Equipotential Surfaces - Relation Between Electric Field and Electrostatic Potential

Step 1 — Setup

Consider two closely spaced equipotential surfaces A and B:

• Potential of surface A = V
• Potential of surface B = V + δV (where δV = small change in potential)
• Let P be a point on surface B
• Let δl = perpendicular distance from P to surface A (along the direction of E)

Step 2 — Work Done Against the Electric Field

• Move a unit positive charge from surface B (point P) to surface A, against the electric field.
• Work done by external force: W = ∣E∣ ⋅ δl
  (Force on unit charge = |E|; displacement = δl; both along same direction)

Step 3 — Work Done = Potential Difference

By definition, this work equals the potential difference V_A − V_B:

Step 4 — Derive the Relation

Step 5 — Vector Form

In three dimensions, the electric field is the negative gradient of potential:

Two Key Conclusions:

1. The electric field is in the direction in which the potential decreases most steeply (the direction of the maximum negative potential gradient).
2. The magnitude of E equals the change in potential per unit displacement normal to the equipotential surface at that point (i.e., E = potential gradient).

When lightning strikes, it follows the path of steepest potential drop — exactly what this derivation describes. Your phone's touchscreen detects the location of your finger by measuring tiny changes in electric potential across a grid. The direction in which potential drops fastest is always the direction of the electric field.

Analogy: Think of electric potential like altitude on a hill. A ball always rolls downhill (from high to low altitude) — this is the direction gravity acts. Similarly, the electric field always points from high to low potential. The steeper the slope of the hill, the stronger the gravitational force. Likewise, the faster the potential drops over a distance, the stronger the electric field.