Weber electrodynamics

The Weber electrodynamics (also Weber electrodynamics called) is physics a historically relevant approach to the explanation and description of the basic phenomena of electrodynamics . It goes back to Wilhelm Eduard Weber and Carl Friedrich Gauß . The theory assumes that the electric force depends not only on the distance, but also on the speed. It extends the Coulomb law with the aim of including the magnetic force with additional terms. In modern physics, Maxwell electrodynamics is the undisputed basis of classical electromagnetism. Weber electrodynamics, on the other hand, is largely unknown and forgotten.

Weber's electrodynamics, however, has some remarkable properties that make it appear interesting from today's perspective. In particular, their ability to explain the magnetic force in any shaped, current-carrying conductor tracks , as well as their ability to meet the impulse , angular momentum and energy conservation are to be emphasized here.

Basic approach

Weber electrodynamics assumes that the force between two point charges and through the equation ${\displaystyle {\vec {F}}}$ ${\displaystyle q_{s}}$ ${\displaystyle q_{d}}$

${\displaystyle {\vec {F}}={\frac {q_{s}q_{d}}{4\pi \varepsilon _{0}}}{\frac {\vec {r}}{r^{3}}}\left(1-{\frac {{\dot {r}}^{2}}{2c^{2}}}+{\frac {r{\ddot {r}}}{c^{2}}}\right),}$

given is. Here is the distance vector from the point charge to the point charge and the amount of the distance vector. Correspondingly, the first and second time derivatives represent the magnitude of the distance vector. Is the speed of light in a vacuum. The force formula is also known as Weber's law . ${\displaystyle {\vec {r}}={\vec {r}}_{d}-{\vec {r}}_{s}}$ ${\displaystyle q_{s}}$ ${\displaystyle q_{d}}$ ${\displaystyle r}$ ${\displaystyle {\dot {r}}}$ ${\displaystyle {\ddot {r}}}$ ${\displaystyle c}$

Two years after developing his force formula, Weber presented a formula for a speed-dependent potential energy in 1848:

${\displaystyle U(r,{\dot {r}})={\frac {q_{s}q_{d}}{4\pi \varepsilon _{0}r}}\left(1-{\frac {{\dot {r}}^{2}}{2c^{2}}}\right).}$

That this is actually the potential energy of the force postulated by Weber will be shown below.

The Weber force also differs for small velocities ( ) and accelerations from the force obtained by solving Maxwell's equations for a point charge. Only for and do both approaches lead to the same experimental predictions. ${\displaystyle {\dot {r}}\ll c}$ ${\displaystyle {\dot {r}}=0}$ ${\displaystyle {\ddot {r}}=0}$

Historical context

The defining characteristic of Weber electrodynamics is the assumption that the electrical force depends not only on the distance between the charges, but also on their speed of difference to one another. The idea for this can be found for the first time in the work of Carl Friedrich Gauß in 1835. Wilhelm Eduard Weber, who was a close friend of Gauss, probably took up this idea and carried out a series of complex experiments which were based on the work of André- Marie Ampère and Hans Christian Ørsted from the years 1820 to 1822. In 1846 Weber published Weber's law named after him and generalizing Coulomb's law.

In 1861 James Clerk Maxwell introduced the displacement current into Ampère's law . This gave the Maxwell equations their current form. Four years later he derived a wave equation from Maxwell's equations, in which the electromagnetic wave propagates in all inertial systems at the speed of light. This then established the connection between electromagnetic waves and light. Because of this success, the Maxwell equations quickly became the standard theory. Weber electrodynamics, on the other hand, fell into oblivion because it is not compatible with the first Maxwell equation, which requires the invariance of the electrical charge independent of the reference system.

properties

Conservation of momentum, conservation of angular momentum and conservation of energy in Weberscher and Maxwellian electrodynamics

In Maxwell's electrodynamics , Newton's third axiom does not apply to point charges. Instead, particles influence surrounding electromagnetic fields and the fields exert forces on particles. In Maxwell's electrodynamics, however, there are no direct forces between particles. For this reason, the forces that neighboring particles mutually exert on each other are not always inversely the same in Maxwell's electrodynamics, especially not when the particles move very slowly compared to the speed of light and accelerations are negligible.

