In high school physics, how is the formula about electric potential φ=kq/r derived?

The derivation of the electric potential formula φ = kq/r for a point charge in high school physics is a foundational exercise that directly applies the work-energy theorem to Coulomb's law. It begins by defining electric potential at a point as the work done per unit charge by an external agent in bringing a small positive test charge q₀ from infinity to that point without acceleration. The core of the derivation is calculating this work against the repulsive Coulomb force F = kqq₀/r² exerted by the source charge q on the test charge. Because the force varies with distance, this calculation requires integration. The work done to move the test charge an infinitesimal distance dr against the force is dW = F·dr = (kqq₀/r²) dr. The total work is the integral of this expression from r = ∞ to the finite distance r, yielding W = kqq₀ ∫_{∞}^{r} (1/r'²) dr' = kqq₀ [-1/r']_{∞}^{r} = kqq₀ (1/r). The electric potential φ is then this total work divided by the test charge q₀, resulting in φ = W/q₀ = kq/r.

This mathematical process embeds several critical physical assumptions and conventions. The choice of infinity as the reference point where potential is defined to be zero is not arbitrary; it is the only point where the Coulomb force theoretically becomes zero, providing a natural and universal baseline. The derivation explicitly assumes the source charge q is stationary and isolated, ensuring the force field is conservative, which is a prerequisite for defining a scalar potential function. The path of integration is taken along a radial line, but because the electrostatic force is conservative, the result is path-independent, making the potential a property of the point in space alone. The sign of the resulting potential is inherently tied to the sign of the source charge q: a positive q yields a positive potential (repulsive work done on a positive test charge), while a negative q yields a negative potential, indicating the field itself does attractive work.

In the pedagogical context of high school, this derivation serves as a concrete bridge from the vector concept of electric field (E = kq/r²) to the scalar concept of potential, emphasizing energy over force. It establishes the mechanism—integration of the inverse-square law—that connects these two central ideas. The simplicity of the resulting formula, however, belies its specific limitations; it applies only to point charges or spherically symmetric charge distributions outside their radius, a nuance often highlighted through contrast with more complex configurations like dipoles or charged plates. Mastery of this derivation is typically a student's first encounter with the powerful technique of using calculus to sum variable forces, a method immediately extended to calculate gravitational potential.

The primary implication of understanding this derivation is that it frames electric potential not as a standalone formula but as a direct consequence of the inverse-square nature of the Coulomb force. It explains why potential falls off as 1/r, unlike the field which falls off as 1/r², a distinction crucial for solving problems involving work, energy, and the motion of charges. Furthermore, it provides the logical foundation for the principle of superposition for potential, as the scalar sum φ_total = Σ kq_i/r_i follows directly from the work-energy theorem being linear. This derivation is therefore not a mere algebraic exercise but a compact model of field theory reasoning, setting the stage for more advanced concepts like voltage, capacitance, and potential energy in electrostatic systems.