Solenoid and Coil Inductance Formulas

Solenoid inductance formulas: the inductance from the turns, the core area, the length and the core permeability, the field in the coil and the stored energy.

The formulas

μ0 = 4π × 10⁻⁷ H/m, μr the relative permeability of the core (1 for air), N the number of turns, A the core area in m², l the length of the winding in metres, d the diameter and I the current. The formula is for a long solenoid, where the length is several times the diameter, and it gives a value that is a little too high for a short coil. A magnetic core raises L by μr only until it saturates, and then the inductance falls, so the iron core figure is the value below saturation. The sizes here are examples.

Worked example: 500 turns on a coil of 100 mm length and 20 mm diameter

1. A = π × (0.02 / 2)² = 314.2 mm²
2. In air, L = 4π × 10⁻⁷ × 500² × 0.0003142 / 0.1 = 0.987 mH
3. At 2 A, B = 4π × 10⁻⁷ × 500 × 2 / 0.1 = 12.57 mT
4. E = ½ × 0.987 mH × 2² = 1.974 mJ
5. With a core of μr = 200 below saturation, L = 200 × 0.987 = 197.4 mH

Inductance in air for the same coil by number of turns

Frequently asked questions

Why does the inductance rise with the square of the turns?

More turns make a stronger field for the same current, and each turn also links that field. Both effects scale with N, so L scales with N².

What does the core do?

A magnetic core raises the flux for the same current by its permeability, so the inductance rises by μr until the core saturates.

Why is the value of a short coil lower?

The field of a short coil spreads out at the ends, so less flux links the turns. A correction factor (Nagaoka) applies, which the formula leaves out.

How is this linked to the energy?

The energy is stored in the field, and it equals ½ × L × I². A larger inductance stores more for the same current.

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