Calculate and visualize magnetic fields produced by current distributions using the Biot-Savart law, Ampere's law, and magnetic dipole approximations. Use when computing B-fields from arbitrary current geometries, exploiting symmetry with Ampere's law, analyzing superposition of multiple sources, or characterizing magnetic materials through permeability, B-H curves, and hysteresis behavior.
Calculate the magnetic field produced by a given current distribution by characterizing the source geometry, selecting the appropriate law (Biot-Savart for arbitrary geometries, Ampere's law for high-symmetry configurations), evaluating field integrals, checking limiting cases, incorporating magnetic material effects where relevant, and visualizing the resulting field-line topology.
Fully specify the source before selecting a method:
## Source Characterization
- **Current type**: [line I / surface K / volume J]
- **Geometry**: [parametric description]
- **Coordinate system**: [and justification]
- **Symmetries**: [translational / rotational / reflection]
- **Nonzero B-components by symmetry**: [list]
- **Current continuity**: [verified / issue noted]
Expected: A complete geometric description of the current distribution with coordinate system chosen, symmetries cataloged, and current continuity verified.
On failure: If the geometry is too complex for a closed-form parametric description, discretize into short straight segments (numerical Biot-Savart). If current continuity is violated, add displacement current or return charge accumulation terms before proceeding.
Choose the method that matches the problem's symmetry and complexity:
Ampere's law (high symmetry): Use when the current distribution has sufficient symmetry that B can be pulled out of the line integral. Applicable cases:
Biot-Savart law (general): Use for arbitrary geometries where Ampere's law cannot simplify:
Magnetic dipole approximation (far field): Use when the observation point is far from the source (r >> source dimension d):
Superposition: For multiple sources, compute B from each independently and sum vectorially. Linearity of Maxwell's equations guarantees this is exact.
## Method Selection
- **Primary method**: [Ampere / Biot-Savart / dipole]
- **Justification**: [symmetry argument or distance criterion]
- **Expected complexity**: [closed-form / single integral / numerical]
- **Fallback method**: [if primary fails or for cross-validation]
Expected: A justified choice of method with a clear statement of why the chosen law is appropriate for the problem's symmetry level.
On failure: If Ampere's law is chosen but the symmetry is insufficient (B cannot be extracted from the integral), fall back to Biot-Savart. If the source geometry is too complex for analytic Biot-Savart, discretize numerically.
Execute the calculation using the method selected in Step 2:
Ampere's law path: For each Amperian loop:
Biot-Savart integration: For each field point r:
Dipole calculation:
Superposition assembly: Sum contributions from all sources at each observation point. Track components separately to preserve cancellation accuracy.
## Field Calculation
- **Integral setup**: [explicit expression]
- **Evaluation method**: [analytic / numeric with N segments]
- **Result**: B(r) = [expression with units]
- **Convergence check** (if numerical): [N vs. 2N comparison]
Expected: An explicit expression for B(r) at the observation points, with correct units (Tesla or Gauss) and a convergence check for numerical results.
On failure: If the integral diverges, check for a missing regularization (e.g., the field on the wire itself diverges for an infinitely thin wire -- use finite wire radius). If numerical results oscillate with N, the integrand has a near-singularity that requires adaptive quadrature or analytical subtraction of the singular part.
Verify the result against known physics before trusting it:
Far-field dipole limit: At large r, any localized current distribution should produce a field that matches the magnetic dipole formula. Compute B from your result in the limit r -> infinity and compare with (mu_0 / 4 pi) * [3(m . r_hat) r_hat - m] / r^3.
Near-field infinite-wire limit: Close to a long straight segment of the conductor (distance rho << length L), the field should approach B = mu_0 I / (2 pi rho). Check this for the relevant portion of your geometry.
On-axis special cases: For loops and solenoids, the on-axis field has simple closed forms:
Symmetry consistency: Verify that components predicted to vanish by symmetry (Step 1) are indeed zero in the computed result. A nonzero forbidden component indicates an error.
Dimensional analysis: Verify that B has units of Tesla. Every term should carry mu_0 * [current] / [length] or equivalent.
## Limiting Case Verification
| Case | Condition | Expected | Computed | Match |
|------|-----------|----------|----------|-------|
| Far-field dipole | r >> d | mu_0 m / (4 pi r^3) scaling | [result] | [Yes/No] |
| Near-field wire | rho << L | mu_0 I / (2 pi rho) | [result] | [Yes/No] |
| On-axis formula | [geometry] | [known result] | [result] | [Yes/No] |
| Symmetry zeros | [component] | 0 | [result] | [Yes/No] |
| Units | -- | Tesla | [check] | [Yes/No] |
Expected: All limiting cases match. The field has the correct units, symmetry, and asymptotic behavior.
On failure: A failed limit indicates an error in the integral setup or evaluation. The most common causes are: wrong sign in the cross product, missing factor of 2 or pi, incorrect limits of integration, or a coordinate system mismatch between source and field point parameterizations.
Extend the analysis to include material effects and produce field visualizations:
Linear magnetic materials: Replace mu_0 with mu = mu_r * mu_0 inside the material. Apply boundary conditions at material interfaces:
Nonlinear materials (B-H curves): For ferromagnetic cores:
Demagnetization effects: For finite-geometry magnetic materials (e.g., short rods, spheres), the internal field is reduced by the demagnetization factor N_d: H_internal = H_applied - N_d * M.
Field visualization:
Physical intuition check: Confirm that the field pattern makes qualitative sense. The field should be strongest near the current source, should circulate around currents (right-hand rule), and should decay with distance.
## Material Effects and Visualization
- **Material model**: [vacuum / linear mu_r / nonlinear B-H / hysteretic]
- **Boundary conditions applied**: [list interfaces]
- **Visualization**: [field lines / magnitude contour / both]
- **Div B = 0 check**: [field lines close / verified numerically]
Expected: A complete field solution including material effects where relevant, with a visualization that shows closed field lines consistent with div B = 0 and qualitative behavior matching physical intuition.
On failure: If field lines do not close, the calculation has a divergence error -- recheck the integral or numerical method. If the material introduces unexpected field amplification, verify that mu_r is applied only inside the material volume and that boundary conditions are correctly enforced at every interface.
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