Work in moving a charge in a field

In a uniform field, moving a charge q a distance d along a field line the field does work A = qEd; if the angle between the force and the displacement is α, A = qEs·cosα = qEΔx, where Δx is the projection of the displacement on the field direction. The work depends not on the shape of the path but only on the initial and final points: the electrostatic field is a potential field and its force is conservative; around a closed loop the work is zero. In terms of potentials: A = q(φ₁ – φ₂) = qU = –ΔWₚ. If a charged particle is accelerated from rest through a voltage U, the field’s work becomes kinetic energy: qU = mυ²/2, so υ = √(2qU/m). In atomic physics energy is measured in electronvolts: 1 eV = 1.6·10⁻¹⁹ J (the energy gained by the elementary charge e = 1.6·10⁻¹⁹ C across 1 V). The mass of an electron is 9.1·10⁻³¹ kg, of a proton 1.67·10⁻²⁷ kg.