Calculate magnetic field B = μ₀nI inside a solenoid. Includes field intensity H, magnetic flux, inductance, stored energy, field strength scale, and parameter sensitivity table.
The magnetic field inside an ideal solenoid is remarkably uniform and depends only on the turns density and current: B = μ₀nI, where n is the number of turns per unit length and I is the current. This simple relationship makes solenoids the go-to geometry for generating controlled, uniform magnetic fields in laboratory and industrial applications.
By inserting a ferromagnetic core, the field is amplified by the relative permeability: B = μ₀μᵣnI. This is the principle behind electromagnets, relays, solenoid valves, and magnetic actuators. The field outside the solenoid is approximately zero for a long solenoid, making it an efficient geometry for containing magnetic flux.
This calculator computes the magnetic field B, field intensity H, total magnetic flux, inductance, and stored energy for a solenoid with specified geometry and current. A field strength scale provides context by comparing your solenoid to familiar magnetic fields, and a sensitivity table shows how changes to current, turns, and length affect the results.
Solenoid field calculations involve multiple physical quantities (B, H, Φ, L, E) and unit conversions between SI and CGS systems (Tesla vs Gauss, Wb vs Maxwell). This calculator handles all conversions and provides both the magnetic field data and the associated electrical parameters (inductance, stored energy) in one comprehensive view.
Magnetic Field: B = μ₀ × μᵣ × n × I B = μ₀ × μᵣ × (N/l) × I Field Intensity: H = n × I = N × I / l Magnetic Flux: Φ = B × A = B × π(d/2)² Inductance: L = μ₀ × μᵣ × N² × A / l Stored Energy: E = ½ × L × I² Where: μ₀ = 4π×10⁻⁷ H/m n = N/l (turns per meter)
Result: B = 12.57 mT
With N = 500 turns over l = 100 mm = 0.1 m, n = 5000 turns/m. At I = 2 A, B = 4π×10⁻⁷ × 1 × 5000 × 2 = 0.01257 T = 12.57 mT (about 126 gauss). This is roughly 250 times stronger than Earth's magnetic field.
Ampere's law states that the line integral of B around a closed path equals μ₀ times the enclosed current. For an ideal solenoid, applying this law to a rectangular path that runs along the solenoid axis inside and returns outside (where B ≈ 0) gives B × l = μ₀NI, yielding the familiar B = μ₀nI. This elegant derivation shows why the field depends only on turns density and current.
Real solenoid design must account for wire resistance (which generates heat), available power supply voltage and current, thermal management (air cooling, water cooling, or cryogenic cooling), and mechanical stress on the windings. At high fields, the outward electromagnetic force on the conductors can be enormous — high-field magnets use reinforced structures of steel and composites.
While solenoids are prized for uniform fields, the field gradient near the solenoid ends is also useful. Magnetic particle separators, magnetophoresis devices, and magnetic levitation experiments exploit the strong field gradient region. Gradient coils in MRI machines are specialized solenoids designed to create precisely controlled non-uniform fields for spatial encoding of the MR signal.
B (magnetic flux density, in tesla) is the total field including the core contribution. H (magnetic field intensity, in A/m) is the external driving field from the current. In free space, B = μ₀H. With a core, B = μ₀μᵣH. H depends only on current geometry; B depends on the medium.
A bench-top lab solenoid with a few hundred turns and a few amps produces fields in the 1-100 mT range. Strong research magnets achieve 1-20 T. For comparison, Earth's field is about 50 μT.
By symmetry and Ampere's law, the contributions from different turns add uniformly along the axis. The field lines run parallel to the solenoid axis inside. This is strictly true only for infinite solenoids; real solenoids have weaker, non-uniform fields near the ends.
Three main limits: (1) resistive heating in the wire (I²R losses), (2) core saturation for cored solenoids, and (3) mechanical forces — at high fields, the coil experiences enormous expansive forces. Superconducting magnets bypass limit (1).
1 Tesla = 10,000 Gauss. Tesla (T) is the SI unit; Gauss (G) is the older CGS unit still commonly used in some industries. 1 mT = 10 G.
No, the magnetic field depends only on the current and geometry (NI/l). However, wire material affects resistance (and thus achievable current for a given voltage) and thermal limits. Copper gives low resistance; superconductors allow zero-loss high currents.