2.6 Electric Field and Potential

Field as the Negative Gradient of Potential

The electric field is the negative spatial derivative of electric potential: E = -dV/dr (or E = -∇V in 3D). This means the field points in the direction of steepest decrease in potential, and its magnitude reflects how fast V changes with position. Regions of closely spaced equipotential surfaces have strong fields; widely spaced ones have weak fields.

Getting Potential from the Field

Conversely, potential difference is found by integrating the field: ΔV = -∫E·dr along a path between two points. This is path-independent since E is a conservative field. For a uniform field, this simplifies to ΔV = -E·d, where d is displacement along the field direction.

Equipotentials and Field Lines

Equipotential surfaces are always perpendicular to electric field lines, since moving along an equipotential does no work (E·dr = 0 means E ⊥ dr). Field lines point from high to low potential. No work is done moving a charge along an equipotential surface.

Key terms

E = -dV/dr

Electric field is the negative gradient of potential; field points toward decreasing potential.

ΔV = -∫E·dr

Potential difference is the negative line integral of the electric field along any path between two points.

Equipotential surface

A surface where potential is constant; always perpendicular to electric field lines.

Uniform field potential formula

ΔV = -Ed, where d is displacement along the direction of the field.

Conservative field property

Work done by the electric field (and thus ΔV) is path-independent, depending only on start and end points.

Try an AP-style question

V(x) = 3x² (V). What is E_x at x = 2 m?

1. 12 V/m
2. −6 V/m
3. −12 V/m
4. 6 V/m