Circular Current Loop Research Articles

Current loops fitted to geomagnetic model spherical harmonic coefficients.

One hundred-sixty circular current loops with radial axes were fitted by least squares to the 899 spherical harmonic coefficients of a 29th degree model. In the first case, the parameters that were fitted for each loop were the normalized magnetic moment, the distance from the center of the Earth to the current element, the colatitude and E. longitude of the loop axis, and one-half of the central apex angle of the loop. For this case, two of the loops converged near the inner-core outer-core boundary. They accounted for most of the dipolar field. Twenty of the loops, all with much smaller magnetic moments than the two deep loops, converged in the distance range of 0.42 to 0.67 Earth's radius from the center of the Earth. The other 138 loops, after many iterations, were located at distances between 0.81 and 1.0 Earth's radius from the center of the Earth. The loops with radial distances between 0.21 and 0.67 Earth's radius from the center of the Earth are referred to as “core” loops and those at distances greater than 0.81 Earth's radius as “crustal” loops. The spherical harmonic coefficients from these 160 loops, when subtraced from the 899 coefficients of the original model, left a root-mean-square residual of only 0.2nT. A second case was tried which constrained the 138 “crustal” loops to be at 0.996 Earth's radius (25.5km depth). In this case, the root-mean-square residual of the spherical harmonic coefficients from the original model was 0.9nT.

A combined form of the laws of Ampere and Biot–Savart

This article gives a combined form of the laws of Ampere and Biot–Savart which allows us to calculate the infinitesimal magnetic circulation around a closed loop ∑δB⋅δl in terms of the current flow i through an infinitesimal current path δs, and the change in the solid angle subtending the closed loop δΩ as the apex of the solid angle traverses δs:∑δB⋅δl=iδΩ×10−7. After giving its derivation from the law of Biot–Savart, the combined form is applied to a variety of magnetostatic problems. The illustrative examples, none of which requires the use of calculus for its solution, include the usual introductory course examples such as determining the magnetic field inside a toroidal coil, and at the center and along the axis of a circular current loop. In addition, the potentialities of the combined law are demonstrated by determining the field anywhere in the plane and along the axis of a rectangular current loop. Appendices include discussions of solid angles and relativistic effects.

Force on a parallel circular loop moving above a conducting slab and the eddy-current distribution

The purpose of this work is the determination of eddy-currents induced within a conducting slab, when a parallel circular current loop is moving above it. The field owing to the eddy-currents acts on the loop and produces forces on the loop. This geometrical configuration can approach problems of the electrical machines related to the magnetic suspension, when the geometry of these problems is compared with the above configuration. The rigorous analytical determination contributes either to the calculation of similar configurations or to the comparison of approximate solutions of more complicated configurations.

Field Distribution Due to a Circular Current Loop Placed in an Arbitrary Position above a Conducting Plate

The calculation of eddy currents within a conducting slab when the excitation is a circular current loop in an arbitrary position above the slab is examined. The magnetic flux density in regions outside the slab is also determined. The diffusion equation of the magnetic vector potential is written in a coordinate system attached to the geometry of the configuration by using two Dirac functions. The field vectors are expressed as double Fourier integrals.

Electric fields and charges in elementary circuits

In an effort to clarify the role of surface charges on the conductors of elementary electric circuits and the electric fields in the space around them, we present a quantitative analysis of (two-dimensional) circular current loops. It is also noted that, in general, lines of Poynting flux lie in the equipotential surfaces of quasistatic systems.

Transient electromagnetic fields due to a circular current loop perpendicular or parallel to a conducting half-space

Currents having the form of a step function or a pulse flowing in a circular loop placed above a conducting medium occupying the half-space give rise to eddy currents within the conducting material. The loop is located either perpendicular or parallel to the dividing the two half-spaces surface. For the transient state the distribution of eddy currents is determined inside the conducting medium, the distribution of the magnetic field is also determined in the air region which occupies the other half-space.

Force on a vertical circular current loop moving above a conducting slab and eddy current distribution

The purpose of this work is the determination of the force on a vertical circular current loop which is moving above a conducting slab. The loop is located vertically to the surface which divides the air and the conducting medium. Eddy currents are induced within the conducting slab and the field of the eddy currents is acting on the moving loop. This configuration can approach problems of the electrical machines related to the magnetic suspension, when the geometry of these problems is compared with the above configuration. The rigorous analytical determination contributes either to the calculation of the similar configurations or to the comparison of approximate solutions of more complicated configurations.

Magnetic multipoles in theory and practice

The general expression for the magnetic moment of arbitrary order of a circular current loop about any point on its axis is derived, and used to obtain the magnetic moments of systems of current loops. A magnetic multipole apparatus suitable for the physics teaching laboratory is described. The apparatus enables the student to measure the magnetic field configuration of a single large coil and of systems of one or more small coils, and results obtained with it are presented. The potential role of the nuclear magnetic octupole moment in nuclear structure studies is indicated by deriving the M3/M1 ratio of three spheroidal current distributions.

Circular Current Loops, Magnetic Dipoles and Spherical Harmonic Analyses

Circular Current Loops, Magnetic Dipoles and Spherical Harmonic Analyses

Interpreting regional geomagnetic anomalies and their secular variation with the aid of radial dipoles and current loops.

