Thomson's Theorem Research Articles

A variational proof of Thomson's theorem

Thomson's theorem of electrostatics, which states the electric charge on a set of conductors distributes itself on the conductor surfaces to minimize the electrostatic energy, is reviewed in this letter. The proof of Thomson's theorem, based on a variational principle, is derived for a set of normal charged conductors, with and without the presence of external electric fields produced by fixed charge distributions. In this novel approach, the variations are performed on both the charge densities and electric potentials, by means of a local Lagrange multiplier associated with Poisson's equation, constraining the two variables.

The integral invariant of the equations of motion of a viscous gas

An expression for the velocity of motion of a simple vortex contour for which the circulation of the fluid velocity is preserved in it is obtained in the general three-dimensional case for the flow of a viscous gas or liquid. The velocity of motion of the contour at each point is calculated from the values of the flow parameters and their derivatives at the same point. This result extends Thomson's theorem, which is well known for an ideal barotropic fluid. A previously unknown conservation property is found, which consists of the fact that the circumferential circulation of the swirling axisymmetric flow in the potential field of mass forces is the first integral of the equations of unsteady flow of a non-barotropic ideal gas.

Toward the Landau–Lifshitz version of the Thomson electrostatics theorem

In the classical textbook (Landau and Lifshitz, 1963), Landau and Lifshtz suggested their version of the famous Thomson variational principle (a.k.a Thomson theorem.) So far, their version has not gained the interest it deserves, either among physicists or among applied mathematicians. Partially, the lack of interest can be explained because of the quality of the suggested proof of the principle. It is considerably lower than the standards accepted in classical electrostatics and mathematical physics. Even more importantly, Landau and Lifshitz did not demostrate the minimum property of the electrostatic energy at equilibrium. In this note, we, first, modify and specify the Landau–Lifshitz formulation of the principle presenting it as the isoperimetric variational problem. Then, for this isoperimetric problem we calculate the first and second variations, and we prove that the first variation vanishes, whereas the second variation appears to be positive.

Accuracy of magnetic energy computations

For magnetically driven events, the magnetic energy of the system is the prime energy reservoir that fuels the dynamical evolution. In the solar context, the free energy is one of the main indicators used in space weather forecasts to predict the eruptivity of active regions. A trustworthy estimation of the magnetic energy is therefore needed in three-dimensional models of the solar atmosphere, eg in coronal fields reconstructions or numerical simulations. The expression of the energy of a system as the sum of its potential energy and its free energy (Thomson's theorem) is strictly valid when the magnetic field is exactly solenoidal. For numerical realizations on a discrete grid, this property may be only approximately fulfilled. We show that the imperfect solenoidality induces terms in the energy that can lead to misinterpreting the amount of free energy present in a magnetic configuration. We consider a decomposition of the energy in solenoidal and nonsolenoidal parts which allows the unambiguous estimation of the nonsolenoidal contribution to the energy. We apply this decomposition to six typical cases broadly used in solar physics. We quantify to what extent the Thomson theorem is not satisfied when approximately solenoidal fields are used. The quantified errors on energy vary from negligible to significant errors, depending on the extent of the nonsolenoidal component. We identify the main source of errors and analyze the implications of adding a variable amount of divergence to various solenoidal fields. Finally, we present pathological unphysical situations where the estimated free energy would appear to be negative, as found in some previous works, and we identify the source of this error to be the presence of a finite divergence. We provide a method of quantifying the effect of a finite divergence in numerical fields, together with detailed diagnostics of its sources.

MAGNETIC FIELD AND CURRENT ARE ZERO INSIDE IDEAL CONDUCTORS

We prove a theorem on the magnetic energy minimum in a system of perfect, or ideal, conductors. It is analogous to Thomson's theorem on the equilibrium electric fleld and charge distribution in a system of conductors. We flrst prove Thomson's theorem using a variational principle. Our new theorem is then derived by similar methods. We flnd that magnetic energy is minimized when the current distribution is a surface current density with zero interior magnetic fleld; perfect conductors are perfectly diamagnetic. The results agree with currents in superconductors being conflned near the surface. The theorem implies a generalized force that expels current and magnetic fleld from the interior of a conductor that loses its resistivity. Examples of solutions that obey the theorem are presented.

Proof of Thomson's theorem of electrostatics

A 100 years old formula that was given by Thomson [Footnote on p. 154 of J.J. Thomson, Maxwell's Treatise on Electricity and Magnetism, Dover, New York, NY, 1954.] recently found numerous applications in computational electrostatics and electromagnetics. Thomson himself never gave a proof for the formula; but a proof based on Differential Geometry was suggested by Jackson [Classical Electrodynamics, Wiley, New York, NY, 1962, p. 51] and later published by Pappas [Differential geometric solution of a problem in electrostatics, SIAM Rev. 28 (1986) 225]. Unfortunately, Differential Geometry, being a specialized branch of mathematics, is normally inaccessible to the majority of scientists and engineers. This paper provides for the first time a proof that does not depend on Differential Geometry.

