Inductance and magnetic energy

For each closed winding orient its spanning surface by the right-hand rule relative to the chosen positive current direction. Its magnetic flux is Φ=∫B·n̂dS; reversing the winding orientation reverses signed Φ. With N identical tightly coupled turns, flux linkage is λ=NΦ; for unequal turns use λ=ΣkΦk.

For fixed geometry, linear vacuum magnetostatics implies B∝I, hence λ=LI and L=λ/I for nonzero I. Inductance describes that complete current-return geometry and surrounding material, not an isolated undefined open segment. In general a multiconductor structure requires an inductance matrix.

Faraday's induced emf opposes the change in linked flux: ε_ind=−dλ/dt for a fixed oriented winding. To build the current quasistatically, an external source must supply power I dλ/dt against that back-emf, neglecting resistive dissipation, motion and radiation. Therefore incremental STORED magnetic energy is dU=I dλ.

With fixed linear L, λ=LI, so U=∫₀ᴵ i L di=½LI². The ½ arises from the linear growth of flux linkage during charging. If L varies with current, geometry or magnetic history, return to the work integral and explicitly account for mechanical/source work.

Consider a tightly wound, long vacuum solenoid of length ℓ, N turns, transverse cross-section area A and current I. Assume ℓ much greater than its transverse size, uniform turn density n=N/ℓ and negligible end/fringe field. Ampère's rectangular path with one long leg inside and the return outside then gives approximately Binside=μ₀nI along the solenoid axis and Boutside≈0.

If all N turns intercept approximately the same interior flux Φ≈BA, the linkage is λ=NBA. Dividing by I gives L≈μ₀N²A/ℓ. This approximation is not exact for finite solenoids or tightly packed coils with strong external/return fields.

For localized finite-energy stationary currents in vacuum, the magnetic field energy can be written U=(1/(2μ₀))∫all space B²d³r, with boundary terms and wire self-field singularities treated correctly. The ideal solenoid assumes field concentrated in interior volume Aℓ, so its integral becomes U≈(B²/(2μ₀))Aℓ.

Substituting B≈μ₀NI/ℓ gives U≈μ₀N²AI²/(2ℓ)=½LI². These are TWO REPRESENTATIONS of the SAME energy and must not be added. A real coil has finite exterior/fringing and finite wire fields; its total-energy integral includes those regions.

Take N=1000 turns, ℓ=0.500 m, A=1.00×10⁻³ m² and I=0.200 A. The long-solenoid approximation gives B≈5.03×10⁻⁴ T, L≈2.51×10⁻³ H, and U=½LI²≈5.03×10⁻⁵ J.

The interior volume Aℓ=5.00×10⁻⁴ m³ and B²/(2μ₀) reproduce the same ≈50.3 μJ. The unit check [B²/μ₀] is J/m³, since 1 T=A-compatible N/(A·m) and μ₀ has N/A².

Consider an ideal densely wound toroidal vacuum coil with N turns, inner radius a, outer radius b>a and axial core height h. With rotational symmetry and an Ampèrian circle at radius r within its annular cross-section, Bφ(r)(2πr)=μ₀NI. Treat external leakage and finite winding gaps as negligible.

The flux per turn through the rectangular cross-section is Φ=∫aᵇBφ(r)h dr=μ₀NIh/(2π)ln(b/a). Multiply by N and divide by I to obtain L=μ₀N²h/(2π)ln(b/a). Using a single B across a broad annulus instead of integrating its 1/r variation is generally wrong.

Choose the winding orientations and currents once. For a fixed linear two-coil system write λ₁=L₁I₁+M₁₂I₂ and λ₂=M₂₁I₁+L₂I₂. In a passive linear reciprocal magnetic environment with compatible orientations, M₁₂=M₂₁=M, which may be positive or negative depending on the winding convention.

Incremental reversible work dU=I₁dλ₁+I₂dλ₂ then integrates to U=½L₁I₁²+½L₂I₂²+MI₁I₂. Nonnegative field energy for all current pairs requires L₁≥0, L₂≥0 and M²≤L₁L₂. A nonreciprocal, hysteretic or time-modulated material cannot be presumed to satisfy the same scalar-M model without checking its assumptions.

An ideal filament produces B≈μ₀I/(2πs) near its axis. Its magnetic energy per unit length in an annulus scales as ∫(B²/(2μ₀))2πs ds∝∫ds/s and diverges logarithmically as the inner radius tends to zero. An infinite isolated wire also creates an unbounded far-field energy per length when no return path sets a finite outer scale.

Physical conductors have finite cross-sections, spatial current distributions and return-current geometry. A numerical self-inductance calculation must state those dimensions and its high-frequency skin-current assumptions. Do not identify a divergent ideal self-field with a measured finite coil inductance.

In a local, linear, lossless, nondispersive medium under reversible quasistatic excitation, B=μH and u=½B·H. If the magnetization curve is nonlinear but reversible, the charging-work density follows u=∫₀ᴮ H(B′)·dB′ along the chosen path. Hysteresis can make the work path dependent and include dissipated heat.

A frequency-dependent complex permeability μ(ω) cannot simply be substituted into ½B·H as a generic THz stored-energy formula. Dispersion, loss, kinetic/material degrees of freedom and electromagnetic power flow require a broader treatment; neither the static L nor the static u is automatically an exact high-frequency device parameter.

1. Derive L≈μ₀N²A/ℓ for a long vacuum solenoid from B≈μ₀NI/ℓ and oriented N-turn flux linkage.
2. For N=1000, ℓ=0.500 m, A=1.00×10⁻³ m², I=0.200 A, calculate B, L, U and verify field/circuit energy equality.
3. For an ideal vacuum toroid with N=500, a=0.020 m, b=0.040 m and h=0.010 m, compute L from the logarithmic cross-section integral.
4. Given L₁=4.00 mH and L₂=9.00 mH, what is the allowed |M| for a passive reciprocal linear system? Does M=7.00 mH satisfy the energy requirement?
5. For L₁=4.00 mH, L₂=9.00 mH, M=+3.00 mH, I₁=1.00 A and I₂=+0.500 A, calculate U; repeat for I₂=−0.500 A.
6. Explain why the self-energy of a mathematical zero-radius current filament diverges and why using a complex μ(ω) in static ½B·H is not a general energy calculation.

EM-02 now spans force laws, steady-current source integrals, Ampère's circulation, vector potential and magnetic energy. EM-03 will derive the complete time-dependent Maxwell system, including Faraday induction, displacement current, continuity and electromagnetic power flow.