P
PromptSchola Prompt-based STEM subjects mastery
Sign In Register for Free
Electricity · Physics

Equipotentials are “level curves” of voltage.

An equipotential surface has the same potential everywhere. Moving a charge along it requires no work by the electric field, and field lines must cross equipotentials at right angles.

This topic

Equipotential Surfaces

Define equipotentials, relate them to field lines, and interpret common shapes for typical charge distributions.

Definition
What an equipotential is
Equipotential surfaces are sets of points where the electric potential has the same value. They are surfaces in 3D (and curves in 2D diagrams).
  • Same V everywhere on the surface
  • Only differences in V matter physically
  • Equipotentials can be surfaces or contours
  • Useful for visualization and energy reasoning
Start nano-lesson Equipotentials are “voltage contours.”
Geometry
Relation to electric field lines
Electric field lines are perpendicular to equipotential surfaces. This follows from the fact that the field points in the direction of greatest decrease of V.
  • Field points “downhill” in V
  • No work along an equipotential: ΔV = 0
  • Therefore E has no tangential component on an equipotential
  • In 1D: E = −dV/dx
Start nano-lesson Perpendicular always: E ⟂ equipotential.
Examples
Equipotentials for common charge distributions
Symmetry determines equipotential shapes. For a point charge, equipotentials are concentric spheres. For a uniform field, they are parallel planes.
  • Point charge: concentric spheres (circles in 2D)
  • Dipole: distorted closed curves near charges
  • Uniform field: equally spaced parallel planes
  • Conductors in electrostatic equilibrium: the conductor is an equipotential
Start nano-lesson Symmetry gives you the shape.
Motion
Motion of charges on equipotentials
If a charge moves along an equipotential, the electric field does no work. However, that does not mean the charge cannot move—other forces or initial kinetic energy can drive motion.
  • Along equipotential: ΔV = 0 ⇒ ΔU = qΔV = 0
  • Electric work is zero along the surface
  • Charge motion depends on net force (could be constrained)
  • Interpretation: equipotential is not “force-free” everywhere
Start nano-lesson No work by E along the surface.
Practice
Practice & Exercises
Practice reading equipotential maps, connecting spacing to field strength, and using perpendicularity to infer field direction.
  • Infer E direction from equipotential maps
  • Relate closer spacing to larger |E|
  • Compute ΔU = qΔV for moves between surfaces
  • Identify paths with zero electric work
  • Exam-style equipotential reasoning sets
Open practice sets Draw equipotentials first, then E.