Toroid Magnetic Field Calculator

Introduction

A toroid is a coil wound around a ring (often described as a “doughnut-shaped” core). When current flows through the windings, the magnetic field is largely confined to the core, which helps reduce stray magnetic fields and electromagnetic interference. For an ideal toroid (tightly wound, symmetric, and evaluated inside the core), Ampère’s law leads to a simple relationship between magnetic field strength B, number of turns N, current I, and radius r.

This page provides a single calculator that can solve for B, N, I, or r. Enter any three values and leave exactly one field blank; the calculator will compute the missing quantity using the standard vacuum permeability constant μ₀ = 4π × 10⁻⁷ T·m/A.

How to use

1. Choose which quantity you want to compute: B, N, I, or r.
2. Fill in the other three fields with numeric values (decimals are allowed).
3. Leave exactly one field empty.
4. Select Compute Missing Quantity to see the result in the results box.

Units matter: enter B in tesla (T), I in amperes (A), and r in meters (m). If you have centimeters or millimeters, convert first (for example, 5 cm = 0.05 m). If you accidentally fill all four fields or leave more than one blank, the calculator will ask you to correct the inputs.

Formula (Ampère’s law for an ideal toroid)

For a circular Amperian loop of radius r inside the toroid, Ampère’s circuital law gives:

Formula: B = (μ_0 N I) / (2 π r)

$B=\frac{{\mu }_{0}NI}{2\pi r}$

Where B is magnetic flux density (tesla), N is the number of turns, I is current (amperes), and r is the radius from the toroid’s center to the point where the field is evaluated (meters). This calculator rearranges the same equation to solve for whichever variable is left blank.

Assumptions used by the calculator:

• μ = μ₀ (vacuum permeability). If your toroid has a magnetic core (ferrite/iron), the actual field can be higher by approximately the relative permeability factor, until saturation effects appear.
• The winding is sufficiently uniform that the field is approximately tangential and constant along the chosen circular path.
• The point of evaluation is inside the toroid (within the core region), not far outside the windings.

Worked example

Suppose you have a toroidal coil with N = 500 turns carrying I = 2 A, and you want the magnetic field at a mean radius of r = 0.05 m (5 cm). Leave the Magnetic Field B input blank and enter the other three values.

Using the formula:

Formula: B = ((4 π × 10 −7) N I) / (2 π r)

$B=\frac{(4\pi \times 10{\mathrm{-7}}^{\mathrm{}})NI}{2\pi r}$

Numerically, this is approximately 0.004 T (4 mT). If instead you measured B and wanted to find the required current, you would leave I blank and enter B, N, and r.

Limitations and practical notes

The ideal toroid model is a strong starting point, but real hardware can differ. Use these notes to interpret results responsibly:

• Core materials: Many toroids use ferrite or powdered iron. In that case, the permeability is μ = μ₀ μ_r, and μ_r can vary with frequency, temperature, and flux density. At high fields, cores can saturate, making the relationship non-linear.
• Thick toroids: If the toroid has a large cross-sectional thickness, the field varies with radius (B ∝ 1/r). Engineers often use a mean radius (average of inner and outer radii) as an approximation, but detailed designs may integrate across the cross-section.
• Leakage fields: The field outside an ideal toroid is near zero, but real windings have gaps and finite geometry, so some leakage exists—important for sensitive circuits and EMC considerations.
• Heating and safety: High current increases copper losses (I²R) and temperature rise. Ensure insulation, wire gauge, and duty cycle are appropriate, and keep strong fields away from magnetically sensitive devices.
• Units and rounding: The calculator outputs in scientific notation for very small/large values. Double-check unit conversions (cm to m, mT to T) to avoid errors by factors of 10 or 1000.

Background: magnetic field in a toroid

A toroid is a coil of wire wound into a donut-shaped ring. Unlike a solenoid, whose field extends outward from its ends, a toroid confines its magnetic field almost entirely within its core. This confinement makes toroidal inductors and transformers highly efficient, minimizing electromagnetic interference with nearby circuitry. The magnetic field inside an ideal toroid is uniform along circular paths centered on the ring and is given by Ampère's law as $B=\frac{{\mu }_{0}NI}{2\pi r}$ . Here N is the number of turns, I is the current through each turn, and r is the radial distance from the center of the toroid to the point where the field is evaluated.

Deriving the field expression uses Ampère's circuital law. By choosing a circular path of radius r inside the toroid and integrating the magnetic field around it, we obtain $\oint B·dl={\mu }_{0}NI$ . Because B is approximately constant along the chosen path and parallel to dl, the integral simplifies to $B\left(2\pi r\right)={\mu }_{0}NI$ , yielding the familiar formula. Outside the toroid, the magnetic field ideally cancels because the net enclosed current is zero for loops that lie entirely outside the windings.

Real toroids deviate from the ideal due to finite core permeability, wire thickness, and spacing between turns. Nonetheless, the equation provides an excellent approximation when the winding is dense and the evaluation radius is within the core region. For a toroid with significant thickness, using the average radius $r=\frac{r_{\text{inner}}+r_{\text{outer}}}{2}$ often gives a good estimate.

Applications of toroidal coils span multiple disciplines. In electrical power systems, toroidal transformers provide efficient voltage conversion with minimal hum. In audio equipment, they are favored for their low magnetic leakage, preventing interference with sensitive circuitry. In power electronics and radio-frequency designs, toroidal inductors help control EMI by keeping flux largely inside the core. In laboratory settings, toroidal magnets can guide charged particles along curved paths, taking advantage of the predictable interior field.

The reference table below shows typical parameter combinations for an air-core (μ = μ₀) estimate. Real cores can produce higher fields for the same N, I, and r.