Calculate solenoid inductance L = μ₀μᵣN²A/l with Nagaoka correction. Compare core materials, view scaling relationships, and compute magnetic field and stored energy.
A solenoid is a coil of wire wound in a helical shape, and its inductance depends on the number of turns, cross-sectional area, length, and core material. The ideal formula L = μ₀μᵣN²A/l assumes the solenoid is much longer than its diameter. For shorter solenoids, the Nagaoka correction factor provides a more accurate result.
Solenoid inductance design is fundamental to electromagnet construction, relay design, transformer winding, RF inductor creation, and power electronics. The inductance scales with the square of the number of turns, so doubling the turns quadruples the inductance. Core material choice dramatically affects inductance — ferrite cores provide thousands of times more inductance than air cores.
This calculator computes both the ideal and Nagaoka-corrected inductance, the magnetic field strength, stored energy, reactance at key frequencies, and wire length estimates. A core material comparison table shows how different materials affect inductance, and a scaling visualization demonstrates the N² relationship.
Solenoid inductance depends on geometry, turns count, and core permeability in a nonlinear way. The Nagaoka correction for short solenoids, core material selection, and practical wire length estimation add complexity that makes manual design tedious. This calculator provides accurate inductance values with practical information like stored energy, reactance, and wire requirements.
Ideal Solenoid Inductance: L = μ₀ × μᵣ × N² × A / l Nagaoka Correction: L_corrected = L × k_N k_N ≈ 1 / (1 + 0.9 × d/l) Magnetic Field: B = μ₀ × μᵣ × (N/l) × I Stored Energy: E = ½ × L × I² Where: μ₀ = 4π × 10⁻⁷ H/m μᵣ = relative permeability A = πd²/4 (cross-section area) N = number of turns, l = coil length
Result: L = 63.2 μH (Nagaoka: 48.7 μH)
A 200-turn air-core solenoid with 10 mm diameter and 50 mm length has A = π(0.005)² = 78.5 mm². Ideal L = 4π×10⁻⁷ × 200² × 78.5×10⁻⁶ / 0.05 = 63.2 μH. The Nagaoka correction factor for d/l = 0.2 is about 0.77, giving L ≈ 48.7 μH.
Designing a solenoid involves balancing inductance, resistance, size, and power handling. More turns increase inductance (N²) but also increase wire length and DC resistance. A larger diameter increases inductance (through area) but makes the solenoid bulkier. A ferrite core dramatically boosts inductance but introduces core losses at high frequencies and saturation limits at high currents.
Single-layer solenoids are simplest to analyze, but practical inductors often use multiple winding layers. Multi-layer coils pack more turns into a given volume but have higher inter-winding capacitance. Toroidal cores confine the magnetic field inside the core, reducing EMI and improving efficiency. Toroid inductance calculation requires different formulas based on core geometry (Al value method).
Beyond their role as inductors, solenoids are widely used as linear actuators — the magnetic field pulls a ferromagnetic plunger into the coil. The force depends on the current, turns, core material, and air gap. Solenoid actuators are found in door locks, automotive starters, fuel injectors, and pinball machines.
Each turn contributes to the magnetic flux, and each turn also links that flux. So doubling N doubles both the flux-generating capability and the number of turns linking the flux, giving a total factor of N².
The ideal solenoid formula assumes infinite length (uniform field inside). Real solenoids have fringing fields at the ends that reduce inductance. The Nagaoka factor (< 1) corrects for this, becoming more significant as the solenoid gets shorter relative to its diameter.
Core permeability μᵣ multiplies the inductance directly. Air has μᵣ = 1. Ferrite typically has μᵣ = 2000, so a ferrite-core solenoid has 2000× the inductance of the same air-core solenoid. However, ferrite cores saturate at lower field levels.
When the magnetic field intensity becomes high enough, all magnetic domains in the core are aligned and can no longer increase B proportionally with H. The core is "saturated" — inductance drops dramatically, and the core acts more like air.
Yes, air-core inductors are common in RF applications where core losses would be unacceptable. They are linear (no saturation) but require more turns or larger size for a given inductance compared to cored inductors.
Energy stored is E = ½LI². This energy is stored in the magnetic field. When current is interrupted, this energy must go somewhere — in relay coils, a flyback diode protects the driving circuit from the voltage spike caused by this energy release.