Classical diamagnetism, magnetic interaction energies, and repulsive forces in magnetized plasmas

The Bohr-van Leeuwen theorem is often summarized as saying that there is no classical magnetic susceptibility, in particular no diamagnetism. This is seriously misleading. The theorem assumes position dependent interactions but this is not required by classical physics. Since the work of Darwin in 1920 it has been known that the magnetism due to classical charged point particles can only be described by allowing velocity dependent interactions in the Lagrangian. Legendre transformation to an approximate Hamiltonian can give an estimate of the Darwin diamagnetism for a system of charged point particles. Comparison with experiment, however, requires knowledge of the number of classically behaving electrons in the sample. A new repulsive effective many-body force, which should be relevant in plasmas, is predicted by the Hamiltonian.Comment: added references, revise

Difference in Coulomb Electrostatic Energy for Localized versus Delocalized Electrons and Electron Pairs—Exact Results Based on Cubic Charge Distributions

Wigner showed that a sufficiently thin electron gas will condense into a crystal of localized electrons. Here, we show, using a model based on cubic charge distributions that gives exact results, that the Coulomb repulsion energy of localized charge distributions is lower than that of delocalized distributions in spite of the fact that the total overall charge distribution is the same. Assuming a simple cubic geometry, we obtain an explicit result for the energy reduction. This reduction results from the exclusion of self-interactions of the electrons. The corresponding results for electron pairs are also discussed

Difference in Coulomb Electrostatic Energy for Localized versus Delocalized Electrons and Electron Pairs—Exact Results Based on Cubic Charge Distributions

Wigner showed that a sufficiently thin electron gas will condense into a crystal of localized electrons. Here, we show, using a model based on cubic charge distributions that gives exact results, that the Coulomb repulsion energy of localized charge distributions is lower than that of delocalized distributions in spite of the fact that the total overall charge distribution is the same. Assuming a simple cubic geometry, we obtain an explicit result for the energy reduction. This reduction results from the exclusion of self-interactions of the electrons. The corresponding results for electron pairs are also discussed

The Magnetic Interaction Energy between an Infinite Solenoid and a Passing Point Charge

The standard expression for the magnetic interaction energy used in the study of the Aharonov-Bohm effect is investigated. We calculate the magnetic interaction energy between a point charge and an infinite solenoid from first principles. Two alternative expressions are used: the scalar products of the currents with the vector potentials and the scalar product of the magnetic fields. The alternatives are seen to agree. The latter approach also involves taking into account surface integrals at infinity, which are shown to be zero. Our model problem indicates no classical Aharonov-Bohm effect, but we also discuss the normally neglected fact of energy non-conservation. The problem is treated from the point of view of Lagrangian and Hamiltonian mechanics.Peer reviewe

Observation of abundant heat production from a reactor device and of isotopic changes in the fuel

New results are presented from an extended experimental investigation of anomalous heat production in a special type of reactor tube operating at high temperatures. The reactor, named E-Cat, is charged with a small amount of hydrogen-loaded nickel powder plus some additives, mainly Lithium. The reaction is primarily initiated by heat from resistor coils around the reactor tube. Measurements of the radiated power from the reactor were performed with high-resolution thermal imaging cameras. The measurements of electrical power input were performed with a large bandwidth three-phase power analyzer. Data were collected during 32 days of running in March 2014. The reactor operating point was set to about 1260 ºC in the first half of the run, and at about 1400 °C in the second half. The measured energy balance between input and output heat yielded a COP factor of about 3.2 and 3.6 for the 1260 ºC and 1400 ºC runs, respectively . The total net energy obtained during the 32 days run was about 1.5 MWh. This amount of energy is far more than can be obtained from any known chemical sources in the small reactor volume. A sample of the fuel was carefully examined with respect to its isotopic composition before the run and after the run, using several standard methods: XPS, EDS, SIMS, ICP-MS and ICP-AES. The isotope composition in Lithium and Nickel was found to agree with the natural composition before the run, while after the run it was found to have changed substantially . Nuclear reactions are therefore indicated to be present in the run process, which however is hard to reconcile with the fact that no radioactivity was detected outside the reactor during the run

The comfortable roller coaster -- on the shape of tracks with constant normal force

A particle that moves along a smooth track in a vertical plane is influenced by two forces: gravity and normal force. The force experienced by roller coaster riders is the normal force, so a natural question to ask is: what shape of the track gives a normal force of constant magnitude? Here we solve this problem. It turns out that the solution is related to the Kepler problem; the trajectories in velocity space are conic sections.Comment: 10 pages, 4 figure

Static deformation of heavy spring due to gravity and centrifugal force

The static equilibrium deformation of a heavy spring due to its own weight is calculated for two cases. First for a spring hanging in a constant gravitational field, then for a spring which is at rest in a rotating system where it is stretched by the centrifugal force. Two different models are considered. First a discrete model assuming a finite number of point masses connected by springs of negligible weight. Then the continuum limit of this model. In the second case the differential equation for the deformation is obtained by demanding that the potential energy is minimized. In this way a simple application of the variational calculus is obtained.Comment: 11 pages, 2 figure