Magnetic Field of a Long Coil

For a long cylindrical coil the interior field is B = μ₀·(N/l)·I with the field constant μ₀ = 1.257×10⁻⁶ V·s/(A·m). Insert the number of turns N, the coil length l in metres and the current I in amperes. Example: N = 500, l = 0.25 m, I = 2 A gives B = 1.257×10⁻⁶ × 2000 × 2 ≈ 5 mT. What matters is the turn density N/l, not the absolute number of turns: 500 turns on a short length act more strongly than on a long one. The result applies in the uniform interior region; with an iron core you additionally multiply by the permeability μ_r.

Each individual turn produces a curved ring field. With many turns packed closely one behind the other, these contributions superpose: inside, the field parts of all turns add up to parallel, equally dense field lines along the coil axis, while the transverse components of neighbouring turns cancel each other. Outside, the magnetic return flux spreads over a huge volume, so the external field is almost zero. The approximation works as long as the length is clearly larger than the diameter and you stay away from the ends: there the field lines splay apart and the flux density falls to about half the interior value.

It boosts the field dramatically: B = μ_r·μ₀·(N/l)·I with the permeability μ_r, which for iron ranges from a few hundred to several thousand depending on the grade. Microscopically the coil field aligns the elementary magnets (Weiss domains) of the iron, whose fields add to the external one. This turns a 5 mT air coil into an electromagnet with effects of several tesla, the principle of relays, lifting magnets and transformer cores. Two limits matter: from about 1.5 to 2 T iron saturates, all domains are aligned and more current yields hardly more field, and μ_r is not a true constant but depends on the operating point. After switch-off some residual magnetism (remanence) also remains.

The formula is a pure product, so rearranging is division: I = B·l/(μ₀·N) and N = B·l/(μ₀·I). Example: B = 10 mT in a coil with N = 1000 and l = 0.5 m requires I = 0.01 × 0.5/(1.257×10⁻⁶ × 1000) ≈ 4 A. Conversely, 2 mT at I = 0.8 A and l = 0.1 m demands N = 0.002 × 0.1/(1.257×10⁻⁶ × 0.8) ≈ 200 turns. Practical note: more turns mean a longer wire and thus more ohmic resistance, so at a fixed voltage the current drops. Designing real electromagnets therefore balances turn count, wire cross-section and heat removal against each other.

A map of orders of magnitude helps to judge results. The Earth magnetic field is about 50 µT = 5×10⁻⁵ T. A school coil without a core typically reaches 1 to 10 mT, twenty to two hundred times that. Strong permanent magnets (neodymium) achieve 0.3 to 1.4 T at their surface. Medical MRI machines run superconducting coils at 1.5 to 7 T, research magnets reach several tens of tesla. For comparison downwards: the magnetic fields of the human heart are around 10⁻¹⁰ T. So if your coil calculation yields 50 T, a power of ten has almost certainly slipped, usually through cm instead of m or mA instead of A.