Solenoid or Toroid Magnetic Field Tool

Physics Electricity and Magnetism • Magnetic Fields and Sources

Written by STEM Calculators Team Published January 17, 2026 Updated February 24, 2026

4. Solenoid/Toroid Magnetic Field Tool

Computes magnetic field using Ampère’s law: $B=\mu_0 n I$ for a (long) solenoid ($n=N/L$), or $B=\mu_0 N I /(2\pi r)$ for a toroid (inside the core region). Includes a finite-length (fringe) on-axis approximation for solenoids and an interactive diagram.

Inputs accept 1e-7, pi, sqrt(2), sin(), cos(), tan(), log(), ln(), abs(). Use * for multiplication.

Mode + Inputs

Geometry: Animation (Play button): Diagram yaw (rotate view): Solenoid model:

Solenoid parameters

Turns $N$:

Total turns on the solenoid.

Length $L$ (m):

Turns per length: $n=N/L$.

Radius $R$ (m):

Used for finite on-axis approximation.

Current $I$ (A):

Sign flips field direction.

On-axis point $z$ (m):

Measured from solenoid center (for finite model).

Axis direction:

This sets the displayed axial direction; reversing $I$ flips it.

Ready

Steps

Choose geometry and click Solve.

Diagram (pan/zoom + animation)

$\mathbf{B}$, $d\mathbf{l}$, and the Amperian loop

Drag to pan • Wheel/trackpad/pinch to zoom • Play animates loop traversal

What the vectors represent

$\mathbf{B}$: magnetic field direction (axial for solenoid, circular for toroid)
$d\mathbf{l}$: tangent direction along the chosen Amperian loop
Source: current $I$ and turns ($n=N/L$ or $N$)
Shaded surface: spanning surface used to define $I_{\text{enc}}$

Solve to render the diagram.

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Theory

Ampère’s law (core idea)

Ampère’s law relates the circulation of the magnetic field around a closed path to the current enclosed by any surface spanning that path:

\[ \oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\text{enc}} \]

where $\mu_0 = 4\pi\times 10^{-7}\,\mathrm{T\cdot m/A}$. The trick is to choose an Amperian loop that matches the symmetry so that $\mathbf{B}\cdot d\mathbf{l}$ becomes simple.

Long solenoid: $B=\mu_0 n I$

For a long solenoid, the field is approximately uniform inside and approximately zero outside. Choose a rectangular Amperian loop with one side of length $\ell$ inside the solenoid and the other outside.

\[ \oint \mathbf{B}\cdot d\mathbf{l} \approx B\ell \]

The enclosed current counts how many turns pierce the surface. The number of turns intersected is $n\ell$, where $n=N/L$ is turns per meter, so:

\[ I_{\text{enc}}=(n\ell)I \] \[ B\ell=\mu_0(n\ell)I \quad\Rightarrow\quad B=\mu_0 n I \]

Direction is along the solenoid axis, set by the right-hand rule; reversing $I$ reverses $\mathbf{B}$.

Finite solenoid (fringe field approximation on axis)

Real solenoids are finite, so the field is weaker near the ends and outside. A common on-axis approximation for a solenoid of radius $R$ and length $L$ is:

\[ B(z)=\frac{\mu_0 n I}{2}\left( \frac{z+L/2}{\sqrt{R^2+(z+L/2)^2}} - \frac{z-L/2}{\sqrt{R^2+(z-L/2)^2}} \right) \]

At the center $z=0$, this reduces to:

\[ B(0)=\mu_0 n I\;\frac{L/2}{\sqrt{(L/2)^2+R^2}} \]

The factor $\dfrac{L/2}{\sqrt{(L/2)^2+R^2}}<1$ captures “fringe reduction”: shorter, fatter solenoids have more fringing.

Toroid: $B=\mu_0 N I /(2\pi r)$

For a toroid (donut-shaped winding), magnetic field lines circulate around the core. Use a circular Amperian loop of radius $r$ concentric with the toroid.

\[ \oint \mathbf{B}\cdot d\mathbf{l} = B(2\pi r) \]

The loop encloses $N$ turns carrying current $I$, so $I_{\text{enc}}=NI$. Therefore:

\[ B(2\pi r)=\mu_0 NI \quad\Rightarrow\quad B=\frac{\mu_0 NI}{2\pi r} \]

This ideal result is mainly valid inside the core region (between the inner and outer radius). Outside, the field is much smaller (often approximated as $0$ in the ideal model).

What the diagram shows

$\mathbf{B}$: axial inside solenoid; circulating inside toroid.
$d\mathbf{l}$: tangent to the selected Amperian loop; the Play button animates traversal.
Shaded surface: one possible spanning surface used for $I_{\text{enc}}$.

University extension (preview): self-inductance

Once you know $B$, you can estimate magnetic flux $\Phi_B$ and self-inductance $L$ using $\Phi_B=\int \mathbf{B}\cdot d\mathbf{A}$ and $L = N\Phi_B/I$. Finite length and core materials (relative permeability) become important at this level.

Steps

Diagram (pan/zoom + animation)

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