Homogeneous Isotropic Conductor Research Articles

Non‐stationary bending of a finite electromagnetoelastic rod

AbstractThe related problem of non‐stationary bending of a finite electromagnetoelastic rod is considered. It is assumed that the material of the rod is a homogeneous isotropic conductor. The problem statement takes into account the initial electromagnetic field, the Lorentz force, Maxwell's equations and the generalized Ohm's law. The unknown functions are assumed to be bounded, and the initial conditions are assumed to be zero. It has noted that it is difficult to construct a solution analytically for a general model. Therefore, a transition is made to simplified equations corresponding to the Bernoulli‐Euler rod and the electromagnetic field is considered quasi‐stationary. For a rod with hinged ends, trigonometric series expansions and the Laplace transform in time are used. The solution of the problem is constructed in integral form with kernels in the form of influence functions. Images of kernels are found in the space of transformations and Fourier in the spatial coordinate. Their original functions are found explicitly. The present study presents examples of calculations for a concentrated load.

Bounds for the electrical resistance for homogeneous conducting body of rotation

A mathematical model is developed for the steady-state electric current flow through in a homogeneous isotropic conductor whose shape is a body of rotation. The body of rotation considered is bounded by the coordinate surfaces of an orthogonal curvilinear coordinate system. The equations of the Maxwell’s theory of electric current flow in a homogeneous solid conductor body are used to formulate the corresponding electric boundary value problem. The studied steady-state conduction problem is axisymmetric. The determination of the steady motion of charges is based on the concept of the electrical conductance of the conductors the inverse of which is the electrical resistance. The exact (strict) value of the electrical resistance is known only for bodies with very simple shapes, therefore, the principles and the methods that can be used for creating lower and upper bounds to the numerical value of electrical resistance (electrical conductance) are important. The derivation of the upper and lower bound formulae for the electrical conductance of axisymmetric ring-like conductor is based on the two types of Cauchy–Schwarz inequality. The condition of equality of the derived lower and upper bounds is examined. Several examples illustrate the applications of the derived upper and lower bound formulae.

Динамический изгиб бесконечного электромагнитоупругого стержня

The problem of non-stationary bending of an infinite electromagnetoelastic rod is considered. It is assumed that the material of the rod is a homogeneous isotropic conductor. The closed-form system of process equations is constructed under the assumption that the desired functions depend only on the longitudinal coordinate and time using the corresponding relations for shells which take into account the initial electromagnetic field, the Lorentz force, Maxwell’s equations, and the generalized Ohm’s law. The desired functions are assumed to be bounded, and the initial conditions are assumed to be null. The solution of the problem is constructed in an integral form with kernels in the form of influence functions. Images of kernel are found in the space of Laplace transformations in time and Fourier transformations in spatial coordinates. It is noted that the images are rational functions of the Laplace transform parameter, which makes it quite easy to find the originals. However, for a general model that takes into account shear deformations, the subsequent inversion of the Fourier transform can be carried out only numerically, which leads to computational problems associated with the presence of rapidly oscillating integrals. Therefore, the transition to simplified equations corresponding to the Bernoulli – Euler rod and the quasistationary electromagnetic field is carried out. The method of a small parameter is used for which a coefficient is selected that relates the mechanical and electromagnetic fields. In the linear approximation, influence functions are found for which images and originals are constructed. In this case, the zeroth approximation corresponds to a purely elastic solution. Originals are found explicitly using transform properties and tables. Examples of calculations are given for an aluminum rod with a square cross section. It is shown that for the selected material the quantitative difference from the elastic solution is insignificant. At the same time, taking into account the connectedness of the process leads to additional significant qualitative effects.

