EM-08 · Electrostatics field

Potential and Equipotential Surfaces

Potential is a scalar field that gives the potential energy per unit charge. Following an equipotential surface, you confirm that V is constant on it and that the electric field crosses it at right angles, pointing from high to low.

Pick an equipotential shell and read its V

The gold dot in the middle is a point charge, and the faint spheres around it are equipotential shells. Pick one shell with the slider and it lights up, and the indicator shows the potential on it. The potential is the same everywhere on one shell, and it drops as you move the shell outward. You can also see at what angle the electric field, drawn as arrows, passes through the shell.

Equipotential radius rr = 1.50

Drag to rotate; the slider changes the radius of the equipotential surface.

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 an electric field takes work. Potential is that work divided by the charge, the potential energy per unit charge, and its unit is the volt (V). Usually the point infinitely far away is taken as the reference (V=0). Around a positive point charge, V = kQ/r, so it is higher up close and approaches 0 farther away. For a negative point charge the sign is reversed, so it is lower up close. Potential is a scalar field with magnitude only (EM-02).

ObserveV = kQr

The potential of a point charge is inversely proportional to distance (1/r).

ChooseV = ?

Potential is the work needed to move a unit charge.

Equipotential surfaces, where V is the same

Joining points of equal potential gives an equipotential surface. Around a point charge these are points at the same distance, that is, concentric spheres. On the same equipotential surface the potential does not change wherever you move, so the electric field does no work. It is 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 electric field always crosses an equipotential surface at right angles. If it were slanted, it would have a component along the surface, and moving a charge along the surface against that component would take work and change the potential. That contradicts the definition of an equipotential surface, so E is exactly perpendicular. Its direction is the one in which the potential falls fastest, from high potential down to low. Picture a ball rolling down at right angles to the contour lines.

Fill in|E| = ?

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

V as 1/r, E as 1/r²

For a point charge, the potential falls off as 1/r and the field as 1/r². The field weakens faster, and this is no accident: the electric field is the rate at which the potential changes with distance, that is, its slope. The more steeply the potential drops in some direction, the stronger the field that way. That is why equipotential surfaces drawn at equal potential steps are crowded near the charge and spread out farther away. Written precisely, this relation is E = -∇V, and the minus sign means the downhill direction. In the next unit you equip this gradient tool.

On your ownE = ?

The electric field is the slope of the potential, flipped to point downhill.

Back to the first screen

On the first screen, the farther out you moved the equipotential shell with the slider, the lower the potential, following 1/r. On one shell the potential is the same everywhere, so moving along it takes no work. The field arrows always pierced the shell at right angles, pointing from high V (inside) to low V (outside). Potential is a scalar field with magnitude only, yet its slope sets the electric field, a vector field.

Potential V is a scalar field giving the potential energy per unit charge. For a point charge, V = kQ/r, inversely proportional to distance. An equipotential surface is a surface of equal potential, and moving a charge on it takes no work. The electric field E points perpendicular to equipotential surfaces, from high V to low V (the steepest way downhill). Written precisely, E = -∇V.

What comes next

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