Theory: Electric Potential (Point Charges)
Electric potential \(V\) is a scalar that measures electric potential energy per unit charge.
With the standard convention \(V(\infty)=0\), the potential due to a single point charge \(q\) at distance \(r\) is:
Here \(k=\dfrac{1}{4\pi\varepsilon_0}\approx 8.99\times 10^9\ \mathrm{N\,m^2/C^2}\).
The sign of \(V\) follows the sign of \(q\): positive charges produce positive potential; negative charges produce negative potential.
Superposition
For multiple point charges \(q_i\) located at positions \(\mathbf{r}_i\), the total potential at \(\mathbf{r}\) is the sum:
Units and interpretation
Singularity at \(r=0\)
Since \(V\propto 1/r\), the potential diverges as \(r\to 0\).
In the calculator, a small guard \(r_{\min}\) is used to avoid numeric blow-ups when drawing the potential map.
This does not change the underlying physics; it simply prevents division by zero.
Conductors
In electrostatic equilibrium, an ideal conductor is an equipotential: \(V\) is constant throughout the conductor.
This calculator focuses on point-charge superposition, but this idea explains why conductors “level out” potential differences internally.
Sample
For \(q=5\times10^{-6}\,\mathrm{C}\) at the origin and the point \((0.2,0)\,\mathrm{m}\),
\(r=0.2\,\mathrm{m}\) so:
