Inductance and Force Calculations in Electrical Circuits by Bueno and Assis

Overview

Based on the book "Inductance and Force Calculations in Electrical Circuits" by Marcelo de Almeida Bueno and Andre Koch Torres Assis (Nova Science Publishers, 2001).

The book's central question is whether the two historically competing force laws between current elements — Ampère's (1826) and Grassmann's (1845) — are physically distinguishable. The answer, proved rigorously here, is no for closed circuits. Along the way it calculates both self-inductances and inter-element forces for a wide range of geometries, demonstrating that four different definitions of the inductance coefficient are also completely equivalent for closed circuits.

Introduction: The Central Equations

The book opens by placing two force laws and four inductance coefficients side by side, motivating the rest of the work. In SI units, with $\mu_0 = 4\pi \times 10^{-7}$ H/m:

Ampère's force (1826) exerted by the current element $I_j d\vec{r}_j$ at $\vec{r}_j$ on $I_i d\vec{r}_i$ at $\vec{r}_i$:

d^2\vec{F}^A_{ji} = \frac{\mu_0 I_i I_j}{4\pi} \frac{\hat{r}_{ij}}{r^2_{ij}} \left[ 3(d\vec{r}_i \cdot \hat{r}_{ij})(d\vec{r}_j \cdot \hat{r}_{ij}) - 2(d\vec{r}_i \cdot d\vec{r}_j) \right] \tag{a}

Grassmann's force (1845), equivalent to $I d\vec{r} \times \vec{B}$ with $\vec{B}$ from Biot-Savart:

d^2\vec{F}^G_{ji} = \frac{\mu_0 I_i I_j}{4\pi} \frac{1}{r^2_{ij}} \left[ (d\vec{r}_i \cdot \hat{r}_{ij}) d\vec{r}_j - (d\vec{r}_i \cdot d\vec{r}_j) \hat{r}_{ij} \right] \tag{b}

Ampère's force satisfies Newton's third law element-by-element ($d^2\vec{F}^A_{ji} = -d^2\vec{F}^A_{ij}$) and points along $\hat{r}_{ij}$. Grassmann's does not satisfy action-reaction in general between individual elements.

The four mutual inductance coefficients — all equal for closed circuits — are:

• Neumann ($k=1$): $M^N_{12} = \frac{\mu_0}{4\pi}\oint_{\Gamma_1}\oint_{\Gamma_2}\frac{d\vec{r}_i \cdot d\vec{r}_j}{r_{ij}}$
• Weber ($k=-1$): $M^W_{12} = \frac{\mu_0}{4\pi}\oint_{\Gamma_1}\oint_{\Gamma_2}\frac{(\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j)}{r_{ij}}$
• Maxwell ($k=0$): arithmetic mean of Neumann and Weber
• Graneau ($k=-5$): related to Ampère's force directly

The general one-parameter family (Helmholtz, 1870):

d^2M_{ij} = \frac{\mu_0}{4\pi} \left[ \frac{1+k}{2}\frac{d\vec{r}_i \cdot d\vec{r}_j}{r_{ij}} + \frac{1-k}{2}\frac{(\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j)}{r_{ij}} \right] \tag{h}

Part I: Inductance

Part I proves that Neumann, Weber, Maxwell, and Graneau inductance coefficients are all equal for any closed circuit of arbitrary shape.

1. Chapter 1: Inductance Coefficients

Four Expressions from Four Force Laws

1.1 The Current Element Model

A current element $Id\vec{r}$ is modeled as two overlapping charge distributions drifting past each other:

Id\vec{r} = dq_+\vec{v}_+ + dq_-\vec{v}_- = dq_+(\vec{v}_+ - \vec{v}_-) \tag{1.1}

where electrical neutrality requires $dq_+ = -dq_-$.

1.2 Neumann's Coefficient (1845)

From the potential energy of two current circuits, Neumann's element-element energy is:

d^2V^N_{ij} = \frac{\mu_0}{4\pi} I_1 I_2 \frac{d\vec{r}_i \cdot d\vec{r}_j}{r_{ij}} \tag{1.4}

Integrating over two closed loops gives Neumann's mutual inductance:

M^N_{12} = \frac{\mu_0}{4\pi}\oint_{\Gamma_1}\oint_{\Gamma_2}\frac{d\vec{r}_i \cdot d\vec{r}_j}{r_{ij}} \tag{1.6}

This is the most widely used form, related to the magnetic flux: $M^N_{12} = \Phi_{12}/I_2$.

