Darwin magnetic interaction energy and its macroscopic consequences

Essén, Hanno

doi:10.1103/physreve.53.5228

Cited by 34 publications

(35 citation statements)

References 27 publications

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“…A book by Podolsky and Kunz [12] is a bit more thorough. Several good fundamental and pedagogical studies can, however, be found in the literature [13,14,15,16,17,18,19,20].…”

Section: The Darwin Lagrangianmentioning

confidence: 99%

From least action in electrodynamics to magnetomechanical energy—a review

Essén

2009

Eur. J. Phys.

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The equations of motion for electromechanical systems are traced back to the fundamental Lagrangian of particles and electromagnetic fields, via the Darwin Lagrangian. When dissipative forces can be neglected the systems are conservative and one can study them in a Hamiltonian formalism. The central concepts of generalized capacitance and inductance coefficients are introduced and explained. The problem of gauge independence of self-inductance is considered. Our main interest is in magnetomechanics, i.e. the study of systems where there is exchange between mechanical and magnetic energy. This throws light on the concept of magnetic energy, which according to the literature has confusing and peculiar properties. We apply the theory to a few simple examples: the extension of a circular current loop, the force between parallel wires, interacting circular current loops, and the rail gun. These show that the Hamiltonian, phase space, form of magnetic energy has the usual property that an equilibrium configuration corresponds to an energy minimum.

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“…A book by Podolsky and Kunz [12] is a bit more thorough. Several good fundamental and pedagogical studies can, however, be found in the literature [13,14,15,16,17,18,19,20].…”

Section: The Darwin Lagrangianmentioning

confidence: 99%

From least action in electrodynamics to magnetomechanical energy—a review

Essén

2009

Eur. J. Phys.

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“…The magnetic interaction described by the Darwin Lagrangian is essential in relativistic many-electron calculations as noted by Breit and others [12][13][14][15]. It has found applications in nuclear physics [16,17], and especially in plasma physics, for numerical simulation [18][19][20][21][22], thermodynamics and kinetics [23][24][25][26][27], as well fundamental theory [28][29][30]. Barcons and Lapiedra [31] noted that the Darwin approach is not valid for a relativistic plasma and therefore used a different approach to its statistical mechanics.…”

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confidence: 99%

The exact Darwin Lagrangian

Essén

2007

Europhys. Lett.

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Darwin (1920) noted that when radiation can be neglected it should be possible to eliminate the radiation degrees-of-freedom from the action of classical electrodynamics and keep the discrete particle degrees-of-freedom only. Darwin derived his well known Lagrangian by series expansion in v/c keeping terms up to order (v/c) 2 . Since radiation is due to acceleration the assumption of low speed should not be necessary. A Lagrangian is suggested that neglects radiation without assuming low speed. It cures deficiencies of the Darwin Lagrangian in the ultra-relativistic regime.PACS numbers: 03.50. De, 11.10.Ef When radiation can be neglected the Lagrangian of classical electrodynamics, putting β = v/c, can be written,(1) In 1920 Darwin [1] expanded the Liénard-Wiechert potentials to second order in β = v/c and thus found that,and (hats are used for unit vectors),give the correct Lagrangian to this order. More recent derivations can be found in a few textbooks [2][3][4]. In particular Jackson [4] notes that using the Coulomb gauge (∇ · A = 0) makes the electrostatic Coulomb potential φ exact and moves all approximation to the vector potential A which obeys the inhomogeneous wave equation with the transverse (divergence free) current as source. [7]), and it is useful in various fundamental studies of electrodynamics [8][9][10][11]. The magnetic interaction described by the Darwin Lagrangian is essential in relativistic many-electron calculations as noted by Breit and others [12][13][14][15]. It has found applications in nuclear physics [16,17], and especially in plasma physics, for numerical simulation [18][19][20][21][22], thermodynamics and kinetics [23][24][25][26][27], as well fundamental theory [28][29][30]. Barcons and Lapiedra [31] noted that the Darwin approach is not valid for a relativistic plasma and therefore used a different approach to its statistical mechanics.Corrections to the Darwin Lagrangian have been discussed. Since a system of particles with identical charge to mass ratio does not dipole radiate a higher order expansion should be meaningful for such systems [32][33][34]. To that order, however, acceleration inevitably enters and must be handled in some way. Others have argued that since radiation is due to acceleration, v/c expansion is irrelevant, and further that radiation can be negligible even if the particle speeds are considerable (Trubnikov and Kosachev [35], Frejlak [36]). We will pursue that lead here.One frequently encounters the statement that the Darwin approach neglects retardation. This may be due to the fact that the, nowadays best known, elegant derivation by Jackson [4] hides the complications due to retardation. Nevertheless it is wrong. The derivations by Darwin [1] and by Landau and Lifshitz [2] show that the contribution of retardation to the Coulomb potential in the Lorenz gauge, is quite large. The main acceleration dependent part, however, vanishes either, as in Darwin's derivation, because it gives a total time derivative term in the Lagrangian, or, as in Landau ...

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“…The vector potential produced by this current density can be found using the methods of Essén [28], see also [29][30][31]. If we introduce ξ = a/R, we find,…”

Section: A02 Current In Sphere Due To Rigidly Rotating Chargementioning

confidence: 99%

Magnetic Field and Current Are Zero Inside Ideal Conductors

Fiolhais¹,

Essén²,

Providência³

et al. 2011

PIER B

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Abstract-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 field and charge distribution in a system of conductors. We first prove Thomson's theorem using a variational principle. Our new theorem is then derived by similar methods. We find that magnetic energy is minimized when the current distribution is a surface current density with zero interior magnetic field; perfect conductors are perfectly diamagnetic. The results agree with currents in superconductors being confined near the surface. The theorem implies a generalized force that expels current and magnetic field from the interior of a conductor that loses its resistivity. Examples of solutions that obey the theorem are presented.

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Darwin magnetic interaction energy and its macroscopic consequences

Cited by 34 publications

References 27 publications

From least action in electrodynamics to magnetomechanical energy—a review

From least action in electrodynamics to magnetomechanical energy—a review

The exact Darwin Lagrangian

Magnetic Field and Current Are Zero Inside Ideal Conductors

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