7. Equipotential Line Or Surface Plottet — Theory
1) Potential from multiple point charges
The electric potential is a scalar and adds by superposition:
\( V(\mathbf{r}) = k\sum_i \dfrac{q_i}{r_i} \),
where \(r_i\) is the distance from the point \(\mathbf{r}\) to the \(i\)-th charge.
Near a point charge, \(V\) becomes very large in magnitude; plots often “clip” extremes to keep contours readable.
2) Equipotential lines and surfaces
An equipotential is the set of points where the potential has the same value:
\(V=\text{constant}\).
In 2D this appears as equipotential lines; in 3D as equipotential surfaces.
3) Why equipotentials are orthogonal to field lines
The electric field is the negative gradient of the potential:
\( \mathbf{E}=-\nabla V\).
Gradients point in the direction of steepest increase and are perpendicular to level sets, so \(\mathbf{E}\)
is perpendicular to equipotential curves/surfaces.
In the calculator, turning on E-arrows helps you visually confirm the perpendicular relationship.
4) Conductors as equipotentials
In electrostatic equilibrium, a conductor is an equipotential: the potential is constant throughout its interior
and over its surface. This is why conductors are often drawn as “equipotential surfaces.”
5) Dipole example
A dipole is a pair of charges \(+q\) and \(-q\) separated by a distance \(d\).
Equipotentials “bulge” around \(+q\) and “pinch” near \(-q\), with \(V=0\) forming a characteristic curve between them.
