Analyze Magnetic Field

Calc B-field from current distribution: characterize src geometry → select law (Biot-Savart arbitrary, Ampere high-sym) → eval integrals → check limits → incorporate material effects → visualize field-line topology.

Use When

• B-field from arbitrary current conductor (wire loop, helix, irregular)
• Exploit cylindrical/planar/toroidal sym → Ampere direct
• Estimate far-field via dipole approx
• Superpose multiple current srcs
• Analyze materials: linear permeability, B-H curves, hysteresis, saturation

In

• Required: Current distribution spec (geometry, current magnitude + direction)
• Required: Region of interest (obs pts or vol)
• Optional: Material props (mu_r, B-H curve data, coercivity, remanence)
• Optional: Accuracy (exact integral, multipole order, numerical res)
• Optional: Viz reqs (2D cross-section, 3D field lines, magnitude contour)

Do

Step 1: Characterize Current + Geometry

Fully spec src pre-method:

1. Current path: Geometry each current element. Line currents → parametric curve r'(t). Surface currents → K (A/m). Volume currents → J (A/m^2).
2. Coord system: Align w/ dominant sym. Cylindrical (rho, phi, z) wires/solenoids. Spherical (r, theta, phi) dipoles/loops far dist. Cartesian planar sheets.
3. Sym analysis: ID translational, rotational, reflection. Src sym = field sym. Doc nonzero B-components + vanish.
4. Current continuity: Verify div(J) = 0 (steady) or div(J) = -d(rho)/dt (time-varying). Inconsistent → unphysical fields.

## 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]

→ Complete geometric desc, coord system chosen, sym cataloged, continuity verified.

If err: Too complex for closed-form → discretize short straight segments (numerical Biot-Savart). Continuity violated → add displacement current or return charge accumulation before proceeding.

Step 2: Select Law

Match method to sym + complexity:

1. Ampere (high sym): Use when B extractable from line integral. Applicable:

   • Infinite straight wire (cylindrical) → circular Amperian loop
   • Infinite solenoid (translational + rotational) → rectangular loop
   • Toroid (rotational about ring axis) → circular loop
   • Infinite planar current sheet (translational in 2 dirs) → rectangular loop

2. Biot-Savart (general): Arbitrary geometries Ampere can't simplify:

   • dB = (mu_0 / 4 pi) * (I dl' x r_hat) / r^2
   • Vol currents: B(r) = (mu_0 / 4 pi) * integral of (J(r') x r_hat) / r^2 dV'

3. Dipole approx (far field): Obs far from src (r >> src dim d):

   • Dipole moment: m = I * A * n_hat (planar loop area A)
   • B_dipole(r) = (mu_0 / 4 pi) * [3(m . r_hat) r_hat - m] / r^3
   • Valid r/d > 5 for ~1% accuracy

4. Superposition: Multi-src → B each independent + sum vectorially. Maxwell linearity → 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]

→ Justified method choice, clear why law appropriate for sym level.

If err: Ampere chosen but insufficient sym (B not extractable) → fallback Biot-Savart. Src geometry too complex analytic Biot-Savart → discretize numerically.

Step 3: Set Up + Eval Field Integrals

Execute calc via Step 2 method:

1. Ampere path: Per Amperian loop:

   • Parameterize path + compute line integral B . dl
   • Compute enclosed I_enc counting currents threading loop
   • Solve: contour_integral(B . dl) = mu_0 * I_enc
   • Extract B via sym (Step 1)

2. Biot-Savart integration: Per field pt r:

   • Parameterize src: dl' = (dr'/dt) dt or express J(r') over vol
   • Compute displacement: r - r' + magnitude |r - r'|
   • Eval cross product: dl' x (r - r') or J x (r - r')
   • Integrate over src (line, surface, vol)
   • Analytic eval: exploit sym reduce dim (on-axis field of loop → 1 integral)
   • Numerical: discretize N segments, sum, check convergence doubling N

3. Dipole calc:

   • Total magnetic moment: m = (1/2) integral (r' x J) dV' vol currents, or m = I * A * n_hat planar loop
   • Apply dipole field formula at each obs pt
   • Err estimate: next multipole (quadrupole) correction (d/r)^4

4. Superposition assembly: Sum all srcs per obs pt. Track components separately → 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]

→ Explicit B(r) expr at obs pts, correct units (Tesla/Gauss), convergence check numerical.

If err: Integral diverges → missing regularization (field on thin wire diverges → use finite wire radius). Numerical oscillates w/ N → near-singularity → adaptive quadrature or analytical subtraction.

