Comprehensive Analysis of Weber Electrodynamics

Abstract

Weber electrodynamics is a 19th-century theory of electromagnetism that preceded Maxwell's field theory. Unlike Coulomb's law, Weber's force includes velocity and acceleration dependence, making it compatible with Newton's third law and conservation of angular momentum. This document presents a systematic analysis of Weber electrodynamics, deriving closed-form solutions where possible and comparing dynamics to the Coulomb case.

Part I: Foundations

1.1 Historical Context

Wilhelm Eduard Weber developed his force law in 1846-1848, extending Coulomb's law to include velocity-dependent terms. Weber's theory successfully explained:

• Ampère's force law between current elements
• Faraday's law of electromagnetic induction
• The propagation of electrical signals in wires at speed $c$

The theory was eventually superseded by Maxwell's field equations, primarily because Weber electrodynamics cannot describe electromagnetic wave propagation in vacuum. However, for quasistatic phenomena and low-velocity particles, Weber's theory remains mathematically equivalent to Maxwell's equations.

1.2 Weber's Force Law

For two charged particles with charges $q_1$ and $q_2$ separated by distance $r$, Weber's force on particle 1 due to particle 2 is:

$\vec{F}_{12} = \frac{q_1 q_2}{4\pi\epsilon_0 r^2}\hat{r}\left(1 - \frac{\dot{r}^2}{2c^2} + \frac{r\ddot{r}}{c^2}\right)$

In absolute (Gaussian) units:

$\vec{F}_{12} = \frac{q_1 q_2}{r^2}\hat{r}\left(1 - \frac{\dot{r}^2}{2c^2} + \frac{r\ddot{r}}{c^2}\right)$

1.3 Weber's Potential Energy

The velocity-dependent potential energy is:

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

Critical observation: At any instant when $\dot{r} = 0$ (turning points, circular orbits), the Weber potential reduces exactly to the Coulomb potential:

$U_{\text{Web}}\big|_{\dot{r}=0} = U_{\text{Coul}} = \frac{q_1 q_2}{4\pi\epsilon_0 r}$

1.4 Comparison: Weber vs. Coulomb vs. Maxwell

| Property | Coulomb | Weber | Maxwell | |----------|---------|-------|---------| | Force depends on velocity | No | Yes | Yes (via Lorentz force) | | Force depends on acceleration | No | Yes | Yes (via retardation) | | Newton's 3rd law | Yes | Yes | No (for particles) | | Momentum conservation | Particles only | Particles only | Particles + fields | | Angular momentum conservation | Yes | Yes | Particles + fields | | Energy conservation | Yes | Yes | Particles + fields | | EM waves in vacuum | No | No | Yes | | Magnetic field concept | N/A | Not needed | Essential |

Part II: Conservation Laws

2.1 Conservation of Angular Momentum

Weber's force is a central force — it acts along the line connecting the two charges. This guarantees conservation of angular momentum:

$\vec{L} = \mu \vec{r} \times \dot{\vec{r}} = \text{constant}$

where $\mu = m_1 m_2/(m_1 + m_2)$ is the reduced mass.

Consequence: Motion is confined to a plane, allowing polar coordinates $(r, \theta)$.

2.2 Conservation of Energy

The total energy in Weber electrodynamics is:

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

This can be rewritten as:

$E = \frac{1}{2}\mu_{\text{eff}}(r)\dot{r}^2 + U_{\text{eff}}(r)$

where the effective mass is:

$\mu_{\text{eff}}(r) = \mu + \frac{|q_1 q_2|}{4\pi\epsilon_0 c^2 r} = \mu + \frac{a}{r}$

with the Weber length scale:

$a = \frac{|q_1 q_2|}{4\pi\epsilon_0 c^2}$

2.3 The Effective Mass Interpretation

The position-dependent effective mass $\mu_{\text{eff}}(r)$ is a unique feature of Weber electrodynamics. It arises because the kinetic energy acquires a correction from the velocity-dependent potential:

$T_{\text{eff}} = \frac{1}{2}\mu\dot{r}^2 + \frac{\kappa\dot{r}^2}{2c^2 r} = \frac{1}{2}\left(\mu + \frac{\kappa}{c^2 r}\right)\dot{r}^2$

Physical interpretation: The effective inertia of a charged particle depends on its distance from other charges. This is reminiscent of electromagnetic mass contributions in relativistic electrodynamics.

2.4 Helmholtz's Criticism: Negative Mass

Helmholtz criticized Weber's theory by noting that for a charge inside a uniformly charged spherical shell at potential $\Phi$, the effective mass becomes:

$m_{\text{eff}} = m\left(1 - \frac{q\Phi}{mc^2}\right)$

For sufficiently large $\Phi$, this could become negative or zero, leading to unphysical behavior. Modern analyses (Assis & Caluzi, 1997) have shown this criticism can be addressed by incorporating gravitational interactions or using modified kinetic energy expressions.

