Solenoid Calculator (B = μ₀nI)
A solenoid is a coil of wire that creates a uniform magnetic field inside when current flows through it. Solenoids are the building blocks of electromagnets, relay switches, MRI machines, and particle accelerators. This calculator finds the internal field strength and inductance from the coil's dimensions and current.
Solenoid Formulas
Magnetic Field Inside
B = mu0 x n x I
Where mu0 = 4 pi x 10 to the negative 7 T m/A (permeability of free space), n = N/l (turns per meter), and I = current in amps. The field inside a long solenoid is remarkably uniform.
Inductance
L = mu0 x n squared x A x l
Inductance depends on the square of the turn density, so doubling the number of turns quadruples the inductance.
Worked Example
A solenoid with 500 turns, 10 cm long, 2 cm radius, carrying 1A:
- n = 500 / 0.10 = 5,000 turns/m
- A = pi x 0.02 squared = 1.257 x 10 to the negative 3 m squared
- B = 4 pi x 10 to the negative 7 x 5,000 x 1 = 6.28 mT
- L = 4 pi x 10 to the negative 7 x 25,000,000 x 0.001257 x 0.10 = 3.95 mH
Increasing the Field Strength
Three ways to get a stronger field:
- More turns per meter: Wind the coil tighter (B scales with n)
- More current: Double the current doubles the field (B scales with I)
- Add a ferromagnetic core: An iron core multiplies mu0 by the material's relative permeability (hundreds to thousands for iron)
Applications
| Application | Typical Field | Notes |
|---|---|---|
| Door buzzer solenoid | 1-10 mT | Pulls a plunger to strike |
| Car starter relay | 10-50 mT | Engages the starter motor |
| Lab electromagnet | 0.1-2 T | With iron core |
| MRI scanner | 1.5-3 T | Superconducting solenoid |