In Maxwell's electrodynamics the true momentum conservation , the conservation of angular momentum and energy conservation only if both sources of power, as their fields are also considered. An exclusive consideration of the sources alone is not sufficient, for example the total momentum of all point charges in a system is not necessarily constant, since the point charges can transfer momentum to the electromagnetic field and this, conversely, can transfer momentum to the point charges. The well-known phenomenon of radiation pressure shows that electromagnetic waves are actually able to transmit momentum to matter.

The Weber force, by contrast, is quite different. Since the Weber force is a symmetrical central force, all particles, regardless of their mass and size, obey exactly the principle of action equal to reaction . Hence, in Weber's electrodynamics, conservation of momentum applies . The conservation of the angular momentum follows from the property of the Weber force to be a central force .

The total energy of a particle system is basically retained in Weber electrodynamics. To show this, the Weber force is multiplied by. If you write in spherical coordinates , it can be shown that the relationship holds. This leads to ${\displaystyle {\dot {\vec {r}}}}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {r}}\cdot {\dot {\vec {r}}}=r{\dot {r}}}$

${\displaystyle {\vec {F}}\cdot {\dot {\vec {r}}}={\frac {q_{s}q_{d}}{4\pi \varepsilon _{0}}}{\frac {\dot {r}}{r^{2}}}\left(1-{\frac {{\dot {r}}^{2}}{2c^{2}}}+{\frac {r{\ddot {r}}}{c^{2}}}\right).}$

Deriving the Weber potential according to time results

${\displaystyle {\dot {U}}={\frac {q_{1}\,q_{2}}{4\,\pi \,\varepsilon _{0}}}\left(-{\frac {\dot {r}}{r^{2}}}+{\frac {{\dot {r}}^{3}}{2r^{2}c^{2}}}-{\frac {{\dot {r}}{\ddot {r}}}{rc^{2}}}\right)=-{\frac {q_{1}\,q_{2}}{4\,\pi \,\varepsilon _{0}}}{\frac {\dot {r}}{r^{2}}}\left(1-{\frac {{\dot {r}}^{2}}{2c^{2}}}+{\frac {r{\ddot {r}}}{c^{2}}}\right).}$

A comparison of the two equations shows that . This makes it clear that energy conservation must also apply, because it follows from this by inserting the basic equation of mechanics . Except for the sign, the right-hand side corresponds to the time derivative of the kinetic energy. This means that every change in potential energy over time is precisely compensated by a change in kinetic energy over time. The total energy, i.e. the sum of potential energy and kinetic energy, is therefore a conservation quantity . ${\displaystyle {\dot {U}}=-{\vec {F}}\cdot {\dot {\vec {r}}}}$ ${\displaystyle {\dot {U}}=-m\,{\ddot {\vec {r}}}\cdot {\dot {\vec {r}}}}$

Relative velocity dependence of the strength of the force in Weber electrodynamics

The absolute strength of the Weber force depends not only on the distance between the point charges, but also on the radial velocity and the radial acceleration . This becomes clear when one calculates the amount of Weber force: ${\displaystyle r}$ ${\displaystyle {\dot {r}}}$ ${\displaystyle {\ddot {r}}}$

${\displaystyle \Vert {\vec {F}}\Vert ={\sqrt {{\vec {F}}\cdot {\vec {F}}}}={\frac {q_{s}q_{d}}{4\pi \varepsilon _{0}}}{\frac {1}{r^{2}}}\left(1-{\frac {{\dot {r}}^{2}}{2c^{2}}}+{\frac {r{\ddot {r}}}{c^{2}}}\right).}$

From this it follows that a neutral plasma would have to lose its electrical neutrality when the temperature rises, since the electrons are significantly lighter than the ions and thus gain disproportionately in speed when the plasma temperature rises. Nothing like this has yet been observed.

