Around 1820, André-Marie Ampère carried out numerous systematic experiments with direct currents. Eventually in 1823 he developed the force law

$${\displaystyle d^{2}\mathbf {F} =-{\frac {\mu _{0}}{4\,\pi }}\,I_{1}\,I_{2}\,{\frac {\mathbf {r} }{r^{3}}}\left(2\left(d\mathbf {s} _{1}\cdot d\mathbf {s} _{2}\right)-{\frac {3}{r^{2}}}\left(\mathbf {r} \cdot d\mathbf {s} _{1}\right)\left(\mathbf {r} \cdot d\mathbf {s} _{2}\right)\right),}$$

(1)

which can be used to calculate the force ${\displaystyle d^{2}\mathbf {F} }$  that a current element ${\displaystyle I_{2}\,d\mathbf {s} _{2}}$  exerts on another current element ${\displaystyle I_{1}\,d\mathbf {s} _{1}}$ . Here, ${\displaystyle \mathbf {r} }$  is the vector that points from the current element ${\displaystyle I_{2}\,d\mathbf {s} _{2}}$  to the current element ${\displaystyle I_{1}\,d\mathbf {s} _{1}}$ . A current element ${\displaystyle I\,d\mathbf {s} }$  should be interpreted as a very short segment of the length ${\displaystyle s}$  of a conductor with a direct current ${\displaystyle I}$  flowing in the direction of ${\displaystyle \mathbf {s} }$ .^[1]

In 1835, Carl Friedrich Gauss realized that Ampère's force law can be interpreted by a minor generalization of Coulomb's law.^[2] He postulated that the electric force exerted by a point charge ${\displaystyle q_{2}}$  on another point charge ${\displaystyle q_{1}}$  depends not only on the distance ${\displaystyle \mathbf {r} =\mathbf {r} _{1}-\mathbf {r} _{2}}$ , but also on the relative velocity ${\displaystyle \mathbf {v} ={\dot {\mathbf {r} }}_{1}-{\dot {\mathbf {r} }}_{2}}$ :

$${\displaystyle \mathbf {F} ={\frac {q_{1}\,q_{2}}{4\,\pi \,\epsilon _{0}}}\,\left(1+{\frac {v^{2}}{c^{2}}}-{\frac {3}{2}}\left({\frac {\mathbf {r} }{r}}\cdot {\frac {\mathbf {v} }{c}}\right)^{2}\right)\,{\frac {\mathbf {r} }{r^{3}}}.}$$

(2)

Importantly, Gauss's force law is a significant generalization of Ampere's force law, since moving point charges do not represent direct currents. In fact, today Ampere's force law is no longer presented in its original form, as there are equivalent representations for direct currents such as the Biot-Savart law in combination with the Lorentz force. It is important to emphasize that this is the point at which Weber electrodynamics and Maxwell electrodynamics take different paths, because James Clerk Maxwell decided to base his theory on the Biot-Savart law, which was originally also only valid for closed conductor loops.^[3]

Wilhelm Eduard Weber's contribution to Weber electrodynamics was that he extended Gauss's force formula in such a way that it was possible to provide a formula for the potential energy.^[4] He presented his formula in 1848 which reads

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

(3)

with ${\displaystyle {\dot {r}}}$  beeing the radial velocity. Weber also carried out numerous experiments and documented the state of knowledge at this time in his substantial work.^[5][6][7]

Weber electrodynamics and Gauss's hypothesis fell gradually into oblivion after the introduction of the displacement current around 1870, since the full set of Maxwell equations made it possible to describe electromagnetic waves for the first time.

From around 1880, experiments such as the Michelson-Morley experiment showed that electromagnetic waves propagate at the speed of light regardless of the state of motion of the transmitter or receiver in a vacuum, which is not consistent with the predictions of Maxwell's equations, since these describe wave propagation in a medium. To overcome this problem, the Lorentz transformation was developed. As a result, Gauss's hypothesis that the electric force depends on the relative velocity was added back in a modified form.

Weber force edit

In Weber electrodynamics, the electromagnetic force ${\displaystyle \mathbf {F} }$  that a point charge ${\displaystyle q_{2}}$  with trajectory ${\displaystyle \mathbf {r} _{2}(t)}$  exerts on another point charge ${\displaystyle q_{1}}$  with trajectory ${\displaystyle \mathbf {r} _{1}(t)}$  at time ${\displaystyle t}$  is given by equation ^[8]

$${\displaystyle \mathbf {F} ={\frac {q_{1}q_{2}}{4\pi \epsilon _{0}}}{\frac {\mathbf {\mathbf {r} } }{r^{3}}}\left(1-{\frac {{\dot {r}}^{2}}{2c^{2}}}+{\frac {r{\ddot {r}}}{c^{2}}}\right).}$$

(4)

Here, ${\displaystyle \mathbf {r} =\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t)}$  is the displacement of ${\displaystyle q_{1}}$  relative to ${\displaystyle q_{2}}$  and ${\displaystyle r=\Vert \mathbf {r} \Vert }$  is the distance. Note that

$${\displaystyle {\dot {r}}={\frac {\mathrm {d} }{\mathrm {d} t}}\Vert \mathbf {r} \Vert ={\frac {\mathrm {d} }{\mathrm {d} t}}{\sqrt {\mathbf {r} \cdot \mathbf {r} }}={\frac {\mathbf {r} \cdot {\dot {\mathbf {r} }}}{r}}}$$

(5)

is the radial velocity and

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

(6)

is the radial acceleration. If one substitutes this into the Weber force (4), one obtains with ${\displaystyle {\dot {\mathbf {r} }}=\mathbf {v} }$  and ${\displaystyle {\ddot {\mathbf {r} }}=\mathbf {a} }$  the alternative representation

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

(7)

For ${\displaystyle \mathbf {a} \approx \mathbf {0} }$  one obtains equation (2) as postulated by Gauss in 1835.