Part III: Closed-Form Solutions

3.1 Solution 1: Zero-Energy Radial Infall (Attractive)

Setup:

• Attractive charges: $q_1 q_2 = -\kappa < 0$
• Zero angular momentum: $L = 0$
• Zero total energy: $E = 0$
• Initial separation: $r_0$, infalling

Energy equation:

$0 = \frac{1}{2}\mu\dot{r}^2 - \frac{\kappa}{r} + \frac{\kappa\dot{r}^2}{2c^2 r}$

Velocity:

$\dot{r}^2 = \frac{2\kappa}{\mu r + a}$

where $a = \kappa/c^2$.

Closed-form solution:

$\boxed{r(t) = \frac{1}{\mu}\left[\left((\mu r_0 + a)^{3/2} - \frac{3\mu\sqrt{2\kappa}}{2}t\right)^{2/3} - a\right]}$

Collision time:

$t_{\text{coll}} = \frac{(\mu r_0 + a)^{3/2} - a^{3/2}}{3\mu\sqrt{\kappa/2}}$

Coulomb limit ($c \to \infty$, $a \to 0$):

$r(t) \to \left(r_0^{3/2} - \frac{3}{2}\sqrt{\frac{2\kappa}{\mu}}t\right)^{2/3}$

3.2 Solution 2: Radial Infall from Rest (Attractive)

Setup:

• Released from rest at $r_0$
• Energy: $E = -\kappa/r_0$

Velocity equation:

$\dot{r}^2 = \frac{2\kappa(1 - r/r_0)}{\mu r + a}$

Time as function of position:

Define:

• $v = 1 - r/r_0$ (progress variable)
• $\beta = \frac{\mu r_0}{\mu r_0 + a}$ (dimensionless, $0 < \beta < 1$)

$\boxed{t(r) = \frac{\mu r_0 + a}{\sqrt{2\kappa\mu r_0}}\left[\arcsin(\sqrt{\beta v}) + \sqrt{\beta v(1-\beta v)}\right]}$

Collision time:

$t_{\text{coll}} = \frac{\mu r_0 + a}{\sqrt{2\kappa\mu r_0}}\left[\arcsin(\sqrt{\beta}) + \sqrt{\beta(1-\beta)}\right]$

Coulomb limit ($\beta \to 1$):

$t_{\text{coll}} \to \frac{\pi}{2}\sqrt{\frac{\mu r_0^3}{2\kappa}}$

This matches the classical result (quarter period of a degenerate ellipse).

3.3 Solution 3: Circular Orbits

Key insight: For circular orbits, $\dot{r} = 0$ and $\ddot{r} = 0$, so Weber's force reduces exactly to Coulomb's force.

Force balance (attractive case):

$\mu r\omega^2 = \frac{\kappa}{r^2}$

Kepler's third law (unchanged):

$\boxed{\omega^2 = \frac{\kappa}{\mu r^3}}$

Orbital period:

$T = 2\pi\sqrt{\frac{\mu r^3}{\kappa}}$

Circular orbit energy:

$E_{\text{circ}} = -\frac{\mu\kappa^2}{2L^2} = -\frac{\kappa}{2r}$

Physical interpretation: Circular orbits in Weber electrodynamics are identical to Keplerian circular orbits because the velocity-dependent correction vanishes when $\dot{r} = 0$.

3.4 Solution 4: Turning Points

At any turning point where $\dot{r} = 0$:

$E = \frac{L^2}{2\mu r_{\text{turn}}^2} + \frac{q_1 q_2}{4\pi\epsilon_0 r_{\text{turn}}}$

This is identical to the Coulomb turning point equation. Therefore:

Repulsive turning point (closest approach):

$\boxed{r_{\min} = \frac{k + \sqrt{k^2 + 2EL^2/\mu}}{2E}}$

for $q_1 q_2 = k > 0$.

Key result: Turning point radii are the same in Weber and Coulomb dynamics.

3.5 Solution 5: Asymptotic Scattering Velocity

For repulsive charges released from rest at $r_0$:

$E = \frac{k}{r_0}$

As $r \to \infty$, $\dot{r}^2 \to 2E/\mu$:

$\boxed{\dot{r}_\infty = \sqrt{\frac{2k}{\mu r_0}}}$

This is identical to the Coulomb result — energy conservation guarantees the same asymptotic velocity.

3.6 Solution 6: Small Oscillations About Circular Orbit

For small radial perturbations $\xi = r - r_{\text{circ}}$:

$\ddot{\xi} + \omega_r^2\xi = 0$

The radial frequency equals the orbital frequency:

$\omega_r = \omega_\phi = \sqrt{\frac{\kappa}{\mu r^3}}$

Consequence: Near-circular orbits close after one revolution (no first-order precession), just as in Coulomb/Kepler dynamics.

