1) Electric field of point charges (superposition)
For point charges \(q_i\) at positions \(\mathbf{r}_i\), the electric field at \(\mathbf{r}\) is
For sketching field lines, the constant \(k\) only scales the magnitude, not the direction, so the plot focuses on direction.
2) What a field line means
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The field line direction at each point is tangent to \(\mathbf{E}\).
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Lines generally start on positive charge and end on negative charge (or at infinity).
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Line density (how many lines) is proportional to \(|q|\) in many textbook sketches.
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Field lines never cross (because \(\mathbf{E}\) has a single direction at each point, except where \(\mathbf{E}=\mathbf{0}\)).
3) How the sketch is generated (direction-field integration)
A field line can be viewed as a solution curve of a direction-field ODE. A convenient choice is to integrate the
unit direction of the field:
Using the unit direction keeps steps stable (you move a fixed distance each step), while still following the field direction.
Seeding
To produce multiple lines, the tool places seed points on a small circle around each source charge.
The number of lines is based on your “Lines per unit \(|q|\)” setting:
Stopping
A traced line stops when it gets close to any charge (to avoid singular behavior) or leaves the visible window.
This is controlled by “Stop radius near charges” and “Max steps per line”.
4) Classic example: dipole \((+q,-q)\)
If you place \(+q\) at \((-1,0)\) and \(-q\) at \((1,0)\), lines emerge from the positive charge and curve into the negative charge.
The pattern is symmetric about the \(x\)-axis.
If you increase \(|q|\) or the “Lines per unit \(|q|\)”, you will see a denser field-line sketch (more “flux tubes”).
5) Limitations (important)
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This is a qualitative visualization. It is not a replacement for precise field magnitude plots.
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Near charges, the true field magnitude diverges; the tool uses practical stopping/softening rules so the plot remains usable.
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When \(\mathbf{E}\approx \mathbf{0}\) (neutral points), direction becomes numerically sensitive and line behavior may look noisy.
Tips for your website UI
- Drag charges live and re-solve to show superposition intuitively.
- Show/hide direction arrows to clarify flow from \(+\) to \(-\).
- Add a “Flux tube count” callout: total lines launched \(\propto \sum |q_+|\).
