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Equipotential Surfaces

Source lecture(s): SC134 Lec5

Intuition

Surfaces where potential is constant need no work to travel along; they are always perpendicular to field lines.

Formal Definition

A surface on which \(V=\) constant everywhere.

Mathematical Formulation

$\(\vec{E} = -\nabla V\)$ Equipotential spacing: closer where \(E\) is stronger.

Derivation

For nearby points on equipotential, \(\Delta V=0\), so \(\vec{E}\cdot\Delta\vec{s}=0\); thus \(\vec{E}\perp\Delta\vec{s}\).

Worked Example

Uniform field has parallel planar equipotentials; point charge has concentric spheres.

Common Mistakes

  • Assuming equipotentials are physical objects.
  • Confusing equipotential spacing with field strength directly.

Quiz

Q1. What direction is \(\vec{E}\) relative to equipotentials?

Answer

Perpendicular, pointing to lower potential.