1) Superposition principle (vector addition)
If several independent charge sources are present, the total electric field is the sum of the individual fields:
Because \(\mathbf{E}\) is a vector, you add components: \(E_x=\sum_i E_{x,i}\), \(E_y=\sum_i E_{y,i}\).
2) Point charge model (Coulomb field)
A point charge \(q_i\) at position \(\mathbf{r}_i\) produces:
Direction: away from \(+\) charges and toward \(-\) charges, along the line connecting the charge to the point.
3) Infinite line charge perpendicular to the plane
For an infinite line charge (density \(\lambda\)) perpendicular to the 2D plotting plane, the magnitude is
\(E=\lambda/(2\pi\varepsilon_0 r)\). Using \(k=1/(4\pi\varepsilon_0)\), this can be written as:
Here \(\mathbf{r}_i\) is the point where the line pierces the plane (its “location” in the plot).
4) Magnitude and direction at a query point
Once the net components are found:
\(\theta\) is measured from the \(+x\) axis toward \(+y\) (standard math convention).
5) Field maps and “neutral points”
A field map shows \(\mathbf{E}\) as arrows across a grid.
A neutral point (in 2D) is a location where the net field is (approximately) zero:
Neutral points may be multiple or outside your view window. Numerically, the tool searches for a minimum of \(|\mathbf{E}|\) in the current view.
6) Numerical notes (why “softening” exists)
- Point-charge fields blow up as \(r\to 0\). Line-charge fields also grow large near the line.
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For stable visualization, the calculator uses a small “softening radius” so arrows don’t explode near sources.
This is mainly a plotting/robustness trick.
- For high-accuracy work near charges, use analytic work or refined numerical methods (and physical charge distributions).
