18.3 Electric Potential, Potential Energy, and Equipotential Geometry

Electric potential Ve at a point is work done per unit positive test charge to bring that charge from infinity to the point. Potential energy Ep belongs to a system of charges, and for a specific charge q at a location of potential Ve, Ep = qVe. Potential is scalar, so addition is algebraic rather than vectorial.

Sign handling matters. Positive source charge gives positive potential nearby; negative source gives negative potential. Potential energy sign then depends on both source potential and the sign of the inserted charge. This is why electron and proton scenarios at the same location can produce opposite Ep signs.

Potential methods are usually faster than force-integration methods in exam settings because path details disappear for electrostatic fields. If the field is conservative, the work depends only on start and end potentials. This is the same strategic advantage you used in gravitation.

In uniform fields, potential changes linearly with distance along field direction. In point-charge fields, potential varies as 1/r. That difference in functional form changes graph shape and gradient behavior, so graph interpretation questions can often be solved with qualitative reasoning before any substitution.

Electric field magnitude equals the spatial rate of potential decrease. The field points toward lower potential for a positive test charge model. On a potential-versus-position graph, steeper slope means larger field magnitude.

Equipotential surfaces connect locations with equal potential. Moving charge along one equipotential requires zero work because DeltaV = 0. Field lines are always normal to equipotentials; if they were not, the field would have a tangential component and would do work along the surface, contradicting the definition.

For isolated point charge, equipotentials are concentric spheres. For two-charge systems they become distorted surfaces whose spacing still encodes field strength. Regions where equipotentials are tightly packed correspond to stronger fields.