Magnetism in Weber Electrodynamics

In order to arrive at the magnetic force of a current-carrying wire on a moving test charge, the Weber force between each individual charge in the wire and the test charge must be calculated and added up.

Weber's goal in developing his force formula was to trace back the magnetic force effects of a wire through which current flows to the electrical forces of the charge carriers contained in the wire. It is remarkable that he succeeded in doing this and that Weber's law, which seems inconspicuous, actually seems to contain magnetic force. In Maxwell's electrodynamics, however, the magnetic force is independent and the magnetic field, together with the electric field, forms the four-dimensional electromagnetic field strength tensor . The actual force effect is then calculated using the Lorentz force law .

In order to gain access to Weber's idea, it is helpful to convert the Weber force into vector notation using and . With ${\displaystyle {\dot {\vec {r}}}:={\vec {v}}}$ ${\displaystyle {\ddot {\vec {r}}}:={\vec {a}}}$

${\displaystyle {\dot {r}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\sqrt {{\vec {r}}\cdot {\vec {r}}}}={\frac {{\vec {r}}\cdot {\dot {\vec {r}}}}{r}}={\frac {{\vec {r}}\cdot {\vec {v}}}{r}}}$

and

${\displaystyle {\ddot {r}}={\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}{\sqrt {{\vec {r}}\cdot {\vec {r}}}}={\frac {{\vec {r}}\cdot {\ddot {\vec {r}}}}{r}}-{\frac {({\vec {r}}\cdot {\dot {\vec {r}}})^{2}}{r^{3}}}+{\frac {{\dot {\vec {r}}}\cdot {\dot {\vec {r}}}}{r}}={\frac {{\vec {r}}\cdot {\vec {a}}}{r}}-{\frac {({\vec {r}}\cdot {\vec {v}})^{2}}{r^{3}}}+{\frac {v^{2}}{r}}}$

is the Weber force

${\displaystyle {\vec {F}}={\frac {q_{s}q_{d}}{4\pi \varepsilon _{0}}}{\frac {\vec {r}}{r^{3}}}\left(1+{\frac {v^{2}}{c^{2}}}-{\frac {3}{2}}\left({\frac {\vec {r}}{r}}\cdot {\frac {\vec {v}}{c}}\right)^{2}+{\frac {{\vec {r}}\cdot {\vec {a}}}{c^{2}}}\right).}$

If one defines as the angle between the connecting axis of the two point charges and the difference in speed and if existing differential accelerations are assumed to be negligible, the Weber force is simplified ${\displaystyle \alpha }$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {a}}}$

${\displaystyle {\vec {F}}={\frac {q_{s}q_{d}}{4\pi \varepsilon _{0}}}{\frac {\vec {r}}{r^{3}}}\left(1+{\frac {v^{2}}{c^{2}}}\left(1-{\frac {3}{2}}\cos(\alpha )^{2}\right)\right).}$

This makes it clear that the Weber force depends on the direction in which the point charges move towards each other. If the point charges move directly towards or away from each other ( or ), the Weber force is weaker than is predicted by the Coulomb force. However , if the charges move laterally past each other ( ), the Weber force is stronger than the Coulom force would suggest. ${\displaystyle \alpha =0^{\circ }}$ ${\displaystyle 180^{\circ }}$ ${\displaystyle \alpha =\pm 90^{\circ }}$