By adjusting the depth and strength of a radial magnetic dipole so that the intensity of the field at the focal point of Z and the distance between the positive and negative focal points of the X and Y isolines correspond to those around a regional anomaly obtained from non-dipole maps, it is demonstrated that the dipole is located at a radial distance of 0.25 Re from the geocentre. A circular current loop at the core-mantle boundary corresponding to this dipole has a radius of 0.35 Re.Using a conducting Earth model, it was found that the effect of the conductivity of the mantle is negligible if the period of the source dipole is more than 50 years.The secular change in Europe, Asia, Africa, and part of North America was found to follow the change of two radial dipoles, one lying under the Asian anomaly and the other under the African anomaly. As interpreted by this two-dipole model, the strong centres of the secular variation observed in the Atlantic Ocean are caused by the rapid westerly drift of the African anomaly. In Asia, the change of the large positive Z-cell to a negative one during the last 30 years was interpreted as having been caused by changes in the intensity of the Asian anomaly.The observed secular change at 29 observatories in Eurasia and part of North America for the epoch 1965.0 can be explained by a two-dipole model with an rms error of 7.1nT/y, which is clearly better than the rms error of the IGRF (14.5nT/y), and roughly the same as that of the definitive model for 1965 (6.1nT/y) obtained.

Resistance of a circular current loop in a cylindrical waveguide

The dependence of the resistance of a circular current loop situated coaxially inside a cylindrical waveguide on the size of the loop relative to the radius of the waveguide is examined.

A finite-circuit-element code for modeling the compression of a gyrating charged-particle beam

A method is described for calculating the interaction between an imploding liner, a magnetically confined charged-particle ring (Astron e-layer, ion ring) and a target plasma, based on the equations of the equivalent circuit. Expressing the electrodynamical behavior in terms of inductive coupling between circular current loops, so that changes in geometry and plasma parameters are described by changes in the induction coefficients, means that only ordinary differential equations arise, in contrast with fluid descriptions. Induced electron currents are conveniently included in the model. Application to a beam-target fusion system driven by the compression of an ion ring is described as an illustration of the utility of the technique.

The Onset of Coronal Transients

Recently Anzer (1978) has discussed a model for the motion of transients in the high corona. In this model the transient is pictured as a circular current loop with a changing radius r, and having two fixed footpoints separated by a distance 2r0at the solar surface. The transient starts off with zero velocity at the point where r = r0. At that point however the acceleration, which is given byis generally non-zero. In (1) M⊙is the solar mass; G is the gravitational constant; b (which equals a/4, where a was introduced by Anzer (1978)) is a constant on a short time scale, which however may change on a long time scale when mass or electric current is supplied to or lost by the system; f is a function describing the distribution of mass and magnetic flux, qualitatively displayed in Figure la; R is the distance of the top of the current circuit to the solar center; r2/R2as a function of R is shown in Figure lb.

Pulsed loop antennas

A rather simple and general analysis allows calculation of the transient response of many different radiators of practical interest directly in the time domain without the knowledge of the corresponding time harmonic field. Here, we present exact expressions for the transient near-and far-fields around thin circular current loops of arbitrary dimensions involving arbitrarily time-varying excitation. The resulting convolution integrals have been numerically evaluated for “step”-function currents with finite rise time, thus allowing interpretation of the different influence of small and large loops on the transient radiation field. The physical background of the time history of the ideal step-function response is discussed in detail.

Transient fields of small loop antennas

A rather simple and general analysis allows us to calculate directly the transient response of many different radiators of practical interest in the time domain without the knowledge of the corresponding time harmonic solution. For example, we present exact expressions for the transient near- and far-field around small circular current loops involving arbitrarily time-varying excitation. The resulting convolution integrals have been numerically evaluated for step function currents with finite rise time, thus showing the influence of finite loop radius on the radiation field in comparison with the ideal magnetic dipole.

Stationary axisymmetric electromagnetic fields around a rotating black hole

The form of stationary electromagnetic fields which are axisymmetric around a rotating black hole is derived, using Teukolsky's perturbation equation. The field of a circular equatorial charged current loop around the black hole is obtained, and it is shown that in a minimum-energy configuration the black hole and the current loop contain an equal but opposite net charge. The value of this equilibrium net charge is calculated.

Subsurface electromagnetic fields of a circular loop of current located above ground

The electromagnetic fields produced by a circular loop of current is considered for a homogeneous half-space model of the earth. The integral representations for the subsurface field are evaluated numerically and presented in graphical form.

Magnetic field of thick finite dc solenoids

The expressions of the magnetic components in or around any thick finite dc solenoid have been obtained from the expression of the vector potential of a circular current loop. The field component derivatives with respect to the spatial coordinates have also been determined. From these general expressions the field of semi-infinite solenoids with zero inner radius has been deduced as a particular case.

Magnetic field of dc coils of cylindrical configuration

Starting from the differential form of the Biot-Savart law, the expressions of the magnetic field components for three general types of dc coils of cylindrical configuration called respectively annular sectorial axial, radial and toroidal solenoids are obtained. The determination is based on the supposition that a fine wire is used for setting up the coils. From these expressions it is easy to obtain as a particular case the magnetic field components for more simple cylindrical configurations as circular current loops, circular cylindrical coils, toroids, etc.

Eddy-current losses in reactor flux shields

The magnetic-field distribution is analysed for the case of an aluminium plate excited by a circular current loop, and an expression for the eddy-current loss in the plate is deduced. This analysis is tested by comparison with measurements on a model which closely resembles the theoretical geometry, and the results are shown to be in close agreement. A method for measuring the eddy-current losses in plates is described. It is found that the losses in thin plates can be abnormally high. It is shown that the image theory may be used with nonmagnetic conducting plates for all plate thicknesses greater than 0.8 times the characteristic depth of flux penetration in the material. The magnetic field of large air-cored reactors is analysed, and expressions are obtained for the inductance and the loss in the flux shields. It is believed that this method is capable of giving more accurate results than have been hitherto obtained.