2,500
turns/m
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T
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mT
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G
2,500
turns/m
—
T
—
mT
—
G
The Solenoid Magnetic Field Calculator computes the magnetic field inside a solenoid using $$B = \mu_0 \mu_r n I$$ where n = N/l is the turn density (turns per meter), I is the current, and μᵣ is the relative permeability of the core material.
Solenoids are the workhorses of electromagnetism — used in electromagnets, relays, inductors, MRI machines, and particle accelerators. Their key advantage is producing a nearly uniform magnetic field inside the coil, with field strength easily controlled by adjusting the current.
A solenoid is a tightly wound helical coil. Applying Ampère's law to a rectangular loop partly inside and partly outside an ideal (infinitely long) solenoid gives:
$$B = \mu_0 n I = \mu_0 \frac{N}{l} I$$
where N is the total number of turns and l is the solenoid length. Key properties:
With a ferromagnetic core (iron, ferrite, etc.), the field is amplified by the relative permeability:
$$B = \mu_0 \mu_r n I$$
Iron cores can have μᵣ from 100 to 100,000, dramatically increasing the field. This is how electromagnets in motors, transformers, and relays achieve strong fields with moderate currents.
The solenoid is a practical approximation of a uniform field — the longer the solenoid relative to its diameter, the better the approximation. For length/diameter ratios above 10, the field is quite uniform over the central 80% of the length.
The result gives the magnetic field inside the solenoid's central region. For an air-core solenoid (μᵣ = 1), fields of a few millitesla are typical with modest current. With an iron core, the same coil can produce fields hundreds to thousands of times stronger. For practical design, ensure the current doesn't exceed the wire's rated capacity (overheating) and that the core doesn't saturate (iron typically saturates around 1.5–2 T).
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A 1000-turn, 30 cm air-core solenoid at 3 A produces ~12.6 mT — a moderate lab field for experiments.
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With an iron core (μᵣ ≈ 5000), even 1 A through 500 turns over 10 cm gives ~3.14 T — near iron saturation. In practice, saturation would limit the actual field.
The magnetic field inside an ideal solenoid is $$B = \mu_0 n I$$, where $$n = N/l$$ is the number of turns per unit length and I is the current. With a magnetic core material, multiply by the relative permeability: $$B = \mu_0 \mu_r n I$$.
By symmetry and Ampère's law, the field from each small ring of current adds constructively along the axis and cancels laterally. In the infinite-solenoid limit, this produces a perfectly uniform axial field. Real solenoids approximate this well in the central region when length ≫ diameter.
At the open ends of a finite solenoid, the field is approximately half the central value: $$B_{end} \approx \frac{\mu_0 n I}{2}$$. The field lines spread outward (fringing), and the field is no longer uniform. This is why long, thin solenoids provide better field uniformity.
Relative permeability (μᵣ) measures how much a core material amplifies the magnetic field compared to vacuum. Air/vacuum: μᵣ = 1. Soft iron: μᵣ ≈ 1,000–5,000. Mu-metal: μᵣ ≈ 50,000–100,000. The core's atomic magnetic dipoles align with the applied field, greatly enhancing it.
Magnetic saturation limits the field in ferromagnetic cores. When all atomic dipoles are aligned, increasing current no longer increases B proportionally. For silicon steel, saturation occurs around 1.5–2 T. For stronger fields, superconducting solenoids (air/vacuum core) are used, reaching 10–20 T or more.
Increase the turn density (more turns, shorter solenoid) and use a high-permeability core. However, more turns increase resistance and self-inductance, and the core will eventually saturate. Optimal design balances these trade-offs based on the application's field, uniformity, and response-time requirements.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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