Equipotential surfaces

An equipotential surface is the geometric locus of points where the potential $V$ takes the same value. In two dimensions we more properly speak of equipotential lines. If field lines show the direction of the force, equipotentials show the “relief” of the potential: they are like the contour lines of a topographic map, joining points at the same height.

The fundamental properties are three:

• Equipotential surfaces are perpendicular to field lines at every point. If this were not so, the field would have a component along the surface and the potential would not be constant on it.
• Moving a charge along an equipotential requires no work: $\Delta V = 0 \Rightarrow W = 0$. Moving along an equipotential is like walking on flat ground, neither climbing nor descending.
• Where equipotentials are denser (closer together), the field is stronger, because the same potential jump occurs over a smaller distance.

This last property is particularly useful: it allows us to read the field strength directly from an equipotential diagram, without having to compute any derivative.

Rule of thumb

$|\vec{E}| \approx \left|\frac{\Delta V}{\Delta d}\right|$ where $\Delta d$ is the distance between two nearby equipotentials with potential difference $\Delta V$.

Positive point charge: the equipotentials (dashed circles, values in volts) are perpendicular to the field lines (blue arrows). The equipotentials spread out as $r$ increases because $V = kQ/r$ decreases slowly.

Dipole: the equipotentials (golden dashed) are closed curves around each pole. The median plane perpendicular to the axis is the $V=0$ equipotential. The field lines (blue arrows) always intersect the equipotentials perpendicularly.

Parallel-plate capacitor: the field is uniform (parallel, equally spaced lines). The equipotentials (golden dashed) are planes parallel to the plates. The distance between equipotentials is constant because $|\vec{E}| = |\Delta V / \Delta d|$ is constant.

Collegamenti

Argomenti: Campo elettrico e potenziale · Elettrostatica Concetti: Potenziale elettrico · Campo elettrico

Esercizi collegati: True or false on field and potential · Cloud-to-ground potential difference · Field lines never cross