1.3 Weber's Coefficient (1846)

Weber's potential energy between two point charges (the unification of electrostatics and electromagnetism):

V^W_{12} = \frac{q_1 q_2}{4\pi\epsilon_0 r_{12}}\left(1 - \frac{\dot{r}^2_{12}}{2c^2}\right) \tag{1.7}

where $\dot{r}_{12} \equiv dr_{12}/dt$ is the radial velocity. Applied to current elements this gives:

d^2V^W_{ij} = \frac{\mu_0}{4\pi} I_1 I_2 \frac{(\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j)}{r_{ij}} \tag{1.14}

Weber's mutual inductance:

M^W_{12} = \frac{\mu_0}{4\pi}\oint_{\Gamma_1}\oint_{\Gamma_2}\frac{(\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j)}{r_{ij}} \tag{1.16}

1.4 Darwin's Lagrangian and Maxwell's Coefficient

Darwin's Lagrangian energy (second-order relativistic expansion) for two charges:

U^D_{12} = \frac{q_1 q_2}{4\pi\epsilon_0 r_{12}}\left(1 - \frac{\dot{\vec{r}}_1 \cdot \dot{\vec{r}}_2 + (\dot{\vec{r}}_1 \cdot \hat{r}_{12})(\dot{\vec{r}}_2 \cdot \hat{r}_{12})}{2c^2}\right) \tag{1.18}

Maxwell's element-element energy is the arithmetic mean of Neumann's and Weber's:

d^2V^M_{ij} = \frac{\mu_0}{4\pi} I_1 I_2 \frac{(d\vec{r}_i \cdot d\vec{r}_j) + (\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j)}{2\,r_{ij}} \tag{1.26}

Maxwell's mutual inductance:

M^M_{12} = \frac{\mu_0}{4\pi}\oint_{\Gamma_1}\oint_{\Gamma_2}\frac{(d\vec{r}_i \cdot d\vec{r}_j) + (\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j)}{2\,r_{ij}} \tag{1.28}

1.5 Graneau's Coefficient

Derived from Ampère's force (the force is the gradient of the energy), Graneau's element energy is:

d^2V^G_{ij} = \frac{\mu_0}{4\pi} I_1 I_2 \frac{3(\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j) - 2(d\vec{r}_i \cdot d\vec{r}_j)}{r_{ij}} \tag{1.29}

Graneau's mutual inductance:

M^G_{12} = \frac{\mu_0}{4\pi}\oint_{\Gamma_1}\oint_{\Gamma_2}\frac{3(\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j) - 2(d\vec{r}_i \cdot d\vec{r}_j)}{r_{ij}} \tag{1.31}

1.6 The Unified $k$-Parameter Family

All four inductance definitions are special cases of the one-parameter Helmholtz family. The element mutual inductance is:

d^2M_{ij} = \frac{\mu_0}{4\pi}\left[\frac{1+k}{2}\frac{d\vec{r}_i \cdot d\vec{r}_j}{r_{ij}} + \frac{1-k}{2}\frac{(\hat{r}_{ij}\cdot d\vec{r}_i)(\hat{r}_{ij}\cdot d\vec{r}_j)}{r_{ij}}\right] \tag{1.32}

| $k$ | Expression | |-----|-----------| | $+1$ | Neumann | | $-1$ | Weber | | $0$ | Maxwell | | $-5$ | Graneau |

For open (filiform/linear) circuits, these give different numerical results. The book's main goal is to show they are identical for closed circuits.

2. Chapter 2: Self-Inductance

Numerical Method and Standard Comparisons

2.1 Linear, Surface, and Volumetric Elements

For surface current density $\vec{K}$ (A/m) or volumetric current density $\vec{J}$ (A/m²), the substitution is:

Id\vec{r} \;\leftrightarrow\; \vec{K}\,da \;\leftrightarrow\; \vec{J}\,dV \tag{2.1, 2.2}

This upgrade from linear to surface or volumetric currents is essential for treating wires in contact (where linear elements give divergences).

