Solenoid Inductance Calculator

The inductance of a solenoid is its ability to store energy in a magnetic field, defined as L = N·Φ/I (flux linkage per ampere) and measured in henries. For an air-core single-layer solenoid it depends only on geometry: the number of turns N, the coil radius r, and the coil length l. Enter those three values above and the calculator returns the inductance via three methods.

Wheeler's 1928 formula is L(µH) = (r²·N²)/(9r+10l), where r is the coil radius in inches, l is the coil length in inches, and N is the number of turns. It is an empirical fit that already accounts for end effects, so it stays accurate for short coils where the ideal long-solenoid formula overestimates. This calculator evaluates it alongside two physics-based methods.

For an ideal (infinitely long) solenoid, L = µ₀·N²·A/l, with µ₀ = 4π×10⁻⁷ H/m, A = π·r² the cross-sectional area in m², and l the length in meters. It assumes a perfectly uniform field and no end effects, so it always overestimates a real finite coil — the Nagaoka correction fixes exactly that.

The Nagaoka coefficient K is a number between 0 and 1 that corrects the ideal long-solenoid formula for end effects: L = µ₀·N²·A/l·K. It depends only on the length-to-diameter ratio l/d — about 0.69 for l/d = 1 and 0.96 for l/d = 10, approaching 1 for very long coils. You need it whenever the coil is not much longer than its diameter. This calculator computes K with Lundin's six-figure closed-form approximation of Nagaoka's table.

Wheeler derived his formula as an empirical fit to measured coils, and it typically agrees with measurement within a few percent for ordinary single-layer air-core coils. Because it is empirical rather than derived, its error is not uniform across all shapes — that is why this calculator also shows the Nagaoka-corrected value, so you can see both estimates side by side and judge the spread.

Inductance grows with the square of the number of turns: doubling N quadruples L, because each turn both carries the field-producing current and links the resulting flux. This N² law holds for the Wheeler, long-solenoid, and Nagaoka-corrected formulas alike, and the design mode inverts it to find the turns needed for a target inductance.

Practically no. Inductance is set by the current-sheet geometry — turns, radius, and length — so the wire diameter barely matters as long as the winding stays single-layer. Gauge matters enormously for everything else though: thicker wire means lower DC resistance, higher current handling, and a longer coil for the same turn count. Use the AWG selector above to estimate DC resistance.

Approximate each turn as one circumference of the coil and add the helical pitch: length ≈ √((N·π·D)² + l²), where D is the coil diameter and l the coil length. This calculator shows that helical estimate automatically — for example, 100 turns on a 2-inch-diameter, 2-inch-long form needs about 16 meters of wire. Buy 10–15% extra for leads and mistakes.

Use R = ρ·l/A with copper's resistivity ρ = 1.7241×10⁻⁸ Ω·m at 20 °C, l the wire length from the estimate above, and A = π/4·d² the wire cross-section from the AWG table (d = 0.127·92^((36−n)/39) mm). Pick your gauge in the AWG selector and the calculator does it for you. Remember resistance rises about 0.4% per °C as the coil warms up.

A solenoid plus a capacitor forms an LC tank that resonates at f₀ = 1/(2π√(LC)) — Thomson's formula. Enter your capacitance above and the calculator returns f₀ and the free-space wavelength λ = c/f₀. Keep in mind every real coil also has stray self-capacitance between turns, so the measured self-resonant frequency is always a bit lower than the ideal LC value.

No — every formula here assumes an air core (relative permeability µr = 1). A magnetic core multiplies the inductance roughly by µr, but µr is nonlinear: it varies with current, frequency, and temperature, and the core saturates. Air-core results are the honest baseline; cored-coil design needs the core manufacturer's AL value or measured data.

These are single-layer formulas. In a multi-layer winding the layers couple to each other and the current sheet is thick, so Wheeler's single-layer formula underestimates. Wheeler published a separate multi-layer formula, and Rosa/Grover tables cover the general case — both are outside this calculator's scope, which is stated in the scope boundary below.

They are two different approximations of the same physical quantity: the inductance of a cylindrical current sheet. Wheeler is an empirical fit to measured coils; Nagaoka corrects the exact long-solenoid theory for end effects. For typical single-layer coils they agree within about 1–2%. A large spread usually means an extreme shape (very short or very long coil) where you should trust the Nagaoka-corrected value.

Switch to Design mode, enter your target inductance plus the coil radius and length your form allows, and the calculator inverts the formulas exactly — N = √(L(9r+10l))/r for Wheeler and N = √(L·l/(µ₀·A·K)) for Nagaoka (K depends only on l/d, so no iteration is needed). Round up to a whole turn, pick an AWG that fits the length, then verify with Calculate mode.