Equipotentials, E = -nabla V, and capacitors
The electric potential V is a scalar field whose negative gradient gives the electric field: E = -∇V. Equipotential surfaces are perpendicular to field lines everywhere. Working with V rather than E simplifies many problems because scalars are easier to add than vectors.
Contour lines of constant voltage are always perpendicular to electric field lines. Place charges and see both simultaneously.
Click to place charges. Contour lines show equipotentials (blue = negative, white = zero, purple = positive). E-field arrows are always perpendicular to equipotential lines and point from high to low potential.
Key insight: No work is done moving a charge along an equipotential. Work is only done moving across equipotentials, which is why E is perpendicular to them.
E = -∇V means the electric field points in the direction of steepest voltage decrease. On a 3D potential surface, the field arrows point "downhill."
The electric field E = −∇V always points perpendicular to equipotential lines in the direction of steepest voltage decrease. Toggle 3D to see V as a height surface with E arrows pointing downhill.
Key insight: The gradient relationship turns a vector problem (find E) into a scalar problem (find V, then differentiate). This is a massive simplification for complex charge distributions.
Two parallel plates with opposite charges create a nearly uniform electric field between them. The voltage drops linearly from one plate to the other. Capacitance C = εA/d stores energy U = ½CV².
Uniform E-field between parallel plates with fringing at edges. Dashed horizontal lines are equipotentials. Adjust separation and charge to see how E, V, C, and stored energy U change.
Key insight: Capacitors store energy in the electric field itself. Doubling the voltage quadruples the stored energy. This is why capacitors are essential in electronics.