EM-08 · Electrostatics field

Potential and Equipotential Surfaces

The potential is a scalar field, the potential energy per unit charge. Following an equipotential surface, see that V stays constant and that the field crosses it perpendicularly, pointing from high to low.

Pick an equipotential shell and read V

Use the slider to pick an equipotential shell around the point charge. Everywhere on one shell the potential V is the same. As the shell moves outward, what happens to V? At what angle does the field cross the shell?

Equipotential radius rr = 1.50

Drag to orbit. The slider sets the equipotential radius.

The potential V of this equipotential

V = kQ/r · potential energy per unit charge (scalar)

V constant on the shell · E ⊥ surface, high V → low V

Relative potential · 60%

Middle

Potential, energy per unit charge

Moving a charge through a field takes work. The potential V is that work divided by the charge, the potential energy per unit charge, measured in volts. The reference is usually infinitely far away (V=0). Around a point charge V = kQ/r: high up close, approaching zero far away. Potential is a scalar field, magnitude only (EM-02).

ObserveV = kQr

A point charge's potential is inverse to distance (1/r).

ChooseV = ?

Potential is the work to move a unit charge.

Equipotential surfaces, where V is the same

Joining points at the same potential traces an equipotential surface. Around a point charge these are points at the same distance r, concentric spheres. Moving anywhere on one equipotential leaves V unchanged, so no work is done, like a contour line on a hiking map: walk along the same height and you neither climb nor descend.

E is perpendicular to equipotentials, downhill

The field always crosses an equipotential at a right angle. If it were slanted, it would have a component along the surface, changing the potential with no work — a contradiction. So E is exactly perpendicular. Its direction is where the potential falls fastest: from high V to low V, downhill. Picture a ball rolling straight down, at right angles to the contour lines.

Fill in|E| = ?

The field falls as 1/r², faster than the potential.

V as 1/r, E as 1/r²

For a point charge the potential falls as 1/r and the field as 1/r²; the field weakens faster. This is no accident: the field is the rate at which the potential changes with distance, its slope. The more steeply the potential drops in some direction, the stronger the field that way. Written precisely, E = -∇V, where the minus sign means downhill. The next unit equips this gradient tool.

On your ownE = ?

The field is the slope of the potential, flipped downhill.

Back to the first screen

On the first screen, sliding the equipotential shell outward lowered the potential V as 1/r. On one shell V is the same everywhere, so moving along it takes no work. The field arrows always pierced the shell at a right angle, pointing from high V (inside) to low V (outside). The potential is a scalar field of magnitude alone, yet its slope sets the vector field that is E.

The potential V is a scalar field, the potential energy per unit charge. For a point charge V = kQ/r, inversely proportional to distance. An equipotential surface is one where V is constant, and moving along it costs no work. The field E is perpendicular to equipotentials, pointing from high V to low V (the steepest descent). Precisely, E = -∇V.

What comes next

The relation that the field is the slope of the potential, E = -∇V, is equipped precisely in the next unit (EM-09, the gradient). The gradient extracts, from a scalar field, the direction of steepest ascent and its steepness. Conversely, the potential is found by integrating the field along a path. Scalar V and vector E pass back and forth like this, by differentiation and integration.