Drag to rotate. Trace the graph below to locate the same axial point in 3D.
Teal: field streamlines of a finite air-core model. Fixed seeds are used; line count and spacing do not measure flux density. At most 90 representative windings are drawn.
Air-core comparison only. The selected core permeability affects the existing calculator, not this field view.
Magnetic field along the axis
Solid: finite length · Dashed: long-solenoid limit · Amber: coil region
Model and experiment
The winding is modeled as a thin, uniformly distributed cylindrical current sheet in air. The axial field integrates the circular-loop formula over the coil length. Streamlines use azimuthal numerical integration of the Biot–Savart law; the singular region near the winding is omitted. This is not finite-element, magnetic-core, saturation or high-frequency analysis. Frequency still affects the existing reactance result, not this magnetostatic view.
Try the air-core preset (D=30 mm, l=50 mm, N=50, I=1 A): about 1.078 mT at the center versus 1.257 mT in the long-solenoid approximation. Their ratio approaches one as length becomes much larger than diameter. The existing B card uses the long-solenoid approximation with the selected permeability; it is distinct from the new finite air-core field.
Background: circular loops and solenoidsDots illustrate current and curves illustrate the magnetic field schematically. Their number, shape and speed are not calculated field distributions or electron velocities. At fixed geometry and μr, L is proportional to N² and does not change with current I. Current changes the long-solenoid estimate B and stored energy E; frequency f changes XL. This constant-μr ideal model does not calculate core saturation or frequency-dependent material properties.
Long-solenoid approximation ($K_n\to1$). $L$ is proportional to the square of the number of turns $N^2$ and the core relative permeability $\mu_r$, and inversely proportional to length $l$.
$$\begin{gathered}B = \mu_0\mu_r \frac{N I}{l}\\ E = \tfrac{1}{2}L I^2\\ X_L = 2\pi f L\end{gathered}$$$B$ = on-axis magnetic flux density, $E$ = stored energy, $X_L$ = inductive reactance.
Wheeler-type SI approximation (not a uniform accuracy guarantee):
$$L \approx \mu_0\mu_r\frac{\pi r^2 N^2}{l + 0.9r}$$$K_n$ is the finite-length correction factor (Hantaro Nagaoka, 1909), given exactly in terms of the elliptic integrals $K(k),E(k)$.