Calculator Inputs
Enter vector field components, a surface normal, surface area, and an optional graph center.
Plotly Graph
The graph shows the field vector, unit normal, and a flat surface patch.
Example Data Table
| Surface | Field Vector F | Normal Vector N | Area | Flux |
|---|---|---|---|---|
| Surface A | (4, 0, 0) | (1, 0, 0) | 3 | 12.000 |
| Surface B | (2, 5, 1) | (0, 0, 2) | 4 | 4.000 |
| Surface C | (3, 3, 0) | (1, 1, 0) | 5 | 21.213 |
| Surface D | (-2, 1, 2) | (0, -3, 0) | 6 | -6.000 |
| Surface E | (1, 2, 2) | (2, 1, 2) | 7 | 14.000 |
Formula Used
For a constant vector field and a flat oriented surface, flux is: Φ = A(F · n̂)
Here, F = (Fx, Fy, Fz) is the field vector, A is the surface area, and n̂ is the unit normal.
If you enter any nonzero normal vector N = (Nx, Ny, Nz), the calculator first converts it into a unit normal: n̂ = N / |N|
The dot product becomes: F · n̂ = Fx·n̂x + Fy·n̂y + Fz·n̂z
The signed flux tells direction. Positive values follow the chosen normal. Negative values move against that normal.
The angle between the field and unit normal is: θ = cos-1((F · n̂) / |F|)
How to Use This Calculator
- Enter a surface name for reporting.
- Type the vector field components Fx, Fy, and Fz.
- Enter the surface normal components Nx, Ny, and Nz.
- Provide the surface area using a positive value.
- Optionally set X₀, Y₀, and Z₀ for graph placement.
- Click Calculate Flux to view results.
- Review flux, angle, dot product, and interpretation.
- Use the CSV or PDF buttons to save results.
Frequently Asked Questions
1. What does 3D flux measure?
3D flux measures how much of a vector field passes through an oriented surface. It combines field direction, surface orientation, and surface area into one signed value.
2. Why does the calculator use a unit normal?
A unit normal keeps orientation separate from surface size. This avoids scaling the flux twice. The surface area already handles the size part of the calculation.
3. What does a negative flux mean?
A negative flux means the field points opposite the chosen normal direction. The field still crosses the surface, but it travels against the orientation you selected.
4. Can I use any normal vector?
Yes. The normal vector can have any nonzero length. The calculator automatically normalizes it before computing the flux.
5. When is the flux equal to zero?
Flux becomes zero when the field is tangent to the surface, when the field magnitude is zero, or when opposite directional effects balance exactly with the chosen normal.
6. Is this suitable for curved surfaces?
This version is designed for a constant field through a flat oriented patch. Curved surfaces usually require integration over many small pieces or a closed-surface theorem approach.
7. What is the angle result showing?
The angle shows the separation between the field vector and the unit normal. Smaller angles produce larger positive flux. Angles above ninety degrees produce negative flux.
8. What do the download buttons save?
The CSV button saves the current numerical results. The PDF button creates a compact report with the main inputs, vector values, and computed flux summary.