Calculate solenoid & toroid self-inductance (ideal & Wheeler short-coil corrected), magnetic core permeability, energy stored, back-EMF, and AC reactance.
Solenoid & Core Geometry
Select a magnetic core material or set custom relative permeability (µᵣ).
Ratio of core permeability to vacuum permeability (µ / µ₀).
Total number of wire turns.
Total axial length of the coil.
Cross-sectional area inside coil.
Inner/mean diameter of the coil loops.
Inductance (L)
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Enter turns, length, and area to compute self-inductance.
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Finite-Length Approximation
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Ideal Solenoid (Infinite)
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Finite-Length Factor (K)
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Coil Diameter (D)
Coil Field Diagram
Toroidal Coil Geometry
Toroids contain virtually all magnetic flux within their core, preventing external interference. Solved via L = (µ₀ µᵣ N² h ln(r₂ / r₁)) / 2π.
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Toroid Inductance (H)
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Mean Radius (r_mean)
Enter toroidal dimensions to estimate inductance.
Energy Stored & Magnetic Field Density
Reuses solenoid geometry + current to find total stored magnetic energy (E = ½ L I²), magnetic flux density (B), and volumetric energy density (u).
DC or peak current flowing through the coil.
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Inductance (L)
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Energy Stored (E)
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Magnetic Field (B)
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Energy Density (u)
Enter geometry (via the Solenoid tab) plus a current to estimate the energy stored.
Induced Back-EMF (Lenz's Law)
Calculates the back-EMF voltage induced across the inductor when current changes over time (EMF = −L · ΔI / Δt).
Net change in current (+ for rise, − for drop).
Switching or rise/fall time.
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Inductance (L)
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Induced EMF (V)
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Slew Rate (dI/dt)
Enter geometry (via the Solenoid tab) plus ΔI and Δt to estimate the induced EMF.
AC Inductive Reactance (X_L)
Computes opposition to alternating current at frequency f via X_L = 2π f L.
AC signal frequency.
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Inductance (L)
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Reactance (X_L)
Enter frequency to compute AC inductive reactance.
3 min read5 steps7 terms3 examples6 FAQsL = µ₀ N² A / l
Every coil of wire resists a change in its own current, and the strength of that resistance — its inductance — depends entirely on geometry: how many turns, how wide, how long.
On the Solenoid tab, enter the number of turns (N), the cross-sectional area (A, in m²), and the coil's length (l, in m). The calculator updates instantly as you type.
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Read the inductance result
The result card shows the self-inductance in henries (H), auto-scaled to mH or µH for readability, plus a plain-language summary of what that inductance means.
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Find the energy stored
Switch to the Energy Stored tab and enter the current flowing through the coil. It reuses the inductance from the Solenoid tab to compute the energy held in the magnetic field, in joules.
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Find the induced EMF
Switch to the EMF tab and enter the change in current (ΔI) and the time interval (Δt) over which it changes. The calculator finds the back-EMF induced across the coil, again reusing the inductance from the Solenoid tab.
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Compare a tighter vs. looser coil
Because L scales with the square of the turns, doubling the turns quadruples the inductance (and the energy stored at a given current) — try it to see how sensitive the result is to N.
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Reference
Formula & Methodology
3 formulas▸
Solenoid self-inductance
L = µ₀ N² A / l
The inductance of a long solenoid depends on the permeability of free space (µ₀ = 4π×10⁻⁷ H/m), the number of turns squared (N²), the cross-sectional area (A, m²), and inversely on the coil's length (l, m). More turns packed into a shorter, wider coil produces a much larger inductance.
Energy stored in the field
E = ½ L I²
An inductor carrying current I stores energy in its magnetic field, in joules. Like kinetic energy's ½mv², the energy grows with the square of the current — doubling the current quadruples the stored energy.
Induced (back-)EMF
EMF = −L · (ΔI / Δt)
A changing current through an inductor induces a voltage that opposes the change (Lenz's law) — the minus sign expresses that opposition. The faster the current changes (smaller Δt) or the larger the inductance, the bigger the induced EMF.
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General reference
Trust, Methodology & Sources
Written by Calculover Editorial Team · Updated 2026-07-31▸
Editorial accountability
Author: Calculover Editorial Team - Editor
Owner: Calculover Editorial Team - Editorial owner
Last reviewed: 2026-07-31
Last verified: 2026-07-31
Methodology
Limitations & guidance
This calculator employs standard Maxwellian electromagnetic equations ($\mu_0 = 4\pi \times 10^{-7} \text{ H/m}$) combined with Harold A. Wheeler's empirical Nagaoka short-coil correction factor ($K = \frac{l}{l + 0.45D}$) for accurate finite-solenoid modelling.
Professional guidance:
Primary sources
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Glossary
Key Terms Explained
7 terms▸
Inductance (L) ↗A coil's opposition to a change in current, measured in henries (H). Higher inductance means a larger induced voltage for the same rate of current change.
