# Weber electrodynamics

Weber electrodynamics is an alternative to Maxwell electrodynamics developed by Wilhelm Eduard Weber. In this theory, Coulomb's Law becomes velocity dependent. The theory is widely rejected and ignored by contemporary physicists, and is not even mentioned in mainstream textbooks on classical electromagnetism.

## Mathematical Description

According to Weber electrodynamics, the force (F) acting simultaneously on point charges $q_1$ and $q_2$, is given by

$\mathbf{F} = \frac{q_1 q_2 \mathbf{\hat{r}}}{4 \pi \epsilon_0 r^2}\left(1-\frac{\dot{r}^2}{2 c^2}+\frac{r\ddot{r}}{c^2}\right)$

where $\mathbf{r}$ is the vector connecting $q_1$ and $q_2$, the dots over $r$ denote time derivatives and $c$ is the speed of light. In the limit that speeds and accelerations are small (i.e. $\dot{r}\ll c$), this reduces to the usual Coulomb's law.[1]

This can be derived from the potential energy

$U_{Web} = \frac{q_1 q_2}{4 \pi \epsilon_0 r}\left(1-\frac{\dot{r}^2}{2 c^2}\right)$

This can be contrasted with the approximate potential energy from Maxwellian electrodynamics (where $v_1$ and $v_2$ are the velocities of $q_1$ and $q_2$, respectively):[1]

$U_{Max} =\frac{q_1 q_2}{4 \pi \epsilon_0 r}\left(1-\frac{\mathbf{v_1}\cdot\mathbf{v_2}+(\mathbf{v_1}\cdot\mathbf{\hat{r}})(\mathbf{v_2}\cdot\mathbf{\hat{r}})}{2 c^2}\right)$

(This only includes terms up to order $(v/c)^2$ and therefore neglects relativistic and retardation effects; see Darwin Lagrangian.)

Using these expressions, the regular form of Ampere's law and Faraday's law can be derived. Importantly, this theory does not predict an expression like the Biot–Savart law and testing differences between Ampere's law and the Biot–Savart law is one way to test Weber electrodynamics.[2]

## Newton's third law in Maxwell and Weber electrodynamics

In Maxwell electrodynamics, Newton's third law does not hold for particles. Instead, particles exert forces on electromagnetic fields, and fields exert forces on particles, but particles do not directly exert forces on other particles. Therefore, two nearby particles need not experience equal and opposite forces. Related to this, Maxwell electrodynamics predicts that the laws of conservation of momentum and conservation of angular momentum are valid only if the momentum of particles and the momentum of surrounding electromagnetic fields are taken into account. The total momentum of all particles is not necessarily conserved, because the particles may transfer some of their momentum to electromagnetic fields or vice-versa. The well-known phenomenon of radiation pressure proves that electromagnetic waves are indeed able to "push" on matter. See Maxwell stress tensor and Poynting vector for further details.

The Weber force law is quite different: All particles, regardless of size and mass, will exactly follow Newton's third law. Therefore, Weber electrodynamics, unlike Maxwell electrodynamics, has conservation of particle momentum and conservation of particle angular momentum.

## Predictions

When applied to gravitation, it has been claimed to predict the perihelion precession of Mercury, and has been used to explain various phenomena such as wires exploding when exposed to high currents.[3]

## Limitations

Despite various efforts, a velocity and/or acceleration dependent correction to Coulomb's law has never been observed, as described in the next section. Moreover, Helmholtz observed that Weber electrodynamics predicted that under certain configurations charges can act as if they had negative inertial mass, which has also never been observed.

## Experimental tests

### Velocity Dependent Tests

Velocity and acceleration dependent corrections to Maxwell's equations arise in Weber electrodynamics. The strongest limits on a new velocity dependent term come from evacuating gasses from containers and observing whether the electrons become charged. However, because the electrons used to set these limits are Coulomb bound, renormalization effects may cancel the velocity dependent corrections. Other searches have spun current-carrying solenoids, observed metals as they cooled, and used superconductors to obtain a large drift velocity.[4] None of these searches have observed any discrepancy from Coulomb's law. Observing the charge of particle beams provides weaker bounds, but tests the velocity dependent corrections to Maxwell's equations for particles with higher velocities.[5][6]

### Acceleration Dependent Tests

Test charges inside a spherical conducting shell will experience different behaviors depending on the force law the test charge is subject to.[7] By measuring the oscillation frequency of a neon lamp inside a spherical conductor biased to a high voltage, this can be tested. Again, no significant deviations from the Maxwell theory have been observed.

## References

1. ^ a b Assis, AKT; HT Silva (September 2000). "Comparison between Weber’s electrodynamics and classical electrodynamics". Pramana - journal of physics 55 (3): 393–404. doi:10.1007/s12043-000-0069-2.
2. ^ Assis, AKT; JJ Caluzi (1991). "A limitation of Weber's law". Physics Letters A 160 (1): 25–30. Bibcode:1991PhLA..160...25A. doi:10.1016/0375-9601(91)90200-R.
3. ^ Wesley, JP (1990). "Weber electrodynamics, part I. general theory, steady current effects". Foundations of Physics Letters 3 (5): 443–469. Bibcode:1990FoPhL...3..443W. doi:10.1007/BF00665929.
4. ^ Lemon, DK; WF Edwards; CS Kenyon (1992). "Electric potentials associated with steady currents in superconducting coils". Physics Letters A 162 (2): 105–114. Bibcode:1992PhLA..162..105L. doi:10.1016/0375-9601(92)90985-U.
5. ^ Walz, DR; HR Noyes (April 1984). "Calorimetric test of special relativity". Physical Review A 29 (1): 2110–2114. Bibcode:1984PhRvA..29.2110W. doi:10.1103/PhysRevA.29.2110.
6. ^ Bartlett, DF; BFL Ward (15 December 1997). "Is an electron's charge independent of its velocity?". Physical Review D 16 (12): 3453–3458. doi:10.1103/physrevd.16.3453.
7. ^ Junginger, JE; ZD Popovic (2004). "An experimental investigation of the influence of an electrostatic potential on electron mass as predicted by Weber’s force law". Can. J. Phys. 82: 731–735. Bibcode:2004CaJPh..82..731J. doi:10.1139/p04-046.