2.2 Solenoid Self-Inductance (Exact)

For a solenoid of length $\ell$, radius $a$, and $n = N/\ell$ turns per unit length, with $p = 2a/\ell$ and $q = p/\sqrt{1+p^2}$, the exact self-inductance is:

L^N_{solenoid} = \frac{2\mu_0 a}{3}\left[ p^2\left(\frac{E(q)}{q} - 1\right) - \frac{dE(q)}{dq} \right] \tag{2.6}

where $E(q)$ is the complete elliptic integral of the second kind. In the long solenoid limit ($a \ll \ell$):

L \approx \frac{\mu_0 \pi a^2 N^2}{\ell}\left(1 - \frac{8a}{3\pi\ell} + \frac{1}{2}\frac{a^2}{\ell^2} + \cdots\right) \tag{2.7}

In the thin-ring limit ($\ell \ll a$):

L \approx \mu_0 a\left(\ln\frac{8a}{\ell} - \frac{1}{2}\right) \tag{2.8}

2.3 Textbook Formula and Its Limitations

The standard textbook result for a long solenoid ($L = \mu_0 \pi N^2 a^2/\ell$) is the leading-order term of (2.7). The exact formula (2.6) shows this overestimates inductance for finite solenoids.

The textbook coaxial cable result:

L^{textbook}_{coaxial} = \frac{\mu_0 \ell}{2\pi}\ln\frac{b}{a} \tag{2.19}

is only valid when the internal inductance of the conductors is negligible.

3. Chapter 3: Inductance in Several Geometries

Where $k$ Matters — and Where It Doesn't

3.1 Mutual Inductance of Two Parallel Conductors (Filiform)

For two parallel straight wires of lengths $\ell_1$ and $\ell_2$ at perpendicular separation $b$, with offset $a$:

M^k_{12} = \frac{\mu_0}{4\pi}\left(\frac{1+k}{2}\right) \cdot \left[ \text{logarithmic and square-root terms in } a,\,\ell_1,\,\ell_2,\,b \right] \tag{3.2}

The Neumann ($k=1$) piece is the only one surviving the standard double line integral.

3.2 Orthogonal Conductors

For two orthogonal straight wires, the Neumann piece vanishes and only the Weber piece ($k=-1$ contribution) survives:

M^k_{orthogonal} \propto \frac{1-k}{2} \tag{3.4}

This means Neumann ($k=1$) gives $M=0$ for orthogonal wires, while Weber ($k=-1$) gives a non-zero result.

3.3 Surface Rectangular Circuit Self-Inductance

For a closed rectangular circuit of sides $\ell_1$ and $\ell_2$ with surface current of width $\omega$, the self-inductance is (to leading order in $\omega/\ell$):

L^N_{\Gamma_{ij}} = L^W_{\Gamma_{ij}} = L^M_{\Gamma_{ij}} = L^G_{\Gamma_{ij}} \approx \frac{\mu_0}{2\pi}\left[2\ell_2\ln\frac{2\ell_2}{\omega} + 2\ell_1\ln\frac{2\ell_1}{\omega} - 2\ell_2\sinh^{-1}\frac{\ell_2}{\ell_1} - 2\ell_1\sinh^{-1}\frac{\ell_1}{\ell_2} + 4(\ell_1^2+\ell_2^2)^{1/2} - \ell_1 - \ell_2\right] \tag{3.11}

This result is independent of $k$ — the self-inductance of any closed rectangular surface circuit is the same regardless of which of the four expressions is used.

3.4 Volumetric Rectangular Circuit Self-Inductance

For a volumetric rectangular circuit (square cross-section of side $\omega$, lengths $\ell_1$ and $\ell_2$):

L^N_{\Gamma_{ij}} = L^W_{\Gamma_{ij}} = L^M_{\Gamma_{ij}} = L^G_{\Gamma_{ij}} \approx \frac{\mu_0}{2\pi}\left[2\ell_2\ln\frac{2\ell_2}{\omega} + 2\ell_1\ln\frac{2\ell_1}{\omega} - 2\ell_2\sinh^{-1}\frac{\ell_2}{\ell_1} - 2\ell_1\sinh^{-1}\frac{\ell_1}{\ell_2} + 4(\ell_1^2+\ell_2^2)^{1/2} + (\ell_1+\ell_2)\left(\frac{1}{6} - \frac{2}{3}\ln 2 - \frac{2\pi}{3}\right)\right] \tag{3.19}

Again $k$-independent.