This presumably relativistic correction of the Coulomb force is sufficient to explain the Lorentz force for any shape, continuous current flowing through, closed conductor tracks. As can be shown, the force effect of the magnetic field results from the sum of all Weber forces between the charge carriers in the conductor track and the test charge. With the summation or integration, it should be noted that the individual distance vector and the correct differential speed are used in each case . ${\displaystyle {\vec {r}}={\vec {r}}_{d}-{\vec {r}}_{s}}$ ${\displaystyle {\vec {v}}={\vec {v}}_{d}-{\vec {v}}_{s}}$

It is important to emphasize that the experimental predictions of Weber electrodynamics and Maxwell's electrodynamics differ for non-closed conductor paths or inhomogeneous current densities. It should also be mentioned that Weber's explanation is Galileo-invariant .

restrictions

Notwithstanding various efforts, a speed or acceleration dependency of the absolute strength of the electrical force has not yet been observed. In addition, Weber's electrodynamics predicts that under certain circumstances charges behave as if they had a negative inertia, which has also never been observed. However, these Helmholtz arguments are questioned by some scientists.

A very obvious limitation of Weber's electrodynamics is that it is not able to describe electromagnetic waves. However, this restriction is not of a fundamental nature. It simply consists of the fact that Weber's electrodynamics was never expanded into a field theory that also takes retardation effects into account. However, there are currently attempts to overcome these limitations.

Weber electrodynamics still does not apply to relativistic particle speeds. There are approaches to these limitations by introducing a relativistic potential energy

${\displaystyle U_{\mathrm {rel} }(r,{\dot {r}})={\sqrt {1-{\frac {{\dot {r}}^{2}}{c^{2}}}}}{\frac {q_{s}q_{d}}{4\pi \varepsilon _{0}r}}}$

to overcome. Weber's original potential formula represents a non-relativistic approximation. It is obtained by a Taylor series expansion with respect to the position 0 and termination after the second order term. ${\displaystyle {\dot {r}}}$

further reading

• André Koch Torres Assis: Weber's electrodynamics. Kluwer Acad. Publ., Dordrecht 1994, ISBN 0-7923-3137-0 .

credentials

1. ↑ Most (perhaps all) current textbooks on classical electromagnetism do not mention Weber electrodynamics. Instead they present the Maxwell equations as the undisputed basis of classical electromagnetism. Five examples are: Classical electrodynamics by JD Jackson (3rd ed., 1999); Introduction to electrodynamics by DJ Griffiths (3rd ed., 1999); Physics for students of science and engineering by D. Halliday and R. Resnick (part 2, 2nd ed., 1962); Electromagnetic field theory by G. Lehner (4th edition, 2004); Feynman lectures on physics by Feynman, Leighton, and Sands, [1]
2. ^ ACT Assis, HT Silva: Comparison between Weber's electrodynamics and classical electrodynamics . In: Pramana . 55, No. 3, September 2000, pp. 393-404. bibcode : 2000Prama..55..393A . doi : 10.1007 / s12043-000-0069-2 .
3. ^ ^{A b} W. Weber: Wilhelm Weber's Works (Volume 3). Galvanism and Electrodynamics. First part. . Royal Society of Göttingen, 1893, pp. 244 and 245.
4. ^ A. O'Rahilly: Electromagnetic Theory: A Critical Examination of Fundamentals, Vol. 2 . Dover Publications, 1965, p. 524.
5. ↑ AKT Assis: Deriving Ampere's Law from Weber's Law . In: Hadronic Journal . 13, 1990, pp. 441-451.
6. ↑ JJ Caluzi, AKT Assis: A critical analysis of Helmholtz's argument against Weber's electrodynamics . In: Foundations of Physics . 27, No. 10, 1997, pp. 1445-1452. bibcode : 1997FoPh ... 27.1445C . doi : 10.1007 / BF02551521 .
7. ^ Anonymous: Advances in Weber and Maxwell Electrodynamics . Amazon Fulfillment, 2018.
8. ^ TE Phipps, Jr .: Toward Modernization of Weber's Force Law . In: Physics Essays . 3, No. 4, 1990, pp. 414-420.