Bio Savart B Field Integrator

Physics Electricity and Magnetism • Magnetic Fields and Sources

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

1. Magnetic Force On Charge Or Wire Calculator

Compute Lorentz force on a moving charge: $\mathbf{F} = q\,\mathbf{v}\times\mathbf{B}$ or on a current-carrying wire: $\mathbf{F} = I\,\mathbf{L}\times\mathbf{B}$ . Returns components, magnitude, and direction (unit vector), plus a right-hand-rule guide and an electron-beam deflection animation.

Inputs support numbers like 1e-19 and expressions like pi, e, sqrt(2), sin(), cos(), tan(), log(), ln(), abs(). Use * for multiplication.

Mode + Inputs

Force type: Electron-beam deflection animation: Right-hand rule guide:

Moving charge inputs

Charge $q$ (C):

Use negative for electron.

Velocity $v_x$ (m/s):

Vector $ \mathbf{v}=(v_x,v_y,v_z) $

Velocity $v_y$ (m/s):

Leave 0 if along axis.

Velocity $v_z$ (m/s):

Example: $ \hat{x} \Rightarrow (v_x,0,0) $.

Wire inputs

Current $I$ (A):

Direction is in $ \mathbf{L} $.

Length $L_x$ (m):

Vector $ \mathbf{L}=(L_x,L_y,L_z) $.

Length $L_y$ (m):

Example: along $+\hat{x}\Rightarrow (L,0,0)$.

Length $L_z$ (m):

Use signs to set direction.

Magnetic field vector

Field $B_x$ (T):

$ \mathbf{B}=(B_x,B_y,B_z) $

Field $B_y$ (T):

Uniform field assumption.

Field $B_z$ (T):

Example: $0.5\hat{z}\Rightarrow (0,0,0.5)$.

Display + diagram controls

Diagram yaw (rotate view): Arrow scale (visual only):

Direction uses the right-hand rule for positive $q$ or $I$. If $q<0$, the force direction reverses.

Ready

Steps

Enter vectors and click Solve.

Right-hand rule guide

Vectors and cross product direction

Solve to render vectors and cross-product direction.

Tip: curl fingers from $\mathbf{u}$ to $\mathbf{B}$; thumb gives $\mathbf{u}\times\mathbf{B}$. Multiply by $q$ or $I$ to get $\mathbf{F}$.

Electron-beam deflection animation

Illustrative motion in uniform $ \mathbf{B} $ (forces from $ \mathbf{v}\times\mathbf{B} $)

Particle: Mass $m$ (kg):

Animation uses moving charge mode. Switch to “Moving charge” and Solve.

This animation is illustrative: it uses your $q$, $\mathbf{v}$, $\mathbf{B}$, and chosen mass.

For uniform $ \mathbf{B} $ and $ \mathbf{v}\perp\mathbf{B} $, the path is circular with radius $ r=\dfrac{m\,v_\perp}{|q|\,|\mathbf{B}|} $. With a component parallel to $ \mathbf{B} $, the path becomes a helix.

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Theory

Biot–Savart law and what the vectors mean

The magnetic field produced by a steady current is obtained by adding (integrating) the contribution from each infinitesimal current element $d\boldsymbol\ell$:

\[ d\mathbf{B}=\left(\frac{\mu_0}{4\pi}\right) I\,\frac{d\boldsymbol\ell\times\mathbf{r}}{\lVert\mathbf{r}\rVert^{3}} \]

$I$: current (A)
$d\boldsymbol\ell$: a tiny vector tangent to the wire, pointing in the current direction
$\mathbf{r}=\mathbf{P}-\mathbf{r}'$: vector from the source point $\mathbf{r}'$ (on the wire) to the observation point $\mathbf{P}$
$\mu_0=4\pi\times 10^{-7}\ \mathrm{T\,m/A}$: permeability of free space

Direction comes from the cross product: $d\boldsymbol\ell\times\mathbf{r}$ sets the direction of $d\mathbf{B}$. Use the right-hand rule.

Infinite straight wire result

For an infinite straight wire along the $z$-axis, the field magnitude at perpendicular distance $\rho=\sqrt{x^2+y^2}$ is:

\[ |\mathbf{B}|=\frac{\mu_0 I}{2\pi\rho} \]

The direction is azimuthal around the wire. For current in $+\hat{z}$, $\hat{\boldsymbol\phi}=(-y/\rho,\ x/\rho,\ 0)$. If the current reverses, $\mathbf{B}$ reverses.

Numerical integration idea

For most geometries, we approximate the line integral by a midpoint sum:

\[ \mathbf{B}\approx \sum_{k=1}^{N}\left(\frac{\mu_0}{4\pi}\right) I\, \frac{\Delta\boldsymbol\ell_k\times\mathbf{r}_k}{\lVert\mathbf{r}_k\rVert^{3}} \]

Larger $N$ increases accuracy (especially near the wire), but costs more computation. Avoid placing the observation point directly on the path where $\lVert\mathbf{r}\rVert\to 0$.

Right-hand rule reminder

Index finger: $d\boldsymbol\ell$ (direction of current element)
Middle finger: $\mathbf{r}$ (from element to observation point)
Thumb: $d\boldsymbol\ell\times\mathbf{r}$ (direction of $d\mathbf{B}$)