4. Chapter 4: Complete Equivalence of Inductance Formulas

The Proof

4.1 Decomposition of a Closed Circuit

Any closed circuit $\Gamma$ can be split into two sub-circuits $\Gamma_a$ and $\Gamma_b$ sharing a common section. The self-inductance satisfies:

L_\Gamma = L_{\Gamma_a} + L_{\Gamma_b} + 2M_{\Gamma_a\Gamma_b} \tag{4.5}

and for $N$ sub-circuits:

L_\Gamma = \sum_{i=1}^{N} L_{\Gamma_i} + \sum_{\substack{i,j=1\\i\neq j}}^{N} M_{\Gamma_i\Gamma_j} \tag{4.6}

4.2 Proof Strategy

Any closed circuit $\Gamma$ (with surface or volumetric current of thickness $\omega$) is approximated to arbitrary precision by $M$ small closed rectangular circuits $\Gamma_{ij}$ (Fig. 4.3). Each rectangular circuit has:

L^N_{\Gamma_{ij}} = L^W_{\Gamma_{ij}} = L^M_{\Gamma_{ij}} = L^G_{\Gamma_{ij}} \tag{4.9}

(from Eq. 3.11). The mutual inductance between any two separate closed rectangular circuits is also the same for all four formulas (as they are external to each other). Therefore the total self-inductance of $\Gamma$ satisfies:

\boxed{L^N_\Gamma = L^W_\Gamma = L^M_\Gamma = L^G_\Gamma} \tag{4.11}

For any closed circuit of arbitrary shape, with surface or volumetric current, all four inductance formulas are equivalent. The equivalence does not hold for open (linear/filiform) circuits, where all four expressions diverge differently.

Part II: Force

Part II proves that Ampère's force law and Grassmann's force law give identical results for the net force on any finite part of a closed circuit due to the remainder.

5. Chapter 5: Force Between Current Elements

Ampère vs. Grassmann

5.1 Ampère's Force (1826)

Derived from careful experiments with closed circuits, Ampère's force between current elements $I_i d\vec{r}_i$ and $I_j d\vec{r}_j$ is:

d^2\vec{F}^A_{ji} = \frac{\mu_0 I_i I_j}{4\pi} \frac{\hat{r}_{ij}}{r^2_{ij}} \left[3(d\vec{r}_i \cdot \hat{r}_{ij})(d\vec{r}_j \cdot \hat{r}_{ij}) - 2(d\vec{r}_i \cdot d\vec{r}_j)\right] \tag{5.1}

This is derived from Weber's force between point charges. Weber's force on charge $q_1$ due to $q_2$:

\vec{F}^W_{21} = \frac{q_1 q_2 \hat{r}_{12}}{4\pi\epsilon_0 r^2_{12}}\left[1 + \frac{1}{c^2}\left(\dot{r}_{12}\ddot{r}_{12} - \frac{\dot{r}^2_{12}}{2}\right)\right] \tag{5.3}

Key property: Ampère's force satisfies Newton's third law between any two elements: $d^2\vec{F}^A_{ji} = -d^2\vec{F}^A_{ij}$, and the force is directed along $\hat{r}_{ij}$ (central force).

5.2 Grassmann's Force (1845)

Grassmann's force is the magnetic part of Lorentz's force, using Biot-Savart for $\vec{B}$:

d^2\vec{F}^G_{ji} = I_i d\vec{r}_i \times d\vec{B}_j(\vec{r}_i) = \frac{\mu_0 I_i I_j}{4\pi} \frac{1}{r^2_{ij}}\left[(d\vec{r}_i \cdot \hat{r}_{ij})d\vec{r}_j - (d\vec{r}_i \cdot d\vec{r}_j)\hat{r}_{ij}\right] \tag{5.10}

Derived from Liénard-Schwarzschild's force (Lorentz force with retardation), Grassmann's force is not central and does not satisfy Newton's third law element-by-element: $d^2\vec{F}^G_{ji} \neq -d^2\vec{F}^G_{ij}$.

