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Work-Energy Theorem (Electrostatics)

The work done in moving a charge against the electric field is equal to the change in its electric potential energy: W = -qΔV.
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The statement of the theorem

Let qq be a test charge moving along a path CC parameterized by r⃗(t)\vec{r}(t) from an initial point AA to a final point BB. The electric force acting on qq is F⃗E=qE⃗\vec{F}_E = q\vec{E}, where E⃗\vec{E} is the electric field. Since E⃗\vec{E} is derived from a scalar potential ϕ\phi such that E⃗=−∇ϕ\vec{E} = -\nabla\phi, the force is conservative: F⃗E=−q∇ϕ\vec{F}_E = -q\nabla\phi. The work done WABW_{AB} by the electric force is defined by the line integral:\n\nWAB=∫CF⃗E⋅dr⃗=∫C(−q∇ϕ)⋅dr⃗W_{AB} = \int_C \vec{F}_E \cdot d\vec{r} = \int_C (-q\nabla\phi) \cdot d\vec{r}\n\nBy the Fundamental Theorem of Calculus for Line Integrals, this simplifies to:\n\nWAB=−qϕ(B)+qϕ(A)=q(ϕ(A)−ϕ(B))W_{AB} = -q\phi(B) + q\phi(A) = q(\phi(A) - \phi(B))\n\nFurthermore, the change in potential energy ΔU\Delta U is defined as ΔU=UB−UA=q(ϕ(B)−ϕ(A))\Delta U = U_B - U_A = q(\phi(B) - \phi(A)). Equating the work done to the negative change in potential energy yields the Work-Energy Theorem in Electrostatics:\n\nWAB=−ΔU=q(ϕ(A)−ϕ(B))W_{AB} = -\Delta U = q(\phi(A) - \phi(B))
Source: Wikipedia