Magnetic Fields & Coils

Magnetic Fields & Coils (magnetic-fields)

Purpose

Estimate solenoid magnetic field, inductance, stored magnetic energy, Lorentz force on conductors, and resistive coil heating. Supports electromagnet and actuator screening before detailed FEA or magnetic circuit design.

Physics & theory

A long solenoid with $N$ turns carrying current $I$ over length $L$ produces uniform field $B={\mu }_{0}NI/L$ in the interior (SI units, ${\mu }_0=4\pi \times 10^{-7}$ H/m). Inductance $L_{\mathrm{ind}}={\mu }_0{N}^{2}A/L$ for cross-sectional area $A$.

Stored magnetic energy $E=\frac{1}{2}{L}_{\mathrm{ind}}{I}^2$. Lorentz force on straight conductor length $\ell$ perpendicular to field: $F=BI\ell$. Resistive heating $P=I^{2}R$ from coil resistance $R$ must be removed to limit temperature rise.

Advanced systems calculators use lumped-parameter screening models suitable for concept trade studies. Each calculator returns explicit assumptions and warnings arrays documenting model limits. Constants such as ( sigma ) (Stefan–Boltzmann), ( mu_0 ), and ( R ) (gas constant) use SI definitions from the solver source.

Results are not certified for regulatory submission without independent verification against detailed analysis or test data.

Governing equations

Numerical method

Closed-form long-solenoid and inductance formulas (advanced-systems/calculators). Lorentz force assumes conductor perpendicular to $B$. No saturation, fringing, or eddy current losses.

Inputs

Outputs

• Magnetic field (T), inductance (H), stored energy (J)
• Lorentz force (N), resistive heating (W).

Design codes & checks

• Indicative: Solenoid field, stored energy, coil heating screening
• IEC: Electrical equipment practice (context)

Assumptions & limitations

• Long-solenoid approximation; fringe fields ignored.
• Linear magnetic circuit; no ferromagnetic saturation or hysteresis.
• DC or quasi-steady; no switching transients or skin effect.
• Structural support for Lorentz loads not analyzed.

References

1. Griffiths, D. J. Introduction to Electrodynamics, 4th ed. Pearson.
2. Feynman, R. P., et al. The Feynman Lectures on Physics, Vol. II.
3. Montgomery, D. C., & Turner, L. R. Principles of Superconducting Magnet Design. Wiley.
4. IEC 60076 series — transformer and reactor design context.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.