Theoretical status: Grassmann's is the only expression compatible with special relativity to second order in $v/c$. Ampère's is derived from Weber's theory and is not compatible with Lorentz/Einstein's theories.

5.3 Extensions to Surface and Volumetric Elements

Replacing $Id\vec{r} \to \vec{K}\,da$ or $\vec{J}\,dV$, Ampère's and Grassmann's forces become:

d^4\vec{F}^A_{ji} = \frac{\mu_0}{4\pi}\frac{\hat{r}_{ij}}{r^2_{ij}}\left[3(\vec{K}_i\cdot\hat{r}_{ij})(\vec{K}_j\cdot\hat{r}_{ij}) - 2(\vec{K}_i\cdot\vec{K}_j)\right]da_i\,da_j \tag{5.35}

d^4\vec{F}^G_{ji} = -\frac{\mu_0}{4\pi}\frac{1}{r^2_{ij}}\left[(\vec{K}_i\cdot\vec{K}_j)\hat{r}_{ij} - (\vec{K}_i\cdot\hat{r}_{ij})\vec{K}_j\right]da_i\,da_j \tag{5.37}

These eliminate the divergences that appear when integrating linear elements over wires in contact.

5.4 Partial Equivalence

When integrated over a closed circuit $\Gamma_1$, Ampère's and Grassmann's expressions give the same force on any external element $I_2 d\vec{r}_j$:

d\vec{F}^A_{1j} = d\vec{F}^G_{1j} \tag{5.3 partial equiv.}

The difference $d\vec{F}^A - d\vec{F}^G$ is an exact differential that integrates to zero over any closed path.

6. Chapter 6: Force in Several Geometries

Explicit Calculations

6.1 Parallel Wires (Linear Elements)

For two parallel wires of lengths $\ell_1$ and $\ell_2$, at perpendicular distance $b$, with axial offset $a$, Ampère's exact force is:

\vec{F}^A_{21} = -\frac{\mu_0 I_1 I_2}{4\pi}\left[\hat{x}\left(\sinh^{-1}\frac{a+\ell_2}{b} - \sinh^{-1}\frac{a}{b} - \sinh^{-1}\frac{a+\ell_2-\ell_1}{b} + \sinh^{-1}\frac{a-\ell_1}{b}\right) + \hat{y}(\cdots)\right] \tag{6.3}

In the collinear limit ($b \to 0$, two aligned segments separated by gap $d = a - \ell_1 > 0$):

\vec{F}^A_{21} = -\hat{x}\frac{\mu_0 I_1 I_2}{2\pi}\left(\ln\frac{a}{d} + \ln\frac{d+\ell_2}{a+\ell_2}\right) \tag{6.4}

This is a repulsion for currents flowing in the same direction. It diverges as $d \to 0$.

For Grassmann's force between collinear wires:

\vec{F}^G_{21} \to 0 \quad \text{as } b \to 0 \tag{6.5}

Grassmann predicts zero force between collinear parallel wires. This is a fundamental experimental difference between the two theories for open circuits.

6.2 Orthogonal Wires

For wires perpendicular to each other, Ampère's force is:

\vec{F}^A_{21} = -\frac{\mu_0 I_1 I_2}{4\pi}\left[\hat{x}(\cdots) - \hat{y}(\cdots)\right] \tag{6.9}

Here $\vec{F}^A_{12} \neq \vec{F}^G_{12}$ and $\vec{F}^A_{21} \neq \vec{F}^G_{21}$ — Ampère and Grassmann differ on the net force from each wire on the other. Moreover, Grassmann's force does not satisfy action and reaction in this case.

6.3 Rectangular Circuit — Linear Elements (the "Bridge" Problem)

Consider a rectangular circuit (Fig. 6.3) with a central segment ("bridge", part 1) and the remainder ("support", parts 2–12). The net force on the bridge due to the support is:

\vec{F}^A_{SB} = \vec{F}^G_{SB} = \hat{y}\frac{\mu_0 I^2}{4\pi}\left(\sinh^{-1}\frac{f}{e-b} - \sinh^{-1}\frac{f}{e-a} + \sinh^{-1}\frac{f}{a} - \sinh^{-1}\frac{f}{b} + \frac{(f^2+b^2)^{1/2}}{f} - \frac{(f^2+a^2)^{1/2}}{f} - \frac{[f^2+(b-e)^2]^{1/2}}{f} + \frac{[f^2+(a-e)^2]^{1/2}}{f}\right) \tag{6.12}

This result is identical for Ampère and Grassmann — the non-trivial first instance of equivalence for non-external elements. Crucially, the force has no longitudinal component ($\hat{x}$), even though Ampère's expression predicts a longitudinal self-force on each piece. The longitudinal forces cancel exactly when the whole circuit is accounted for.