Link between potential energy and force edit

That Weber's potential energy (3) is compatible with force formula (4) can be shown by means of equation (5) and equation

Conservation of energy, momentum and angular momentum edit

In Weber electrodynamics, energy, momentum and angular momentum are conserved quantities. The conservation of momentum results from the property of the Weber force to comply with Newton's third law: If one exchanges source and receiver of the force, only the sign of the force is altered. The conservation of angular momentum is a consequence of the fact that the Weber force is a central force.

The conservation of energy in an isolated system consisting of only two particles is easy to demonstrate. Equation (5) gives ${\displaystyle \mathbf {r} \cdot {\dot {\mathbf {r} }}=r\,{\dot {r}}}$ . This leads to

Lorentz force edit

Maxwell electrodynamics and Weber electrodynamics are equivalent for direct currents and non-relativistic speeds, since direct currents can only flow in closed conductor loops. As Maxwell already demonstrated around 150 years ago, under these conditions the Ampere force law can be represented in several variations.^[3]

Maxwell's electrodynamics follows a two-stage approach, firstly by assigning a magnetic field ${\displaystyle \mathbf {B} }$  to each current element and secondly by defining that the force ${\displaystyle \mathbf {F} }$  on a test charge ${\displaystyle q}$  moving at the speed ${\displaystyle \mathbf {v} }$  can be calculated using the expression ${\displaystyle q\,\mathbf {v} \times \mathbf {B} }$ . In Maxwell's time, the velocity ${\displaystyle \mathbf {v} }$  was interpreted as the velocity of the test charge relative to the medium in which the magnetic field propagates. In Maxwell's electrodynamics, the Lorentz force is a physical law that cannot be traced back to a cause or mechanism.

Weber electrodynamics, on the other hand, does not define a magnetic field or a Lorentz force, but interprets the force of a current on a test charge ${\displaystyle q}$  by postulating that a current-carrying conductor contains negative and positive point charges that move at slightly different relative velocities with respect to the test charge. This in turn produces slight deformations of the force so that, depending on the speed of the test charge, residual forces remain. In sum, these correspond exactly to the Lorentz force.

This means that Weber electrodynamics explains the Lorentz force by means of the principle of relativity, albeit only for relative velocities that are much smaller than the speed of light. Gauss's hypothesis of 1835 therefore already represents an early interpretation of magnetism as a relativistic effect. This interpretation is not included in Maxwell's electrodynamics.

Electromagnetic waves edit

For alternating currents and point charges, the different representations of Ampere's force law are not equivalent. Maxwell was familiar with Weber's electrodynamics and mentioned it positively.^[10] Nevertheless, he decided to build his theory on the Biot-Savart law by generalizing it to cases where the conductor loops contain discontinuities. The significance of the displacement current becomes clear by studying the field of the electromagnetic force that an accelerated electron would generate on a resting test charge. The figures show the field of an electron that is accelerated to 75 percent of the speed of light within 3 nanoseconds.

In the case of the Weber force, it can be recognized that the initially radial field becomes flattened in the direction of motion. This represents an effect that is presently associated with the Lorentz contraction. Something similar can also be seen in the field calculated by means of Maxwell's equations. In addition, however, a wave front can be recognized here. It is also noticeable that in the region of the wave front the force is no longer a central force. This effect is known as bremsstrahlung.

Electromagnetic wave phenomena are therefore not included in Weber electrodynamics. For this reason, Weber's electrodynamics is only applicable in applications in which all involved charges move slowly and uniformly.

Newton's third law in Maxwell and Weber electrodynamics edit

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 do not always 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.^[11] 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.

Potential energy for point charges in Maxwell electrodynamics edit

In Maxwell's equations the force ${\displaystyle \mathbf {F} }$  on a charge from nearby charges can be calculated by combining Jefimenko's equations with the Lorentz force law. The corresponding potential energy is approximately:^[8]

Using these expressions, the regular form of Ampère's law and Faraday's law can be derived. Importantly, Weber electrodynamics 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.^[12]

Weber dynamics has been used to explain various phenomena such as wires exploding when exposed to high currents.^[13]

Despite various efforts, a velocity-dependent and/or acceleration-dependent correction to Coulomb's law has never been observed, as described in the next section. Moreover, Hermann von 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. (Some scientists have, however, disputed Helmholtz's argument.^[14])

Velocity-dependent tests edit

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.^[15] 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.^[16][17]

Acceleration-dependent tests edit

Test charges inside a spherical conducting shell will experience different behaviors depending on the force law the test charge is subject to.^[18] 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.

Relation to quantum electrodynamics edit

Quantum electrodynamics (QED) is perhaps the most stringently tested theory in physics, with highly nontrivial predictions verified to an accuracy better than 10 parts per billion: See precision tests of QED. Since Maxwell's equations can be derived as the classical limit of the equations of QED,^[19] it follows that if QED is correct (as is widely believed by mainstream physicists), then Maxwell's equations and the Lorentz force law are correct too.

Although it has been demonstrated that, in certain aspects, the Weber force formula is consistent with Maxwell's equations and the Lorentz force,^[20] they are not exactly equivalent—and more specifically, they make various contradictory predictions^{[8][12][13][18]} as described above. Therefore, they cannot both be correct.

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