Step 4: Check Limits

Verify vs known physics pre-trust:

1. Far-field dipole limit: Large r, localized src → dipole formula. Compute B → r → infinity, compare (mu_0 / 4 pi) * [3(m . r_hat) r_hat - m] / r^3.

2. Near-field infinite-wire limit: Close to long straight segment (rho << length L) → B = mu_0 I / (2 pi rho). Check for relevant geometry portion.

3. On-axis special cases: Loops + solenoids have simple closed forms:

   • Single circular loop radius R at dist z on axis: B_z = mu_0 I R^2 / [2 (R^2 + z^2)^(3/2)]
   • Solenoid length L, n turns/length: B_interior = mu_0 n I (L >> R)

4. Sym consistency: Verify components predicted vanish (Step 1) are 0. Nonzero forbidden → err.

5. Dim analysis: Verify B = Tesla. Every term mu_0 * [current] / [length].

## 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] |

→ All limits match. Field: correct units, sym, asymptotic behavior.

If err: Failed limit → err in integral setup/eval. Common: wrong sign cross product, missing factor 2/pi, wrong integration limits, coord mismatch src vs field.

Step 5: Materials + Visualize

Extend to material effects + produce viz:

1. Linear materials: Replace mu_0 w/ mu = mu_r * mu_0 inside. Boundary conds at interfaces:

   • Normal: B1_n = B2_n (continuous)
   • Tangential: H1_t - H2_t = K_free (surface free current)
   • No free surface currents: H1_t = H2_t

2. Nonlinear (B-H curves): Ferromagnetic cores:

   • B-H curve relates B + H at each pt
   • Design: piecewise linear segments — linear (B = mu H), knee, saturation (B ~ constant)
   • Hysteresis if op pt cycles: remanent B_r + coercive H_c define loop

3. Demagnetization: Finite-geometry magnetic materials (short rods, spheres) → internal field reduced by N_d: H_internal = H_applied - N_d * M.

4. Viz:

   • Field lines via stream fn or integrating dB/ds along field direction
   • Magnitude contours (|B| color map)
   • 2D cross-sections → current direction (dots out, crosses in)
   • Verify field lines closed loops (div B = 0) — open → viz/calc err

5. Intuition check: Field pattern makes sense qualitatively. Strongest near src, circulates around currents (right-hand rule), decays w/ dist.

## 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]

→ Complete field solution + material effects, viz closed field lines consistent div B = 0, qualitative intuition match.

If err: Field lines don't close → divergence err in calc → recheck integral/numerical. Material → unexpected amplification → verify mu_r only inside material vol + boundary conds enforced each interface.

Check

• Current distribution fully specified: geometry, magnitude, direction
• Current continuity (div J = 0 steady) verified
• Coord system aligned w/ dominant sym
• Method selection (Ampere / Biot-Savart / dipole) justified by sym
• Field integrals setup correct cross products + limits
• Numerical shows convergence (N vs 2N)
• Far-field dipole limit verified
• Near-field + on-axis match known formulas
• Forbidden sym components zero
• Units Tesla throughout
• Material boundary conds correct (if applicable)
• Field lines closed (div B = 0)

Traps

• Wrong cross-product direction: Biot-Savart = dl' x r_hat (src to field), not r_hat x dl'. Backward → flips entire field direction. Right-hand rule quick check.
• Confuse B + H: Vacuum B = mu_0 H, inside material B = mu H. Ampere via H uses free current only; via B includes bound (magnetization). Mix conventions → mu_r errors.
• Ampere no sufficient sym: Always true but only useful when sym extracts B. B varies along loop → single scalar eq for spatially varying fn → underdetermined.
• Ignore finite length of "infinite" wires: Real solenoids + wires have ends. Infinite formula valid only far from ends (dist from end >> radius). Near ends → full Biot-Savart or finite-solenoid corrections.
• Neglect demagnetization in finite: Magnetized sphere/short rod ≠ long rod in same applied field. Demag factor reduces internal field 30-100% depending aspect ratio.
• Non-physical field lines: Begin/end free space (not src or infinity) → calc/plot err. Field lines always closed loops.

→

• solve-electromagnetic-induction

  — use computed B → analyze time-varying flux + induced EMF

• formulate-maxwell-equations

  — generalize → full Maxwell + displacement current + wave propagation

• design-electromagnetic-device

  — apply to electromagnets, motors, transformers

• formulate-quantum-problem

  — quantum magnetic interactions (Zeeman, spin-orbit)