6.4 Rectangular Circuit — Surface Elements

For the same geometry with surface current (width $\omega \ll \ell_1, \ell_2, \ell_3$), Ampère's force on the bridge is:

\vec{F}^A_B = \hat{y}\frac{\mu_0 I^2}{2\pi}\left(\ln\frac{\ell_2}{\omega} - \sinh^{-1}\frac{\ell_2}{\ell_3} + \frac{(\ell_2^2+\ell_3^2)^{1/2}}{\ell_2} + \ln 2 + \frac{1}{2} + O\!\left(\frac{\omega}{\ell}\right)^3\right) \tag{6.23}

For closed lines of current (Fig. 6.5), Grassmann's result with surface elements equals Ampère's:

$\vec{F}^G_B = \vec{F}^A_B \quad \text{(Eq. 6.34)}$

The result is independent of $\ell_1$ (the bridge height) — a non-trivial prediction confirmed experimentally. The logarithmic term $\ln(\ell_2/\omega)$ dominates, showing sensitivity to the wire's cross-section.

6.5 Rectangular Circuit — Volumetric Elements

For a circuit with square cross-section $\omega \times \omega$, the force on the bridge:

\vec{F}^A_B = \hat{y}\frac{\mu_0 I^2}{2\pi}\left(\ln\frac{\ell_2}{\omega} - \sinh^{-1}\frac{\ell_2}{\ell_3} + \frac{(\ell_2^2+\ell_3^2)^{1/2}}{\ell_2} + \frac{2}{3}\ln 2 - \frac{\pi}{3} + \frac{13}{12} + O\!\left(\frac{\omega}{\ell}\right)^3\right) \tag{6.45}

The structure is the same as (6.23) with different numerical constants, due to the 3D integration over the cross-section.

6.6 Solenoid with Poloidal Current (Exact Result)

For a cylinder of length $\ell$ and radius $a$ carrying total poloidal surface current $I_t$, the force on a strip of length $\ell$ and angular width $d\phi$ located at $\phi = \pi/2$:

d\vec{F}^A = d\vec{F}^G = \hat{y}\frac{\mu_0 I^2_t\,a\,d\phi}{\pi\ell^2}\left[(4a^2+\ell^2)^{1/2}\,E\!\left(\frac{2a}{(4a^2+\ell^2)^{1/2}}\right) - 2a\right] \tag{6.57}

where $E$ is the complete elliptic integral of the second kind. This is exact for all $\ell/a$ — an exact equality between Ampère and Grassmann's expressions for a closed surface circuit. In the limits:

• Long solenoid ($\ell \gg a$): $dF \approx \hat{y}\frac{\mu_0 I^2_t\,a\,d\phi}{2\ell}\left(1 - \frac{4a}{\pi\ell} + \cdots\right)$
• Short solenoid / ring ($\ell \ll a$): $dF \approx \hat{y}\frac{\mu_0 I^2_t\,d\phi}{4\pi}\left[\frac{1}{2} + 3\ln 2 - \ln\frac{\ell}{a} + \cdots\right]$

7. Chapter 7: Complete Equivalence of Force Laws

The General Proof

7.1 No Bootstrap Effect

The net force of a closed circuit on itself is zero for both Ampère and Grassmann:

\vec{F}_{\Gamma\Gamma} = \oint_\Gamma\oint_\Gamma d^2\vec{F}_{12} = 0 \tag{7.1}

For Ampère, this follows immediately from Newton's third law. For Grassmann, the proof requires showing two double integrals vanish:

\oint_\Gamma\oint_\Gamma \left(d\vec{r}_1 \cdot \frac{\vec{r}_{12}}{r^3_{12}}\right)d\vec{r}_2 = 0, \qquad \oint_\Gamma\oint_\Gamma (d\vec{r}_1 \cdot d\vec{r}_2)\frac{\vec{r}_{12}}{r^3_{12}} = 0 \tag{7.5, 7.7}

Both vanish because the integrands are either exact differentials around closed loops or antisymmetric. This rules out a "bootstrap effect" — a closed circuit cannot lift itself.