A generalized Milne–Thomson theorem for the case of parabolic inclusion

Complex analysis methods are applied to determine a velocity field of seepage in a heterogeneous infinite planar medium consisting of two dissimilar homogeneous components with a parabolic interface. New cases with arbitrary singularities of the principal part of a required complex potential are considered.

On the Stability Degree

Let A be a nondegenerate operator: |A| 6 = 0 (This is equivalent to the condition |B| 6 = 0.) Then the eigenvalues of A can be of three types: real pairs ±a, pure imaginary pairs ±ib, and quadruples ±a± ib. It was shown in [1] that the linear system (2) with nondegenerate operator A that admits an integral in the form of a nondegenerate quadratic form (1) can always be reduced to a system of Hamiltonian equations. Therefore, the theory developed below applies to this (formally, more general) case. The instability degree u of system (2) is defined as the number of roots (counted according to their multiplicities) of the characteristic equation of A in the right half-plane, and the stability degree s is defined as the number of pure imaginary roots (counted according to their multiplicities) of the characteristic equation of A. By i and i− we denote the positive and negative indices of inertia of the quadratic form (1). By the Thomson theorem [2], u ≡ i−mod 2. (3) In particular, if i− (or i) is odd, then so is u. Consequently, the characteristic polynomial of A necessarily has a positive real root in this case; therefore, x = 0 is an unstable equilibrium. More precisely, the classical Thomson theorem holds for linear mechanical systems subjected to additional gyroscopic and dissipative forces. The congruence (3) was proved in [3] for systems of general form (with Ḣ ≤ 0). If Ḣ = 0, then the Thomson theorem is equivalent to the following assertion on the stability degree: s is even if and only if

Distribution of charges on conductors and Thomson's theorem

A simple numerical method of multidimensional minimisation is applied in order to find the equilibrium configuration of N point charges confined to a given volume or surface. The method is applied to different geometric forms with both Coulomb and nonharmonic interacting potentials. The technique presented in this paper is very useful from a didactic point of view, as it allows the approximate configuration of charges on conductors to be visualised and stresses the importance of the inverse square law for the charge distribution. For this purpose, different three-dimensional and two-dimensional examples are presented.

AN IMPROVEMENT OF A RECENT RESULT OF THOMSON

In \cite{T13}, Brian S. Thomson proved the following result: \emph{Let $f$ be $AC^*G$ on an interval $[a,b]$. Then the total variation measure $\mu = \mu_f$ associated with $f$ has the following properties: a) $\mu$ is a $\sigma$-finite Borel measure on $[a,b]$; b) $\mu$ is absolutely continuous with respect to Lebesgue measure; \linebreak c) There is a sequence of closed sets $F_n$ whose union is all of $[a,b]$ such that $\mu(F_n) < \infty$ for each $n$; d) $\mu(B) = \mu_f(B) = \int_B|f^\prime(x)|\, dx$ for every Borel set $B \subset [a,b]$. Conversely, if a measure $\mu$ satisfies conditions \linebreak a)--c) then there exists an $AC^*G$ function $f$ for which the representation d) is valid.} In this paper we improve Thomson's theorem as follows: in the first part we ask $f$ to be only $VB^*G \cap (N)$ on a Lebesgue measurable subset $P$ of $[a,b]$ and continuous at each point of $P$; the converse is also true even for $\mu$ defined on the Lebesgue measurable subsets of $P$ (see Theorem \ref{T2} and the two examples in Remark~\ref{R1}).

Equilibrium Conditions in Electrostatic Fields

It is shown that Thomson's theorem describes the equilibrium energy of a system. If the charge is held constant, this energy is a minimum, but if the potential is held constant the energy is a maximum. The theorem can therefore be used to give upper and lower bounds to the electrostatic energy.

Some Comments on Thomson's Theorem

In this work some proofs of Thomson's theorem are discussed. Attention is focused on some points which seem to be disregarded by some authors. In this way a more thorough comprehension of the physical situation involved is gained.

Thermodynamik in gegenwart elektromagnetischer felder

Gibbs' fundamental equation for thermodynamics of fluid materials is discussed for the case when electromagnetic fields are present. The conditions of thermodynamic stability are established for such systems. Thomson's theorem in electrostatics is developed from the second law of thermodynamics which at the same time gives the equilibrium conditions for the fluid medium between the charged bodies.

ON THOMSON's THEOREM FOR MEASURING g BY OVERLAPPING TESTS

British Journal of Psychology. General SectionVolume 25, Issue 4 p. 490-493 ON THOMSON's THEOREM FOR MEASURING g BY OVERLAPPING TESTS W. STEPHENSON, W. STEPHENSON Psychology Laboratory, University College, London.Search for more papers by this author W. STEPHENSON, W. STEPHENSON Psychology Laboratory, University College, London.Search for more papers by this author First published: April 1935 https://doi.org/10.1111/j.2044-8295.1935.tb00764.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Volume25, Issue4April 1935Pages 490-493 RelatedInformation