Solution of the Cauchy Problem for the Three-Dimensional Telegraph Equation and Exact Solutions of Maxwell’s Equations in a Homogeneous Isotropic Conductor with a Given Exterior Current Source

For the solution of the Cauchy problem for the linear telegraph equation in three-dimensional space, we derive a formula similar to the Kirchhoff one for the linear wave equation (and turning into the latter at zero conductivity). Additionally, the problem of determining the field of a given exterior current source in an infinite homogeneous isotropic conductor is reduced to a generalized Cauchy problem for the three-dimensional telegraph equation. The derived formula enables us to reduce this problem to quadratures and, in some cases, to obtain exact three-dimensional solutions with a propagating front, which are of great applied importance for testing numerical methods for solving Maxwell’s equations. As an example, we construct the exact solution of the field from a Hertzian dipole with an arbitrary time dependence of the current in an infinite homogeneous isotropic conductor.

A path-independent integral for 2-dimensional cracks in homogeneous isotropic conductive plate

A path-independent integral which is denoted by j e , is introduced for the 2-dimensional crack problems in the homogeneous isotropic conductor in which the steady current flows. By utilizing the j e- integral the distributions of the electric potential, current density and the Joule heating rate near the crack tip are derived. It is shown that the j e- integral provides a parameter which dominates the distributions of the electric potential and current density near the crack tip as the square of the amplitude.

Multipole Representation for an Equivalent Cardiac Generator

One facet of electrocardiography has traditionally dealt with the relationship between body surface potentials and an equivalent generator that could have produced them. A key problem facing investigators today is whether a single fixed-location dipole is a suitable equivalent generator. The equivalent cardiac generator developed here provides a means for resolving this question. It consists of a collection of multipoles which, if placed at a point in a homogeneous isotropic conductor shaped like the human body, will give rise to the same potential distribution on the surface as that observed on the body itself. The multipole components can be evaluated by performing appropriate integrations of the potential over the body surface.

General solutions of equations of some geophysical importance

abstract General solutions are obtained in rectangular, circular cylindrical, and spherical co�nates for the equations of motion of a homogeneous isotropic elastic body (in § 2), the equations of the corresponding statical deformations (§ 3), the equations of motion of an incompressible viscous fluid (§ 4), the equations of the corresponding stationary motion (§ 5), and Maxwell's equations for a homogeneous isotropic conductor (§ 6).

地球物理学的興味のある2, 3の方程式の一般解

General solutions are obtained in rectangular, circular cylindrical, and spherical coordinates for the equations of motion of a homogeneous isotropic elastic body (in § 2), the equations of the corresponding statical deformations (§ 3), the equations of motion of an incompressible viscous fluid (§ 4), the equations of the corresponding stationary motion (§ 5), and Maxwell's equations for a homogeneous isotropic conductor (§ 6).

Superconductors in Alternating Magnetic Fields

A method of studying magnetic properties in alternating fields and its application to the intermediate state of a superconducting sphere are described. It is shown that the behaviour of the sphere in alternating fields of frequencies between 10 and 100 is different from that in very slowly varying fields, the difference becoming less for lower frequencies and for smaller radius. When the specimen is in the intermediate state, energy losses are produced in it by the alternating field, showing that the induced currents are out of phase with the alternating field. The induced currents do not, however, flow in the same way as in a homogeneous isotropic conductor, and this is probably on account of an anisotropic structure of the specimen when in the intermediate state. This structure depends strongly on the amplitude of the alternating field in such a way that the effective average conductivity is increased with decreasing amplitude, and this amplitude effect depends on the size of the specimen. Evidence is given for a time lag in the magnetocaloric effect, and this time lag decreases with decrease of the size of the specimen, being of order sec. for a sphere of about 1 cm. diameter.In conclusion I wish to thank Prof. Lord Rutherford and Dr Cockcroft for their interest in this work; Dr Peierls for suggesting the experiment and for much valuable advice and assistance; Mr Shire for advice on some technical points; and various members of the staff of the Royal Society Mond Laboratory for help in the measurements.

XVI.—Effects of Strain on Electric Conductivity

In the following communication an account is given of some experiments made in the Physical Laboratory of the Glasgow University, whose object was the measurement of the change of conductivity produced by strain in a metal conductor.Two constants are required in order to determine this alteration in the most general case of a homogeneous isotropic conductor, strained in any way, as may be easily inferred from the analogy with elastic strains and stresses, constants expressing the change produced by uniform cubical compression and by simple distortion. The determination of the latter for brass is the object of the present paper.