Ampere's Law B Field Solver

Physics Electricity and Magnetism • Magnetic Fields and Sources

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

3. Ampere’s Law B-Field Solver

Applies Ampère’s law for symmetric paths: $\displaystyle \oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\text{enc}}$ . Computes $\mathbf{B}$ (components), magnitude, and direction, and visualizes an Amperian loop with $d\mathbf l$.

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

Inputs

Symmetry case: Diagram yaw (rotate view):

Infinite wire inputs

Current $I$ (A):

Magnitude of wire current.

Current direction:

Sets $\hat{\phi}$ direction (right-hand rule).

Point $P_x$ (m):

$\rho=\sqrt{P_x^2+P_y^2}$.

Point $P_y$ (m):

Point is in the xy-plane.

Infinite current sheet inputs

Sheet current $K$ (A/m):

Surface current density magnitude.

$\mathbf{K}$ direction:

Sheet is the xy-plane, normal $\hat{n}=+\hat{z}$.

Field side:

Field reverses across the sheet.

Loop length $\ell$ (m):

Used in derivation (cancels out).

Long solenoid inputs

Turns per meter $n$ (1/m):

$\,n=N/L$.

Current $I$ (A):

Ideal: $|\mathbf{B}|=\mu_0 n I$ inside.

Radius $R$ (m):

Inside if $\rho\le R$.

$\rho$ of point (m):

Outside field $\approx 0$ (ideal long solenoid).

Inside field direction:

Set by winding/current (idealized).

Ready

Steps

Choose a case and click Solve.

Diagram (pan/zoom + loop traversal)

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

Drag to pan • Wheel/trackpad/pinch to zoom • Play animates the loop direction.

What the vectors represent

$\mathbf{B}$: magnetic field direction (set by symmetry)
$d\mathbf{l}$: tangent direction along the chosen Amperian loop
Source current: wire $I$, sheet $\mathbf{K}$, or solenoid current $I$
Shaded surface: bounded by the loop; $I_{\text{enc}}$ pierces this surface

Solve to render a case.

The power of Ampère’s law is symmetry: it lets you replace the integral by a simple product like $B(2\pi\rho)$ or $2B\ell$, depending on the geometry.

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Theory

Ampère’s law

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

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

Here:

$\mathbf{B}$: magnetic field (tangent to symmetry lines in many cases)
$d\mathbf{l}$: infinitesimal vector tangent to the chosen closed loop
$I_{\text{enc}}$: net current piercing the surface bounded by the loop
$\mu_0 = 4\pi \times 10^{-7}\,\text{T·m/A}$

Infinite straight wire

Choose a circular Amperian loop of radius $\rho$ around the wire. By symmetry, $|\mathbf{B}|$ is constant on the loop and tangent everywhere:

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

Enclosed current is $I_{\text{enc}}=I$, so:

\[ B(2\pi\rho)=\mu_0 I \quad\Rightarrow\quad B=\frac{\mu_0 I}{2\pi\rho} \]

Direction is azimuthal $\hat{\phi}$, given by the right-hand rule.

Infinite current sheet

For a sheet in the xy-plane carrying surface current density $\mathbf{K}$, symmetry implies uniform $|\mathbf{B}|$ on each side, parallel to the sheet, and opposite directions above vs. below.

A rectangular loop crossing the sheet gives:

\[ \oint \mathbf{B}\cdot d\mathbf{l} = 2B\ell, \qquad I_{\text{enc}}=K\ell \] \[ 2B\ell=\mu_0(K\ell) \quad\Rightarrow\quad B=\frac{\mu_0 K}{2} \]

Direction: above the sheet, $\mathbf{B}$ follows a right-hand rule around $\mathbf{K}$; below, it reverses.

Long solenoid (ideal)

For an ideal long solenoid, $\mathbf{B}$ is approximately uniform inside and nearly zero outside. A rectangular loop with one side inside gives:

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

Enclosed current counts turns: $I_{\text{enc}}=(n\ell)I$, where $n$ is turns per meter. Therefore:

\[ B\ell=\mu_0(n\ell)I \quad\Rightarrow\quad B=\mu_0 n I \]

Outside field is taken as $\approx 0$ in the ideal approximation.

What the diagram shows

$\mathbf{B}$: drawn where symmetry says it is tangent/parallel and constant
$d\mathbf{l}$: drawn tangent to the loop; the Play button animates traversal direction
Shaded region: one possible spanning surface used to define $I_{\text{enc}}$

Steps

Diagram (pan/zoom + loop traversal)

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