Weber electrodynamics

According to Weber electrodynamics, the force (F) acting simultaneously on point charges q₁ and q₂, is given by

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

where r is the vector connecting q₁ and q₂, 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. ${\displaystyle {\dot {r}}\ll c}$), this reduces to the usual Coulomb's law.[1]

This can be derived from the potential energy:

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

To derive Weber's force from the potential energy, we first express the force as ${\displaystyle \mathbf {F} =-\mathbf {\hat {r}} {\frac {dU}{dr}}}$.

Taking the derivative of the potential, we note that ${\displaystyle {\frac {d{\dot {r}}^{2}}{dr}}=2{\dot {r}}{\frac {d{\dot {r}}}{dr}}=2{\dot {r}}{\frac {d{\dot {r}}}{dt}}{\frac {dt}{dr}}=2{\ddot {r}}}$.

In Maxwell's equations, by contrast, the force 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:[1]

${\displaystyle U_{\rm {Max}}\approx {\frac {q_{1}q_{2}}{4\pi \epsilon _{0}r}}\left(1-{\frac {\mathbf {v_{1}} \cdot \mathbf {v_{2}} +(\mathbf {v_{1}} \cdot \mathbf {\hat {r}} )(\mathbf {v_{2}} \cdot \mathbf {\hat {r}} )}{2c^{2}}}\right).}$

where v₁ and v₂ are the velocities of q₁ and q₂, respectively, and where relativistic and retardation effects are omitted for simplicity; see Darwin Lagrangian.

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.[2]

In 1848, only two years after the development of his electrodynamics force (F), Weber presented a velocity-dependent potential energy from which this force might be derived, namely:[1]

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

This result can be achieved using force (F) because the force can be defined as the negative of the vector gradient of the potential field, that is,

${\displaystyle F=-\nabla U_{\rm {Web}}.}$

That considered, the potential energy can be obtained by integrating (F) with respect to ${\displaystyle r}$ and changing the sign:

${\displaystyle U_{\rm {Web}}=-\int F\operatorname {d} \!r}$

where the constant of integration is neglected because the point where the potential energy is zero is arbitrarily chosen.

The last two terms of the force (F) could be united and written as a derivative with respect to ${\displaystyle r}$. By the chain rule, we have that ${\displaystyle {\operatorname {d} \!\left({\operatorname {d} \!r \over \operatorname {d} \!t}\right)^{2} \over \operatorname {d} \!r}=2{\operatorname {d} ^{2}\!r \over \operatorname {d} \!t^{2}}}$, and because of this, we notice that the whole force can be rewritten as

${\displaystyle \mathbf {F} ={\frac {q_{1}q_{2}\mathbf {\hat {r}} }{4\pi \epsilon _{0}r^{2}}}\left(1-{\frac {{\dot {r}}^{2}}{2c^{2}}}+{\frac {r{\ddot {r}}}{c^{2}}}\right)={\frac {q_{1}q_{2}\mathbf {\hat {r}} }{4\pi \epsilon _{0}}}\left({\frac {1}{r^{2}}}-{\frac {1}{2r^{2}c^{2}}}\left({\operatorname {d} \!r \over \operatorname {d} \!t}\right)^{2}+{\frac {1}{c^{2}r}}{\operatorname {d} ^{2}\!r \over \operatorname {d} \!t^{2}}\right)={\frac {q_{1}q_{2}\mathbf {\hat {r}} }{4\pi \epsilon _{0}}}\left({\frac {1}{r^{2}}}+{\frac {1}{2c^{2}}}{\operatorname {d} \!\left({\frac {1}{r}}\left({\operatorname {d} \!r \over \operatorname {d} \!t}\right)^{2}\right) \over \operatorname {d} \!r}\right)}$

where the product rule was used. Therefore, the force (F) can be written as

${\displaystyle \mathbf {F} ={\frac {q_{1}q_{2}\mathbf {\hat {r}} }{4\pi \epsilon _{0}}}\left({\frac {1}{r^{2}}}+{\frac {1}{2c^{2}}}{\operatorname {d} \!\left({\frac {1}{r}}{\dot {r}}^{2}\right) \over \operatorname {d} \!r}\right).}$

This expression can now be easily integrated with respect to ${\displaystyle r}$, and changing the signal we obtain a general velocity-dependent potential energy expression for this force in Weber electrodynamics:

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

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. 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.

Weber dynamics has been used to explain various phenomena such as wires exploding when exposed to high currents.[3]

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.[4])

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