Voltage is the per-unit difference in electrical energy potential between points in a circuit.

Since an electric field exerts a force on a charged particle, the particle should accelerate in the direction of the force. However, a moving charge generates a magnetic force at right angles to the direction of motion. This force acts on the other particle causing it to dodge (rotate away from the path of) the moving charge. Add to this further that gravitational forces are in play as well, and the idea of tracing the path of the charge becomes quite uncertain, especially without knowing initial conditions.

Fortunately, because the electric and magnetic fields are at right-angles to each other, they cannot negate each other's force and they can be treated independently in 3D space. To determine the contribution of a force to the increase or decrease in the speed of a mass, one need only consider how much work was accomplished by the force in the direction of its vector. Since electric force acts along the direction joining the two particles, what matters in electricity is how much progress the force accomplishes in reducing or increasing the distance between the particles.

The work done by a battery or power supply is measured based on the principle described in the previous paragraph. Regardless how the power supply generates the electric field that exists between its positive and negative terminals, the supply is measured based on the increase in energy that a unit charge gets from the force applied to it by that electric field and only that electric field. This increase in energy allows it to arrive at terminal after having consumed the same amount of work inside the components of the circuit.

Assume the battery has a uniform electric field of E between its terminals. The force felt by a charge q is F=Eq. A unit charge is one where q=1. So the force is F=E*1=E. If the shortest internal distance from one terminal to the other is d, then the voltage of the batttery is V=Fd=Ed.