Part IV: Where Weber Differs from Coulomb

4.1 Dynamics Between Turning Points

While turning points coincide, the trajectory between turning points differs:

• Coulomb: $\dot{r}^2 = \frac{2}{\mu}\left(E - U_{\text{eff}}(r)\right)$
• Weber: $\dot{r}^2 = \frac{2(E - U_{\text{eff,static}}(r))}{\mu + a/r}$

The position-dependent effective mass $\mu + a/r$ means particles spend different amounts of time at different radii compared to Coulomb dynamics.

4.2 Orbital Precession (Non-Circular Orbits)

For eccentric orbits, the Weber correction causes orbital precession. Using perturbation theory, the precession angle per orbit is approximately:

$\Delta\phi \approx \frac{\pi\kappa}{\mu c^2 a(1-e^2)}$

where $a$ is the semi-major axis and $e$ is the eccentricity.

Historical note: Gerber (1898) used a similar velocity-dependent gravitational potential to predict Mercury's perihelion advance, obtaining the same formula as Einstein's general relativity:

$\Delta\phi = \frac{6\pi GM}{c^2 a(1-e^2)}$

However, Gerber's derivation was criticized as physically unmotivated, and his theory gives incorrect predictions for light deflection.

4.3 Maximum Velocity and Theory Breakdown

For zero-energy radial infall, as $r \to 0$:

$\dot{r}^2 \to \frac{2\kappa}{a} = 2c^2$

This implies $|\dot{r}| \to c\sqrt{2}$, exceeding the speed of light!

Critical radius where $v = c$:

$r_c = \frac{a}{\mu} = \frac{\kappa}{\mu c^2}$

For $r < r_c$, Weber electrodynamics predicts superluminal velocities, signaling the breakdown of the non-relativistic theory.

For electrons: $r_c \approx 2.8 \times 10^{-15}$ m (classical electron radius)

4.4 Time Dilation and Relativistic Corrections

Weber's velocity-dependent potential can be interpreted as a first-order relativistic correction. The Lagrangian comparison:

Coulomb Lagrangian: $L_C = \frac{1}{2}\mu\dot{r}^2 - U_C(r)$

Weber Lagrangian: (using interaction term $S$) $L_W = \frac{1}{2}\mu\dot{r}^2 - \frac{\kappa}{r}\left(1 + \frac{\dot{r}^2}{2c^2}\right)$

The $\dot{r}^2/c^2$ term mimics relativistic mass increase at $O(v^2/c^2)$.

Part V: Key Physical Insights

5.1 The Turning Point Theorem

Theorem: At any instant when $\dot{r} = 0$, Weber electrodynamics is indistinguishable from Coulomb electrodynamics.

Proof: The Weber potential $U = \frac{q_1q_2}{r}(1 - \dot{r}^2/2c^2)$ equals the Coulomb potential when $\dot{r} = 0$. ∎

Corollaries:

1. Circular orbit radii are identical
2. Turning point radii are identical
3. The Virial theorem applies at turning points

5.2 Energy Partition

In Coulomb dynamics, energy partitions as: $E = T_{\text{radial}} + T_{\text{angular}} + U$

In Weber dynamics, the partition is modified: $E = T_{\text{radial}}\left(1 + \frac{a}{\mu r}\right) + T_{\text{angular}} + U_{\text{static}}$

The radial kinetic energy carries an "enhancement factor" $(1 + a/\mu r)$ that grows at small separations.

5.3 The Weber Length Scale

The characteristic length: $a = \frac{|q_1 q_2|}{c^2} \quad \text{(absolute units)}$

Physical meaning: This is the distance at which the Weber correction becomes comparable to the Coulomb term. For typical atomic/molecular systems, $a \ll$ atomic dimensions, so Weber corrections are small.

Dimensionless parameter: $\epsilon = \frac{a}{\mu r_0} = \frac{\kappa}{\mu c^2 r_0} = \frac{v_0^2}{c^2}$

where $v_0 = \sqrt{\kappa/\mu r_0}$ is the characteristic orbital velocity. Weber corrections are $O(\epsilon)$.