Henry (H) ↗The SI unit of inductance. A 1 H inductor induces 1 volt when current changes at 1 amp per second. Most practical coils measure in millihenries (mH) or microhenries (µH).
Solenoid ↗A coil of wire wound in a tight helix, long compared to its diameter, that produces a nearly uniform magnetic field inside — the standard geometry for the L = µ₀N²A/l formula.
Permeability (µ₀) ↗The vacuum permeability constant, µ₀ = 4π×10⁻⁷ H/m, that sets how strongly a magnetic field builds up per unit of current and geometry in free space (or air, approximately).
Self-inductance ↗The property of a single coil that causes it to induce an EMF in itself when its own current changes — as opposed to mutual inductance between two separate coils.
Energy stored ↗The energy held in an inductor's magnetic field while current flows through it, E = ½LI², measured in joules (J).
Back-EMF ↗The voltage an inductor induces in opposition to a change in its own current, per Lenz's law — it always acts to resist the change, whether the current is increasing or decreasing.
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Scenarios
Real-World Examples
3 worked examples▸
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Coil designer
500-turn solenoid, 5 cm² area, 20 cm long
Turns 500Area 0.0005 m²Length 0.2 m
L = µ₀ × 500² × 0.0005 / 0.2 ≈ 0.785 mH. A modest coil like this is typical of a small inductor used in a power-supply filter.
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Energy check
Same 0.785 mH coil carrying 3 A
Inductance 0.785 mHCurrent 3 A
E = ½ × 0.785 mH × 3² ≈ 3.53 mJ stored in the magnetic field — not much energy, but enough to sustain a brief current spike when the circuit is switched off.
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Switching transient
Same coil, current drops by 2 A in 5 ms
Inductance 0.785 mHΔI 2 AΔt 0.005 s
EMF = −0.785 mH × (2 A / 0.005 s) ≈ −0.314 V. That's the voltage spike an inductor pushes back when its current is interrupted — the faster the interruption, the bigger the spike, which is why switching inductive loads needs snubber circuits or flyback diodes.
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Reference
Cite This Calculator
APA & MLA▸
Use either format to cite this calculator in a paper, report, or resource list.
APA
MLA
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Deep Dive
Inductance, Energy, and the Voltage a Coil Pushes Back
Every coil of wire resists a change in its own current, and the strength of that resistance — its inductance — depends entirely on geometry: how many turns, how wide, how long. This calculator ties that geometry to two practical follow-on questions: how much energy does the coil store while current flows, and how big a voltage spike does it produce when that current changes quickly.
Why inductance depends on turns squared
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The solenoid formula L = µ₀N²A/l has an N² term, not just N, because each additional turn does double duty: it adds another loop for the magnetic field to link through, and it also strengthens the field that every other turn experiences. Wind twice as many turns into the same coil and the inductance quadruples, not doubles — which is why small increases in turn count can matter a lot for a compact inductor design, and why transformer and choke datasheets are so sensitive to winding counts.
What the stored energy is actually doing
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The ½LI² energy isn't dissipated as heat — it's stored in the coil's magnetic field the same way a stretched spring stores mechanical energy. That's precisely why interrupting current through an inductor (opening a switch, a relay contact bouncing) can be dangerous to nearby components: the stored energy has to go somewhere, and without a path to dissipate gradually, it forces current to keep flowing at the moment of interruption — producing the voltage spikes this calculator's EMF tab quantifies.
Lenz's law, in one line
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The minus sign in EMF = −L(ΔI/Δt) is Lenz's law in equation form: the induced voltage always opposes the change that created it. If current is increasing, the induced EMF pushes back against the increase; if current is decreasing, it pushes to sustain it. This calculator reports the magnitude of that EMF and explains the direction in words, since the sign convention depends on which way you've defined positive current flow.
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Questions
Frequently Asked Questions
6 questions▸
What's the formula for solenoid inductance?+
L = µ₀N²A/l, where µ₀ = 4π×10⁻⁷ H/m is the permeability of free space, N is the number of turns, A is the cross-sectional area in m², and l is the coil's length in meters.
How do you calculate the energy stored in an inductor?+
E = ½LI², where L is the inductance in henries and I is the current in amps. The result is in joules. Energy scales with the square of the current, so doubling current quadruples stored energy.
What's the formula for induced EMF?+
EMF = −L·(ΔI/Δt) — the inductance times the rate of change of current, with a minus sign showing (per Lenz's law) that the induced voltage always opposes the change that caused it.
What units does this calculator use?+
Inductance in henries (H, auto-scaled to mH or µH), area in square meters (m²), length in meters (m), energy in joules (J), and EMF in volts (V).
Why does more turns make such a big difference?+
Inductance scales with the number of turns squared (N²), not linearly — doubling the turns on an otherwise identical coil quadruples its inductance, since each turn both adds a flux linkage and strengthens the shared field.
Does adding a magnetic core change the inductance?+
Yes — this calculator assumes an air/vacuum core (relative permeability µᵣ = 1). Adding an iron or ferrite core multiplies the inductance by that core's relative permeability, often by a factor of hundreds or thousands.
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