E-Liberary Pakistan: Electromagnetic mass

Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for some time also considered as a dynamical explanation of inertial mass per se. Today, the relation of mass,momentum, velocity and all forms of energy, including electromagnetic energy, is interpreted on the basis of Albert Einstein's special relativity and mass–energy equivalence. As to the cause of mass of elementary particles, the Higgs mechanism in the framework of the relativistic Standard Model is currently used. In addition, some problems concerning the electromagnetic mass and self-energy of charged particles are still studied.

1 Charged particles
- 1.1 Rest mass and energy
- 1.2 Mass and speed
  - 1.2.1 Thomson and Searle
  - 1.2.2 Longitudinal and transverse mass
  - 1.2.3 Kaufmann's experiments
- 1.3 Poincaré stresses and 4/3 problem
2 Inertia of energy and radiation paradoxes
- 2.1 Radiation pressure
- 2.2 Mass of the fictitious electromagnetic fluid
- 2.3 Momentum and cavity radiation
3 Modern view
- 3.1 Mass–energy equivalence
- 3.2 Relativistic mass
- 3.3 Self-energy
- Charged particles
  
  Rest mass and energy
  
  It was recognized by J. J. Thomson in 1881 that a charged sphere moving in a space filled with a medium of a specific inductive capacity (the electromagnetic aether of James Clerk Maxwell), is harder to set in motion than an uncharged body. (Similar considerations were already made by George Gabriel Stokes (1843) with respect to hydrodynamics, who showed that the inertia of a body moving in an incompressible perfect fluid is increased. So due to this self-induction effect, electrostatic energy behaves as having some sort of momentum and "apparent" electromagnetic mass, which can increase the ordinary mechanical mass of the bodies, or in more modern terms, the increase should arise from their electromagnetic self-energy. This idea was worked out in more detail by Oliver Heaviside (1889),Thomson (1893), George Frederick Charles Searle (1897), Max Abraham (1902), Hendrik Lorentz (1892, 1904), and was directly applied to the electron by using the Abraham–Lorentz force. Now, the electrostatic energy $E_{em}$ and mass $m_{em}$ of an electron at rest was calculated to be
  
  $E_{em}=\frac{1}{2}\frac{e^{2}}{a},\qquad m_{em}=\frac{2}{3}\frac{e^{2}}{ac^{2}}$
  
  where $e$ the uniformly distributed charge, $a$ is the classical electron radius, which must be nonzero to avoid infinite energy accumulation. Thus the formula for this electromagnetic energy–mass relation is
  
  $m_{em}=\frac{4}{3}\frac{E_{em}}{c^{2}}$
  
  This was discussed in connection with the proposal of the electrical origin of matter, so Wilhelm Wien (1900) and Max Abraham (1902), came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass. Wien stated, that if it is assumed that gravitation is an electromagnetic effect too, then there has to be a proportionality between electromagnetic energy, inertial mass, and gravitational mass. When one body attracts another one, the electromagnetic energy store of gravitation is according to Wien diminished by the amount (where $M$ is the attracted mass, $G$ the gravitational constant, $r$ the distance):
  
  $G\frac{\frac{4}{3}\frac{E_{em}}{c^{2}}M}{r}$
  
  Henri Poincaré in 1906 argued that when mass is in fact the product of the electromagnetic field in the aether – implying that no "real" mass exists – and because matter is inseparably connected with mass, then also matter doesn't exist at all and electrons are only concavities in the aether.
  
  Mass and speed
  
  Thomson and Searle
  
  Thomson (1893) noticed that electromagnetic momentum and energy of charged bodies and therefore their masses depend on the speed of the bodies as well. He wrote:
  
  [p. 21] When in the limit v = c, the increase in mass is infinite, thus a charged sphere moving with the velocity of light behaves as if its mass were infinite, its velocity therefore will remain constant, in other words it is impossible to increase the velocity of a charged body moving through the dielectric beyond that of light.
  
  In 1897, Searle gave a more precise formula for the electromagnetic energy of charged sphere in motion:
  
  $E_{em}^{v}=E_{em}\left[\frac{1}{\beta}\ln\frac{1+\beta}{1-\beta}-1\right],\qquad\beta=\frac{v}{c},$
  
  and like Thomson he concluded:
  
  ... when v = c the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light.
  
  Longitudinal and transverse mass
  
  Predictions of speed dependence of transverse electromagnetic mass according to the theories of Abraham, Lorentz, and Bucherer.
  
  From Searle's formula, Walter Kaufmann (1901) and Abraham (1902) derived the formula for the electromagnetic mass of moving bodies:
  
  $m_{L}=\frac{3}{4}\cdot m_{em}\cdot\frac{1}{\beta^{2}}\left[-\frac{1}{\beta^{2}}\ln\left(\frac{1+\beta}{1-\beta}\right)+\frac{2}{1-\beta^{2}}\right]$
  
  However, it was shown by Abraham (1902), that this value is only valid in the longitudinal direction ("longitudinal mass"), i.e., that the electromagnetic mass also depends on the direction of the moving bodies with respect to the aether. Thus Abraham also derived the "transverse mass":
  
  $m_{T}=\frac{3}{4}\cdot m_{em}\cdot\frac{1}{\beta^{2}}\left[\left(\frac{1+\beta^{2}}{2\beta}\right)\ln\left(\frac{1+\beta}{1-\beta}\right)-1\right]$
  
