Relation between electric field and potential

Consider a positive charge q fixed at the origin, and a unit positive test charge moved a small distance dx toward q, in the same direction as the electric field E produced by q. Moving the unit charge against the electric field requires a small amount of work dW = -E dx (negative because work is done against the field when moving in the field's own direction reduces potential energy for this sign convention, or more precisely, dW here represents the work done by an external agent moving against the electrostatic force). This same small amount of work equals, by the earlier definition of potential, the resulting small change in potential dV over the displacement dx. Equating the two gives the fundamental differential relation E = -dV/dx (for motion along a single direction x), or more generally, in three dimensions, E = -(partial V/partial x) i-hat - (partial V/partial y) j-hat - (partial V/partial z) k-hat -- the electric field equals minus the gradient of the electric potential. Physically, this says the electric field always points in the direction along which the potential decreases most rapidly, and its magnitude equals the rate of that decrease per unit distance; equivalently, moving in the direction of E always means moving toward lower potential, while moving against E always means moving toward higher potential. This relation gives a completely equivalent, and often more convenie …