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Magnetic Circuit Calculator (Hopkinson's Law)

Hopkinson's law is Ohm's law with the names changed: MMF drives flux against reluctance. It lets you size a magnetic core the same way you would analyse a resistor network, including the dominant effect of any air gap.

Magnetic Circuit — Hopkinson's Law

The "magnetic Ohm's law": MMF drives flux through a reluctance, just as EMF drives current through a resistance. µr is the core's relative permeability.

MMF (ampere-turns)
Reluctance S
Flux Φ
Flux density B
Field strength H
  1. 1.Given: N = 500, I = 2 A, l = 500 mm, A = 1 mm², µr = 1000
  2. 2.FormulaMMF = N · ISubstituteMMF = 500 × 2ResultMMF = 1000 AT
  3. 3.FormulaReluctance S = l / (µ₀·µr·A)SubstituteS = 0.5 ÷ (4π×10⁻⁷ × 1000 × 0.001)ResultS = 397887.3577 AT/Wb
  4. 4.FormulaFlux Φ = MMF / SSubstituteΦ = 1000 ÷ 397887.3577ResultΦ = 2.5133 mWb
  5. 5.FormulaFlux density B = Φ / ASubstituteB = 0.0025 ÷ 0.001ResultB = 2.5133 T
Common trap: Reluctance is the magnetic analogue of resistance — a high-µr core (soft iron, µr in the thousands) has very lowreluctance, so it carries huge flux for a small MMF. An air gap has µr = 1 and dominates the total reluctance even when it's only a millimetre wide.

The formula

MMF = N × I ; S = l / (µ₀µ_r A) ; Φ = MMF / S ; B = Φ / A

MMF
magnetomotive force (ampere-turns)
S
reluctance (ampere-turns per weber)
Φ
flux (webers)
l, A
mean magnetic path length (m) and cross-section (m²)
µ_r
relative permeability of the core material

Worked example

A 200-turn coil carrying 1 A on a core: l = 0.5 m, A = 4 cm², µ_r = 2000.

  1. MMF = N × I = 200 × 1 = 200 AT
  2. S = 0.5 / (4π×10⁻⁷ × 2000 × 4×10⁻⁴) = 0.5 / 1.005×10⁻⁶ = 4.97 × 10⁵ AT/Wb
  3. Φ = 200 / 4.97 × 10⁵ = 4.02 × 10⁻⁴ Wb
  4. B = Φ / A = 4.02 × 10⁻⁴ / 4 × 10⁻⁴

Φ ≈ 0.40 mWb, giving B ≈ 1.0 T — comfortably below saturation for silicon steel.

Where you'll use it

Transformer and machine core design, relay and solenoid problems. Reluctances add in series just like resistances, and because air has µ_r = 1, even a millimetre of air gap can dominate the whole circuit.

The laws behind it

Parts this applies to

Transformer (power)Relay (electromechanical)Stepper MotorSolenoid / ActuatorLVDT & Displacement TransducersContactorDistribution / Power TransformerInductor / Coil

Related calculators

Faraday's Law EMF Calculatore = −N × (dΦ / dt)DC Machine EMF CalculatorE = (P × Φ × Z × N) / (60 × A), with A = P for lap and A = 2 for waveTransformer Turns Ratio CalculatorN_p / N_s = V_p / V_s = I_s / I_pMagnetic Force Calculator (Lorentz Force)F = B × I × L × sin θ (conductor)