How is the formula for electrical work W = Uq derived?

The formula for electrical work, expressed as W = Uq, where W is work, U is electric potential (voltage), and q is electric charge, is derived directly from the fundamental definition of work in physics and the definition of electric potential. In classical mechanics, work is defined as the product of a force and the displacement of an object in the direction of that force (W = Fd cosθ). For a charge moving in an electric field, the force is the electric force, F = qE, where E is the electric field strength. Substituting this into the work definition gives the differential work dW = qE·dl for a small displacement dl. The total work done in moving a charge between two points is then the line integral W = q ∫ E·dl. By definition, this line integral is the negative of the change in electric potential, ΔU. Therefore, the work done by the external agent against the electric field to move the charge without acceleration is W = qΔU. In its common simplified form, where U represents the potential difference (voltage) and we consider the magnitude of work done or energy transferred, it becomes W = Uq.

This derivation hinges critically on the definition of electric potential as work per unit charge. Electric potential at a point is defined as the work done by an external agent in bringing a unit positive test charge from a reference point (often infinity) to that point without any acceleration. Consequently, the potential difference U between two points is precisely the work required per unit charge, U = W/q. Rearranging this definitional relationship immediately yields W = Uq. This makes the formula less a derived result and more a direct consequence of how the volt is defined: one volt is one joule per coulomb. The formula elegantly encapsulates that the electrical work or energy change associated with moving a charge is simply the charge multiplied by the "electrical pressure" or potential through which it moves.

The practical application and interpretation of W = Uq depend on the context of the system considered. In electrostatics, it typically represents the potential energy gained by a charge when moved against a field or the work an external force must supply. In circuit theory, it calculates the electrical energy transferred when a charge q moves through a component with a voltage drop U; this is the basis for the more common power formula P = UI, since current I is dq/dt. The derivation assumes the charge q is point-like and that the potential U is constant during the movement or is an average value, which is valid for most macroscopic circuit analysis. For a varying potential, the calculation requires integration, reflecting the more general integral form.

Thus, W = Uq is a foundational, almost axiomatic, relationship in electromagnetism. Its derivation is straightforward from first principles, but its profound utility lies in linking the abstract concepts of field and potential to measurable energy transfer. It serves as the critical bridge between the microscopic force on a charge and the macroscopic energy accounting in electrical systems, from capacitors storing energy to the operation of every electrical load. The formula's simplicity belies its comprehensive role in quantifying electrical energy conversion across all scales of physics and engineering.