  On the other hand, already in 1899 Lorentz assumed that the electrons undergo length contraction in the line of motion, which leads to results for the acceleration of moving electrons that differ from those given by Abraham. Lorentz obtained factors of $k^3 \varepsilon$ parallel to the direction of motion and $k\varepsilon$ perpendicular to the direction of motion, where $k = \sqrt{1- v^2 / c^2}$ and $\varepsilon$ is an undetermined factor. Lorentz expanded his 1899 ideas in his famous 1904 paper, where he set the factor $\varepsilon$ to unity, thus:
  
  $m_{L}=\frac{m_{em}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=\frac{m_{em}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ ,
  
  So, eventually Lorentz arrived at the same conclusion as Thomson in 1893: no body can reach the speed of light because the mass becomes infinitely large at this velocity.
  
  Additionally, a third electron model was developed by Alfred Bucherer and Paul Langevin, in which the electron contracts in the line of motion, and expands perpendicular to it, so that the volume remains constant.This gives:
  
  $m_{L}=\frac{m_{em}\left(1-\frac{1}{3}\frac{v^{2}}{c^{2}}\right)}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{8/3}},\quad m_{T}=\frac{m_{em}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{2/3}}$
  
  Kaufmann's experiments
  
  The predictions of the theories of Abraham and Lorentz were supported by the experiments of Walter Kaufmann (1901), but the experiments were not precise enough to distinguish between them.In 1905 Kaufmann conducted another series of experiments (Kaufmann–Bucherer–Neumann experiments) which confirmed Abraham's and Bucherer's predictions, but contradicted Lorentz's theory and the "fundamental assumption of Lorentz and Einstein", i.e., the relativity principle. In the following years experiments by Alfred Bucherer (1908), Gunther Neumann (1914) and others seemed to confirm Lorentz's mass formula. It was later pointed out that the Bucherer–Neumann experiments were also not precise enough to distinguish between the theories – it lasted until 1940 when the precision required was achieved to eventually prove Lorentz's formula and to refute Abraham's by these kinds of experiments. (However, other experiments of different kind already refuted Abraham's and Bucherer's formulas long before.)
  
  Poincaré stresses and 4/3 problem
  
  The idea of an electromagnetic nature of matter, however, had to be given up. Abraham (1904, 1905) argued that non-electromagnetic forces were necessary to prevent Lorentz's contractile electrons from exploding. He also showed that different results for the longitudinal electromagnetic mass can be obtained in Lorentz's theory, depending on whether the mass is calculated from its energy or its momentum, so a non-electromagnetic potential (corresponding to 1/3 of the Electron's electromagnetic energy) was necessary to render these masses equal. Abraham doubted whether it was possible to develop a model satisfying all of these properties.
  
  To solve those problems, Henri Poincaré in 1905 and 1906 introduced some sort of pressure ("Poincaré stresses") of non-electromagnetic nature. As required by Abraham, these stresses contribute non-electromagnetic energy to the electrons, amounting to 1/4 of their total energy or to 1/3 of their electromagnetic energy. So, the Poincaré stresses remove the contradiction in the derivation of the longitudinal electromagnetic mass, they prevent the electron from exploding, they remain unaltered by a Lorentz transformation (i.e. they are Lorentz invariant), and were also thought as a dynamical explanation of length contraction. However, Poincaré still assumed that only the electromagnetic energy contributes to the mass of the bodies.
  
  As it was later noted, the problem lies in the 4/3 factor of electromagnetic rest mass – given above as $m_{em}=(4/3)E_{em}/c^2$ when derived from the Abraham–Lorentz equations. However, when it is derived from the electron's electrostatic energy alone, we have $m_{es}=E_{em}/c^2$ where the 4/3 factor is missing. This can be solved by adding the non-electromagnetic energy $E_{p}$ of the Poincaré stresses to $E_{em}$ , the electron's total energy $E_{tot}$ now becomes:
  
  $\frac{E_{tot}}{c^{2}}=\frac{E_{em}+E_{p}}{c^{2}}=\frac{E_{em}+\frac{E_{em}}{3}}{c^{2}}=\frac{4}{3}\frac{E_{em}}{c^{2}}=\frac{4}{3}m_{es}=m_{em}$
  
  Thus the missing 4/3 factor is restored when the mass is related to its electromagnetic energy, and it disappears when the total energy is considered.
  
  Inertia of energy and radiation paradoxes
  
  Radiation pressure
  
  Another way of deriving some sort of electromagnetic mass was based on the concept of radiation pressure. These pressures or tensions in the electromagnetic field were derived by James Clerk Maxwell (1874) and Adolfo Bartoli(1876). Lorentz recognized in 1895 that those tensions also arise in his theory of the stationary aether. So if the electromagnetic field of the aether is able to set bodies in motion, the action/reaction principle demands that the aether must be set in motion by matter as well. However, Lorentz pointed out that any tension in the aether requires the mobility of the aether parts, which in not possible since in his theory the aether is immobile. This represents a violation of the reaction principle that was accepted by Lorentz consciously. He continued by saying, that one can only speak aboutfictitious tensions, since they are only mathematical models in his theory to ease the description of the electrodynamic interactions.