Hong Life CalculatorsUnderstanding the solenoid inductance calculator — concept and principle
This calculator finds the inductance of a solenoid coil from the number of turns, length, diameter and core permeability using the long-solenoid formula, and adds Wheeler's estimate for air-core coils with end effects. It also gives the field inside, the stored energy, the cross-section area and turns per centimetre. Electronics hobbyists and students use it to wind coils.
How it is calculated
L = μ₀ μ_r N² A ÷ l, with A = π d²/4 and μ₀ = 4π × 10⁻⁷ H/m
Wheeler (air core): L (µH) = r² N² ÷ (9r + 10l), r and l in inches
Field inside B = μ₀ μ_r N I ÷ l; energy = ½ L I²
The principle
Current in a coil creates a magnetic field inside it; in a long solenoid the field is nearly uniform and proportional to turns per length. The flux through each turn is that field times the area, and inductance is the total flux linkage per ampere, N × flux ÷ I. Because both the field and the number of linked turns scale with N, L grows with N². The simple formula ignores the weaker field near the ends, so it overestimates short coils; Wheeler's empirical formula corrects for this in air-core coils.
Worked example
Default inputs: N = 200, length 10 cm, diameter 2 cm, μ_r = 1, I = 1 A.
- A = π × 1² = 3.142 cm²; turns per cm = 20; length ÷ diameter = 5.
- L = 4π × 10⁻⁷ × 200² × 3.142 × 10⁻⁴ ÷ 0.1 = 157.914 µH.
- Wheeler estimate = 144.477 µH, about 9% lower because of end effects.
- B = 4π × 10⁻⁷ × 200 × 1 ÷ 0.1 = 2.513 mT; energy = 0.07896 mJ.
| Number of turns N | Inductance, long-solenoid formula (µH) |
|---|---|
| 50 | 9.8696 |
| 100 | 39.4784 |
| 150 | 88.8264 |
| 200 | 157.914 |
| 250 | 246.74 |
| 300 | 355.306 |
| 400 | 631.655 |
How to read the result
For a coil five times longer than its diameter, the two estimates differ by under 10%; for shorter coils the gap grows and Wheeler's value is the better guide. A ferromagnetic core multiplies L, but its effective permeability depends on the core shape, any air gap and the current, so cored results are rough.
Common mistakes
- Using radius instead of diameter.
- Expecting the long-solenoid formula to hold for short, fat coils.
- Using a material's μ_r for a rod core in open air, where the effective value is much lower.
Frequently asked questions
Why does inductance grow with N²?
More turns make more field and also link more of it.
When is Wheeler's formula accurate?
For single-layer air-core coils with length greater than about 0.8 times the radius, to about 1%.
Does a core always multiply L by μ_r?
Only for a closed core; open rod cores give much less.
How is energy stored?
In the magnetic field, ½LI².
Sources: OpenStax, University Physics Volume 2 (2016), section 14.2 "Self-Inductance and Inductors" · H. A. Wheeler, single-layer coil inductance formula (standard approximation)