Part VI: Summary of Closed-Form Results

| Case | Conditions | Weber Solution | Coulomb Equivalent | |------|------------|----------------|-------------------| | Zero-energy infall | $E=0$, $L=0$, attractive | $r(t) = \frac{1}{\mu}[(A-Bt)^{2/3} - a]$ | $r(t) = (A'-B't)^{2/3}$ | | Infall from rest | $\dot{r}_0=0$, $L=0$, attractive | $t(r)$ via arcsin | $t(r)$ via arcsin | | Circular orbits | $\dot{r}=0$ | $\omega^2 = \kappa/\mu r^3$ | Identical | | Turning points | $\dot{r}=0$ | Same as Coulomb | — | | Asymptotic velocity | $r\to\infty$, repulsive | $v_\infty = \sqrt{2k/\mu r_0}$ | Identical | | Precession per orbit | Eccentric orbit | $\Delta\phi \sim \kappa/\mu c^2 a(1-e^2)$ | Zero |

Part VII: Validity and Limitations

7.1 When Weber Electrodynamics Applies

1. Low velocities: $v \ll c$
2. Large separations: $r \gg a/\mu$ (above critical radius)
3. Quasistatic regime: Retardation effects negligible
4. No radiation: System is not accelerating rapidly

7.2 When Weber Electrodynamics Fails

1. High velocities: Relativistic effects dominate
2. Small separations: Approaches critical radius $r_c = a/\mu$
3. EM wave phenomena: No field propagation in vacuum
4. Rapidly varying fields: Retardation becomes important

7.3 Modern Extensions

Phipps' potential (1990): Modified Weber to avoid negative mass: $U_{\text{Phipps}} = \frac{q_1 q_2}{r}\sqrt{1 - \frac{\dot{r}^2}{c^2}}$

This ensures the potential is real only for $|\dot{r}| < c$, making $c$ a true limiting velocity.

Weber-Maxwell electrodynamics: Modern synthesis combining Weber's particle-based approach with Maxwell's wave equation, satisfying all conservation laws.

Appendix A: Derivation of Arcsin Solution

Starting from: $\dot{r}^2 = \frac{2\kappa(1 - r/r_0)}{\mu r + a}$

Substitute $v = 1 - r/r_0$, $r = r_0(1-v)$, $dr = -r_0 dv$:

$dt = r_0\sqrt{\frac{\mu r_0(1-v) + a}{2\kappa v}}dv = r_0\sqrt{\frac{\mu r_0 + a}{2\kappa}}\sqrt{\frac{1 - \beta v}{v}}dv$

where $\beta = \mu r_0/(\mu r_0 + a)$.

The integral: $\int_0^v \sqrt{\frac{1-\beta u}{u}}du = \frac{1}{\sqrt{\beta}}\left[\arcsin(\sqrt{\beta v}) + \sqrt{\beta v(1-\beta v)}\right]$

can be verified by differentiation or by the substitution $u = \sin^2\theta/\beta$.

Appendix B: Weber Correction in Polar Coordinates

In polar coordinates $(r, \theta)$, the equations of motion are:

Angular momentum: $L = \mu r^2\dot{\theta} = \text{const}$

Radial equation: $\mu\ddot{r} - \frac{L^2}{\mu r^3} = F_{\text{Weber}}(r, \dot{r}, \ddot{r})$

The Weber force introduces coupling between $\ddot{r}$ and the radial coordinate, making the equation implicit. The effective equation becomes:

$\left(\mu + \frac{a}{r}\right)\ddot{r} = \frac{L^2}{\mu r^3} - \frac{\kappa}{r^2}\left(1 - \frac{\dot{r}^2}{2c^2}\right) + \frac{\kappa\dot{r}^2}{c^2 r^2}$

Appendix C: Numerical Parameters

For electron-proton system:

• $\kappa = e^2/4\pi\epsilon_0 \approx 2.31 \times 10^{-28}$ J·m
• $\mu \approx 9.1 \times 10^{-31}$ kg
• $a = \kappa/c^2 \approx 2.57 \times 10^{-45}$ kg·m
• $r_c = a/\mu \approx 2.82 \times 10^{-15}$ m (classical electron radius)

The Weber correction at the Bohr radius ($r_B \approx 5.3 \times 10^{-11}$ m): $\frac{a}{\mu r_B} \approx 5 \times 10^{-5}$

This confirms Weber corrections are tiny at atomic scales.

References

1. Weber, W. (1846). "Elektrodynamische Maassbestimmungen"
2. Assis, A.K.T. (1994). Weber's Electrodynamics. Kluwer Academic Publishers
3. Assis, A.K.T. & Caluzi, J.J. (1997). "A critical analysis of Helmholtz's argument against Weber's electrodynamics". Foundations of Physics 27: 1445-1452
4. Phipps, T.E. (1990). "Toward modernization of Weber's force law". Physics Essays 3: 414-420
5. Wesley, J.P. (1990). "Weber electrodynamics". Foundations of Physics Letters 3: 443-469
6. Clemente, R.A. & Assis, A.K.T. (1991). "Two-body problem for Weber-like interactions". Int. J. Theor. Phys. 30: 537-545
7. Bunchaft, F. & Carneiro, S. (1997). "Weber-like interactions and energy conservation". arXiv:gr-qc/9708047
8. Gerber, P. (1898). "Die räumliche und zeitliche Ausbreitung der Gravitation"