mm
mm
Cycles
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A
kHz
Inductance L vs Number of Turns N
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Inductance L (μH)
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Number of Turns N
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Core Relative Permeability μr
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Stored energy ½LI² (mJ)
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Magnetic flux density B (mT)
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Reactance Xₗ (Ω)
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L（Wheeler）(μH)
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Nagaoka coefficient Kₙ
Inductance L vs Coil Diameter D
Blue dots = current flowing through the coil. Green curves = magnetic field lines threading the coil. Increasing turns N, current I, or core μr makes the field lines denser and shows inductance L∝N²μr increasing.
Theory & Key Formulas
Inductance L vs coil diameter D
Theory & Key Formulas
$$L = \mu_0\mu_r \frac{\pi D^2}{4}\frac{N^2}{l}K_n$$
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 $l$.
$$B = \mu_0\mu_r \frac{N I}{l},\qquad E = \tfrac{1}{2}L I^2,\qquad X_L = 2\pi f L$$$B$ = on-axis magnetic flux density, $E$ = stored energy, $X_L$ = inductive reactance.
Wheeler approximation (error ≤ 1% within its practical range):
$$L \approx \mu_0\mu_r\frac{\pi r^2 N^2}{l + 0.9r}$$$K_n$ is the finite-length correction factor (Hantaro Nagaoka, 1909), given exactly in terms of the elliptic integrals $K(k),E(k)$.