Getting the Potential from the Field

Electric field tells you the force per charge at a point; potential tells you the energy per charge stored there. The two are linked by an integral: V_b - V_a = -integral from a to b of E dot dl. Walking along the field direction, E dot dl is positive, so potential drops. Walking against the field costs you potential, the same way climbing a hill against gravity costs you height. The minus sign is not optional bookkeeping; it is what makes field point from high to low potential, the direction a positive charge actually accelerates.

Mechanically, you pick a path from a reference point to your target point, write E as a function of position along that path, dot it with the direction of travel, and integrate. For a uniform field this is just E times distance. For a field that varies with r, like inside a charged sphere, you integrate piece by piece and the answer naturally depends on r. Since potential only cares about endpoints, not the path taken, you can always choose the easiest route, usually a straight radial line.

This is the direct sequel to treating E as the fundamental vector field: now you build the scalar potential from it, and the next topic reverses the process, getting E back by differentiating V. Keep both directions straight: integrate field to get potential, differentiate potential to get field. The shell problem matters because it shows these two operations are not mirror images in a naive sense: zero field only guarantees constant potential, not zero potential, because the integral of zero from the surface inward adds nothing further.