# Solenoid Calculator (B = μ₀nI)

Calculate solenoid magnetic field (B = mu0 nI) and inductance (L = mu0 n squared Al). Enter turns, length, radius, and current for complete solenoid analysis.

## What this calculates

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.

## Inputs

- **Number of Turns (N)** — min 0
- **Solenoid Length (l)** (cm) — min 0
- **Solenoid Radius (r)** (cm) — min 0
- **Current (I)** (A) — min 0

## Outputs

- **Turns per Meter (n)** (turns/m) — Turn density along the solenoid
- **Magnetic Field (B)** (T) — Magnetic field inside the solenoid
- **Magnetic Field** (mT) — Field in millitesla
- **Cross-Section Area** (m²) — pi r squared
- **Inductance (L)** (mH) — Solenoid self-inductance
- **Stored Energy** (mJ) — Energy stored in the magnetic field

## Details

## 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:
1. **More turns per meter:** Wind the coil tighter (B scales with n)
2. **More current:** Double the current doubles the field (B scales with I)
3. **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 |

## Frequently Asked Questions

**Q: Why is the field uniform inside a solenoid?**

A: Inside a long solenoid, the field from each loop adds up along the axis and cancels near the walls. The result is a nearly uniform field that runs parallel to the axis. This uniformity breaks down near the ends, where the field is about half the interior value. For the formula to be accurate, the solenoid should be much longer than its diameter (at least 10 times for less than 5% error).

**Q: What happens if I add an iron core?**

A: An iron or ferrite core dramatically increases the magnetic field and inductance. The formulas become B = mu0 x mu_r x n x I and L = mu0 x mu_r x n squared x A x l, where mu_r is the relative permeability of the core material. Soft iron has mu_r around 200-5,000, so a 500-turn air-core solenoid producing 6 mT would produce 1.2 T or more with an iron core.

**Q: How does solenoid length affect the field?**

A: A shorter solenoid with the same number of turns has a higher turns-per-meter value (n = N/l), which increases the field. Compressing 500 turns into 5 cm instead of 10 cm doubles n and doubles B. However, making the solenoid very short relative to its diameter reduces uniformity and makes the formula less accurate.

**Q: What limits how strong a solenoid field can be?**

A: Heat and current capacity are the practical limits. More current means a stronger field, but wire resistance generates heat (P = I squared R). Thicker wire helps but adds bulk. For very strong fields (above ~2 T), superconducting solenoids cooled to near absolute zero are used because they carry huge currents with zero resistance. MRI machines use this technology.

---

Source: https://vastcalc.com/calculators/physics/solenoid
Category: Physics
Last updated: 2026-04-08