‐thick Permalloy layer with elliptical geometry.

The post‐acceleration of a 400‐keV, 10‐kA proton beam by a 200‐kV magnetically insulated gap is investigated. The defections from self‐ and applied E and B fields are measured and compared to calculated values. Several important principles of post‐acceleration gap operation are observed for the first time that are important for multigap operation. First, the beam is inadequately space‐charge neutralized without gas puffs or preformed plasma in regions of transverse applied‐B field to allow efficient transport. The beam is also noncurrent neutralized in these regions. Second, the applied‐B field defines equipotential surfaces in the gap allowing the voltage on the gap to steer and focus the beam, and it has an axial field component that acts like a pair of solenoidal lenses to focus the beam. It is also pointed out how azimuthal asymmetries in the beam current density and cathode plasma cause beam self‐field asymmetries that lead to growth of beam emittance. Finally, we discuss a model which can be used to...

At first, the description of the energy cycle of the mature tropical storm is amplified on the basis of recent upper air observations. Air particles passing through such a storm at first undergo isothermal expansion as they move toward a center. Then they ascend with release of condensation. At high levels they move outward and mix with the environment giving off heat to the surrounding colder air. Several requisites for maintenance of the observed temperature field are stated.After a discussion of previous theories of hurricane formation the proposed model is described. The initial intensification of the wind field is brought about by mass divergence at high levels that imposes a pressure reduction on the surface layers. This divergence is the result of interaction between the large‐scale disturbances of the upper air inside and outside the tropics. A solenoidal circulation is initiated that acts in the kinetic energy producing sense. But this circulation contains an internal mechanism for its own destru...

The steady‐state forces and torques acting on a planar current‐carrying winding, moving at constant velocity parallel to a conducting permeable slab of finite thickness and infinite extent are calculated from exact solutions of Maxwell's equations. These calculations are extended to apply to an arbitrary coil consisting of a superposition of planar windings. The special cases of solenoidal coils of finite and infinite height are discussed as approximations to bar, iron core, and horseshoe magnets, and the limiting cases of magnetic monopoles and dipoles are considered. The relativistically exact formulas are simplified for the practical case of a magnetically suspended train, and similarity laws are derived relating the forces measured on scaled‐down models to those on their prototypes. These calculations are valid for levitation over nonpermeable conducting media as well as for the reduction due to eddy currents of the attractive force in the alternatively proposed ferromagnetic suspension systems. Aside...

Multipolar modes in a cylindrical, axially magnetized, cold, collisionless plasma that propagate near the cyclotron frequency are studied quantitatively. Examination of electromagnetic field patterns shows that in addition to the well‐known spectrum of cyclotron modes there exists at positive azimuthal eigenvalues (n>0 in expjnφ) a mode that is distinguished by a different type of field pattern. The electric field of this mode is predominantly solenoidal, in contrast with the cyclotron modes, where a substantial part of the electric field is lamellar. This mode may be of importance in thick‐beam parametric amplifiers at microwave frequencies (X‐band) and in beam—plasma interaction near cyclotron frequency.

Finite‐element vector potential solutions of three‐dimensional magnetic field problems are usually obtained by approximating each component of the vector potential by a separate set of scalar finite‐element approximation functions and by imposing continuity conditions between elements on all three components. This procedure is equivalent to imposing continuity of both the normal and the tangential components of the vector potential. We show in this paper that this procedure is too restrictive: While continuity of the tangential component of the vector potential is required, continuity of the normal components is not essential in the variational formulation. We introduce a new type of vector finite‐element approximation function that has the property that it interpolates not to point values of each component of vector potential, but rather to the tangential projection of the vector potential on each edge of tetrahedral finite elements. With the new basis functions, continuity of the normal component of the vector potential is provided only approximately by means of the natural interface conditions inherent in the variational procedure. This results in a more efficient procedure for the solution of three‐dimensional magnetostatic field problems than is obtained by enforcing normal component continuity exactly.

Three‐dimensional magnetostatics problems, with no a priori assumptions about model geometry or topology, admit two vector potential formulations: the physical formulation which is a consequence of Maxwell’s equations, and an alternate formulation which includes the vector Laplacian formulation as a special case. The lack of uniqueness inherent in the physical formulation is computationally undesirable because numerical discretization may produce singular coefficient matrices. The alternate formulation, on the other hand, can yield a unique solenoidal vector potential and nonsingular finite element matrices. This alternate formulation also satisfies the physical formulation if the physical problem is well posed.

Following Anderson and Arthurs [Int. J. Electron. 56, 571 (1984)], a variational formulation for three‐dimensional magnetostatics based on helicity is considered. The advantages of formulating problems in terms of the magnetic field as opposed to the vector potential are given. It is shown that when the extremal is constrained to be solenoidal and μ=1, the Euler–Lagrange equation of the proposed functional involves a first‐order differential operator whose square is the Laplace–Beltrami operator. This implies that the condition number of any discretization is the square root of the corresponding number for the Laplace–Beltrami operator.

A method for constructing three‐dimensional solenoidal magnetization distributions, with invariant magnitude of m, in arbitrarily shaped objects is presented. The formalism harks back to the theory developed by van den Berg for two‐dimensional m distributions. The space within a general object Ω is partitioned into i subspaces Ωi, described by a family of surfaces to which the magnetization is tangent. A characteristic equation which defines the course of m at each of the surfaces is derived. A boundary condition for m arises naturally, or can be chosen to determine m at the surface. Within the above framework an infinite number of solutions are generated that, in general, exhibit singularities. Special attention, also from the topological point of view, is paid to the m distributions having point defects only.

A little‐known result of Townsend and Donaldson is extended to show that, for a solenoid carrying ac or rf current, the ratio of the average applied longitudinal field Ez to the induced azimuthal field Eθ at the periphery of the solenoid is equal to the ratio of the circumference of the solenoid to its pitch. For conventional solenoidal coils such as are used to excite electrodeless rf discharges, this ratio is typically ≳15–30. Applicability of the Ez/Eθ ratio to the design of coils and shields is discussed.