7.2 Complete Equivalence Proof (Surface Currents)

The proof proceeds in three stages:

Stage 1 — Cylinder: For the cylinder of Fig. 7.6, the force on a surface current element $\delta$ of area $ds$ at $\phi = \pi/2$ due to the whole cylinder:

d^2\vec{F}^A = d^2\vec{F}^G = \hat{y}\frac{\mu_0 I^2_t\,a^2\,ds}{2\pi\ell^2}\,K\!\left(\frac{4ia}{\ell}\right) \tag{7.10}

where $i=\sqrt{-1}$ and $K$ is the complete elliptic integral of the first kind. This exact equality holds for any $\ell/a$.

Stage 2 — Arbitrary point on cylinder: The force on any current element located at distance $d$ from one end of the cylinder of length $L$ is the same for Ampère and Grassmann, because the cylinder can be split into two symmetric sub-cylinders (each is a closed circuit acting on an external element, for which equivalence is already known).

Stage 3 — Arbitrary closed circuit: Any closed circuit $\Gamma$ (Fig. 7.7) can be locally approximated at any element $\delta$ by a cylinder $\Gamma_1$ of radius $a$ equal to the local radius of curvature, plus a far circuit $\Gamma_2$ similar to $\Gamma$ elsewhere (Fig. 7.8). As $d \to 0$ the force on $\delta$ from both parts equals the force from $\Gamma$, and this is the same for Ampère and Grassmann.

Conclusion: For any closed circuit of arbitrary form with surface or volumetric current:

\boxed{\vec{F}^A = \vec{F}^G} \tag{7.10 / Ch. 7}

Moreover, this force is always orthogonal to the current element (a property Grassmann's force has by construction, and Ampère's force acquires upon integration over a closed circuit).

8. Chapter 8: Final Discussion

8.1 Comparison with Experimental Data

Peoglos experiment (rectangular circuit, Fig. 6.3): Measured $F = 3.0513 \times 10^{-7}\,I^2$ N. Theory from Eq. (6.12): same value. ✓

Peoglos experiment (volumetric rectangular circuit, Fig. 6.10, $\ell_1=7.5$ cm, $\ell_2=\ell_3=10$ cm, wire diameter 1.2 mm → $\omega \approx 1.1$ mm): Measured $F = 11.2 \times 10^{-7}\,I^2$ N. Theory from Eq. (6.45): $F = 11.1 \times 10^{-7}\,I^2$ N. ✓

Moyssides experiment ($\ell_2 = 123.7$ cm, $\ell_3 = 48.0$ cm, 2 mm wire → $\omega = 1.77$ mm): Theory: $F = 12.890 \times 10^{-7}\,I^2$; Measured: $F = 12.873 \times 10^{-7}\,I^2$. ✓

8.2 Relation Between Inductance and Force

For closed circuits, the force between two loops can be obtained from the mutual inductance by:

\vec{F}_{12} = I_1 I_2 \vec{\nabla}M^N_{12} \tag{d}

For a single circuit, the force on a finite part can sometimes be obtained from the self-inductance by:

dF = \frac{I_t^2\,d\phi}{4\pi}\frac{dL}{da} \tag{8.1}

where $a$ is a geometric parameter (e.g. radius). Applied to the solenoid self-inductance (2.6), this yields exactly (6.57). However, this method does not always work — deriving (6.12) from (3.11) by differentiation with respect to height $\ell_1$ fails because the force is independent of $\ell_1$.

Caution: Ampère's force cannot be derived as a gradient of Weber's current-element energy, nor can Grassmann's from Maxwell's energy — the approximations made in the current element model prevent this.

8.3 Longitudinal Force via Weber's Electrodynamics

Although Ampère = Grassmann for the net force on any part of a closed circuit, Ampère's force on individual elements has a longitudinal component (along the wire). This motivated Graneau and others to claim Ampère predicts observable longitudinal forces (wire explosions, railgun effects). The book's rebuttal: since Ampère = Grassmann on closed circuits, and Grassmann has no longitudinal components, the net longitudinal force is zero.

