Magnetic Flux Tool

Physics Electricity and Magnetism • Electromagnetic Induction and Inductance

Written by STEM Calculators Team Published January 18, 2026 Updated February 24, 2026

2. Magnetic Flux Tool

Calculates magnetic flux through a surface. For uniform fields: $\Phi_B = \mathbf{B}\cdot\mathbf{A} = B\,A\cos\theta$ . Use the visual to see how the normal component $B_\perp=B\cos\theta$ controls the amount of field “threading” the surface.

Units: tesla (T), meters (m), weber (Wb). Inputs accept 1e-3, pi, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Inputs

Field + angle

Field magnitude $B$ (T):

Uniform magnetic field magnitude.

Angle $\theta$ (deg):

Angle between $\mathbf{B}$ and surface normal $\hat{\mathbf{n}}$.

Turns $N$ (optional):

If $N$ turns, total flux linkage is $N\Phi_B$.

Output mode:

Choose $\Phi_B$ or $N\Phi_B$.

Surface (in the $x$-$y$ plane)

Area shape:

Surface lies in the $x$-$y$ plane; normal $\hat{\mathbf{n}}$ points out of plane (+$z$).

Radius $R$ (m):

Circle area: $A=\pi R^2$.

Visual controls

Line density (visual):

Density is proportional to $\lvert B\cos\theta\rvert$ (threading component).

Animation speed:

Moves markers along the flux lines when Play is ON.

Show guides:

Shows $\hat{\mathbf{n}}$, $\mathbf{B}$ projection, and $\theta$.

Pan/zoom: drag to pan • mouse wheel / trackpad / pinch to zoom • “Reset view” restores default.
Play animates markers along the flux lines (qualitative).

Ready

Steps

Enter values and click Solve.

Flux threading visual

Surface, normal $\hat{\mathbf{n}}$, field $\mathbf{B}$, and qualitative flux lines

What the vectors mean

Surface (circle/rectangle) in the $x$-$y$ plane
Surface normal $\hat{\mathbf{n}}$ (out of plane)
Magnetic field $\mathbf{B}$ (shown by direction + lines)
Angle $\theta$ between $\mathbf{B}$ and $\hat{\mathbf{n}}$
Flux lines (density $\propto \lvert B_\perp\rvert=\lvert B\cos\theta\rvert$, qualitative)

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2. Magnetic Flux Tool — Theory

Magnetic flux through a surface measures how much magnetic field “threads” that surface:

\Phi_B=\int_S \mathbf{B}\cdot d\mathbf{A}

Here, $d\mathbf{A}=\hat{\mathbf{n}}\,dA$ is the directed area element (normal vector $\hat{\mathbf{n}}$ times area). If $\mathbf{B}$ is uniform and makes an angle $\theta$ with $\hat{\mathbf{n}}$, then:

\Phi_B = B\,A\cos\theta

Units

$B$: tesla (T)
$A$: square meters (m$^2$)
$\Phi_B$: weber (Wb), where $1\,\mathrm{Wb}=1\,\mathrm{T\cdot m^2}$

Uniform-field (closed form)

For a simple loop in a uniform field:

Compute the surface area $A$.
Compute $\cos\theta$ (make sure $\theta$ is in radians if using trig directly).
Multiply: $\Phi_B=B\,A\cos\theta$.

Areas used in this tool

Circle (radius $R$): $\;A=\pi R^2$
Rectangle (width $w$, height $h$): $\;A=w\,h$

Non-uniform field (numerical integration)

If the field magnitude varies over the surface, the tool approximates:

\Phi_B \approx \sum_{i} \Big(B(x_i,y_i)\cos\theta\Big)\,\Delta A

This is a grid-based numerical approximation over the chosen shape. Increasing the resolution improves accuracy but may run slower.

Sign convention

Flux can be positive or negative depending on the chosen surface normal direction. This tool uses the normal shown in the diagram. If $\theta=0^\circ$, the field is aligned with the normal and flux is positive; if $\theta=180^\circ$, flux is negative.

Sample

$B=0.4\,\mathrm{T}$, $A=0.05\,\mathrm{m^2}$, perpendicular ($\theta=0$):

\Phi_B = B\,A\cos 0 = 0.4 \cdot 0.05 \cdot 1 = 0.02\ \mathrm{Wb}

Website tip

Flux lines are qualitative: more dense lines indicate larger $|\Phi_B|$. The “Play” animation illustrates “threading” through the surface.

Steps

Flux threading visual

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