Deep Research

Discrete Formulation of Ampère's Force Law

Discrete formulation of Ampère's force law between closed current circuits, with circuit discretization and summation schemes.

Circuit 1 Discretization

Number of segments: N1∈NN_1 \in \mathbb{N}

Position vectors: {r1(i)}i=0N1wherer1(i)∈R3\{\mathbf{r}_1^{(i)}\}_{i=0}^{N_1} \quad \text{where} \quad \mathbf{r}_1^{(i)} \in \mathbb{R}^3

Closure condition: r1(N1)=r1(0)\mathbf{r}_1^{(N_1)} = \mathbf{r}_1^{(0)}

Segment displacement vectors: Δℓ1(i)=r1(i+1)−r1(i)fori=0,1,…,N1−1\mathbf{\Delta\ell}_1^{(i)} = \mathbf{r}_1^{(i+1)} - \mathbf{r}_1^{(i)} \quad \text{for} \quad i = 0, 1, \ldots, N_1-1

Segment center points: c1(i)=12(r1(i)+r1(i+1))fori=0,1,…,N1−1\mathbf{c}_1^{(i)} = \frac{1}{2}\left(\mathbf{r}_1^{(i)} + \mathbf{r}_1^{(i+1)}\right) \quad \text{for} \quad i = 0, 1, \ldots, N_1-1

Circuit 2 Discretization

Number of segments: N2∈NN_2 \in \mathbb{N}

Position vectors: {r2(j)}j=0N2wherer2(j)∈R3\{\mathbf{r}_2^{(j)}\}_{j=0}^{N_2} \quad \text{where} \quad \mathbf{r}_2^{(j)} \in \mathbb{R}^3

Closure condition: r2(N2)=r2(0)\mathbf{r}_2^{(N_2)} = \mathbf{r}_2^{(0)}

Segment displacement vectors: Δℓ2(j)=r2(j+1)−r2(j)forj=0,1,…,N2−1\mathbf{\Delta\ell}_2^{(j)} = \mathbf{r}_2^{(j+1)} - \mathbf{r}_2^{(j)} \quad \text{for} \quad j = 0, 1, \ldots, N_2-1

Segment center points: c2(j)=12(r2(j)+r2(j+1))forj=0,1,…,N2−1\mathbf{c}_2^{(j)} = \frac{1}{2}\left(\mathbf{r}_2^{(j)} + \mathbf{r}_2^{(j+1)}\right) \quad \text{for} \quad j = 0, 1, \ldots, N_2-1

Separation Vectors Between Segment Pairs

Separation vector: Rij=c1(i)−c2(j)∈R3\mathbf{R}_{ij} = \mathbf{c}_1^{(i)} - \mathbf{c}_2^{(j)} \in \mathbb{R}^3

for all pairs (i,j)(i,j) where i∈{0,1,…,N1−1}i \in \{0, 1, \ldots, N_1-1\} and j∈{0,1,…,N2−1}j \in \{0, 1, \ldots, N_2-1\}

Distance: Rij=∣Rij∣=Rij⋅Rij∈RR_{ij} = |\mathbf{R}_{ij}| = \sqrt{\mathbf{R}_{ij} \cdot \mathbf{R}_{ij}} \in \mathbb{R}

Unit vector: R^ij=RijRij∈R3\hat{\mathbf{R}}_{ij} = \frac{\mathbf{R}_{ij}}{R_{ij}} \in \mathbb{R}^3

Discrete Double Sum Formula

The total force on circuit 1 due to circuit 2 is:

F12=I1I2c2∑i=0N1−1∑j=0N2−1R^ijRij2[2(Δℓ1(i)⋅Δℓ2(j))−3(Δℓ1(i)⋅R^ij)(Δℓ2(j)⋅R^ij)]\mathbf{F}_{12} = \frac{I_1 I_2}{c^2} \sum_{i=0}^{N_1-1} \sum_{j=0}^{N_2-1} \frac{\hat{\mathbf{R}}_{ij}}{R_{ij}^2} \left[ 2\left(\mathbf{\Delta\ell}_1^{(i)} \cdot \mathbf{\Delta\ell}_2^{(j)}\right) - 3\left(\mathbf{\Delta\ell}_1^{(i)} \cdot \hat{\mathbf{R}}_{ij}\right)\left(\mathbf{\Delta\ell}_2^{(j)} \cdot \hat{\mathbf{R}}_{ij}\right) \right]

where F12∈R3\mathbf{F}_{12} \in \mathbb{R}^3

Data Structure Summary

| Quantity | Type | Dimension | Index Range | |----------|------|-----------|-------------| | r1(i)\mathbf{r}_1^{(i)} | Array of vectors | (N1+1)×3(N_1+1) \times 3 | i=0,…,N1i = 0, \ldots, N_1 | | r2(j)\mathbf{r}_2^{(j)} | Array of vectors | (N2+1)×3(N_2+1) \times 3 | j=0,…,N2j = 0, \ldots, N_2 | | Δℓ1(i)\mathbf{\Delta\ell}_1^{(i)} | Array of vectors | N1×3N_1 \times 3 | i=0,…,N1−1i = 0, \ldots, N_1-1 | | Δℓ2(j)\mathbf{\Delta\ell}_2^{(j)} | Array of vectors | N2×3N_2 \times 3 | j=0,…,N2−1j = 0, \ldots, N_2-1 | | c1(i)\mathbf{c}_1^{(i)} | Array of vectors | N1×3N_1 \times 3 | i=0,…,N1−1i = 0, \ldots, N_1-1 | | c2(j)\mathbf{c}_2^{(j)} | Array of vectors | N2×3N_2 \times 3 | j=0,…,N2−1j = 0, \ldots, N_2-1 | | Rij\mathbf{R}_{ij} | Matrix of vectors | N1×N2×3N_1 \times N_2 \times 3 | (i,j)(i,j) pairs | | RijR_{ij} | Matrix of scalars | N1×N2N_1 \times N_2 | (i,j)(i,j) pairs | | R^ij\hat{\mathbf{R}}_{ij} | Matrix of vectors | N1×N2×3N_1 \times N_2 \times 3 | (i,j)(i,j) pairs |

Notes

  • Total number of pairwise interactions: N1×N2N_1 \times N_2
  • Computational complexity: O(N1⋅N2)O(N_1 \cdot N_2)
  • The discrete sum converges to the continuous line integral as N1,N2→∞N_1, N_2 \to \infty with proper mesh refinement
  • Units remain Gaussian/CGS as in the continuous formulation