However, the book explores a different mechanism — Weber's force acting on the positive crystal lattice (not the neutral current element). For a rectangular circuit (Fig. 8.1) with asymmetrically placed bridge, the Weber longitudinal force on the positive lattice ions of the left half of the bridge is:

F_+ + F_- = -\frac{\mu_0 I^2}{4\pi}\left[\ln\frac{A+B+L}{A+B} - \ln\frac{B+C+L}{B+C} + \frac{1}{2}\ln\frac{(A+B+L)^2+D^2+(A+B+L)}{(A+B)^2+D^2+(A+B)} - \cdots\right] \tag{8.8}

For a symmetric quadratic circuit ($A = C = D/4$, $L = D/2$), this longitudinal tension is $\approx 0.27\,(\mu_0 I^2/4\pi)$, independent of the circuit size $D$. For $I = 6\times10^3$ A this gives $\approx 0.7$ N — two orders of magnitude below what is needed to break an aluminum wire ($50$–$500$ N/mm²). The exploding-wire phenomenon may therefore require pulsed-current effects or temperature-dependent tensile strength.

8.4 Open Questions

The equivalence between Ampère and Grassmann holds only when:

1. All lines of current form closed loops (no charge accumulation)
2. For Grassmann, the force of each part on itself is included

Open situations where the two forces differ:

• Variable currents in long closed conductors (charge accumulates at some points)
• Capacitor discharge circuits where the gap region carries displacement current but no conduction current
• Antennas (open circuits with ac current — Grassmann predicts a self-torque, Ampère and Weber do not; no torque observed experimentally)
• Moving magnets near wires (oscillating wire in a solenoid field induces alternate current; Ampère predicts longitudinal tension, Grassmann does not)

The fundamental open question: since experiments on closed circuits cannot distinguish Ampère from Grassmann, one must turn to experiments distinguishing Weber's from Lorentz's forces at the level of point charges. Weber's electrodynamics predicts a variable effective inertial mass for a charged particle inside a charged spherical shell — a prediction not shared by Lorentz. An experiment by Mikhailov (1999) reported detecting this effect with the sign and magnitude predicted by Weber.

9. Chapter 9: Conclusions

The two main results of the book:

Part I — Inductance equivalence: The self-inductances of Neumann, Weber, Maxwell, and Graneau are different for open circuits but identical for any closed circuit of arbitrary form: $L^N = L^W = L^M = L^G \quad \text{(closed circuits)}$

Part II — Force equivalence: Ampère's and Grassmann's expressions always yield the same net force on any part of a closed circuit due to the whole circuit, provided:

• The circuit carries current along closed lines
• The force of each part on itself is included (essential for Grassmann)

These results settle a 150-year controversy. The experiment of Cavalleri et al. (1998), cited as evidence for Grassmann/Lorentz, is equally explained by Ampère/Weber — confirming the main thesis of the book.

Synthesis: The Dual Equivalence Theorem

The book establishes a dual equivalence theorem with a beautiful symmetry:

| Property | Ampère | Grassmann | |----------|--------|-----------| | Action-reaction (element pairs) | ✓ always | ✗ in general | | No bootstrap (closed circuit on itself) | ✓ (trivial from 3rd law) | ✓ (requires proof) | | Compatible with Weber's electrodynamics | ✓ | ✗ | | Compatible with special relativity ($v/c$) | ✗ | ✓ | | Net force on part of closed circuit | Same | Same | | Force orthogonal to element (closed circuit) | ✓ | ✓ (always, by construction) | | Longitudinal force on individual element | ✓ (predicts it) | ✗ (always zero) |

The equivalence holds because the difference $d^2\vec{F}^A - d^2\vec{F}^G$ is an exact differential in the closed loop integration variable, integrating to zero. The four inductance formulas are equivalent because the $k$-dependent cross terms integrate to zero around any closed path.

The practical import: in engineering calculations, use whichever formula is most convenient — they all give the same answer for real (closed) circuits. The theoretical import: the force between current-carrying wires cannot, by any closed-circuit experiment, distinguish Weber's from Maxwell-Lorentz electrodynamics.