Formula and Method for Electric Potential
Electric potential (also called voltage) is the electric potential energy per unit charge that would be felt by a small positive test charge placed at a point in space. For a single point charge Q, the potential at a distance r away is V = kQ / r, where k is Coulomb's constant (about 8.9875 × 10⁹ N·m²/C² in a vacuum). Unlike electric force, potential is a scalar — it has a sign (positive or negative, matching the sign of Q) but no direction. This calculator also derives the electric field at that point, the potential energy of an optional test charge, and the force that test charge would feel.
Deriving potential, field, energy, and force
Potential follows directly from Coulomb's law by dividing out the test charge: the electric field of a point charge is E = kQ/r², and potential is the field integrated (summed) from infinity in to r, giving V = kQ/r. Equivalently, E is the negative rate of change of V with distance, so for a radial field E = V/r near a point charge. If you place a second, smaller "test" charge q at that location, it acquires potential energy U = qV and feels a force F = qE = kQq/r² (this force expression is identical to Coulomb's law applied to Q and q). When the charges sit in an insulating medium instead of vacuum or air, both V and E are reduced by the medium's relative permittivity εᵣ, so the calculator uses k/εᵣ in place of k throughout.
Working with units and signs
- Potential is measured in volts (V), where 1 V = 1 joule per coulomb (J/C); the electric field is in volts per meter (V/m), identical to newtons per coulomb (N/C).
- Keep the sign of Q: a positive source charge gives positive potential everywhere around it, while a negative source charge gives negative potential — there is no "distance squared" to erase the sign the way there is with force.
- Convert charge to coulombs and distance to meters before plugging into V = kQ/r; this calculator handles that conversion for you from the unit dropdowns (common charges are in the microcoulomb to nanocoulomb range, not whole coulombs).
- The test charge is optional — leave it at a small reference value if you only care about the potential itself, since V does not depend on q at all.
Practical notes and limits
- The point-charge formula assumes Q behaves as if all its charge were concentrated at a single point (or is a uniformly charged sphere, measured from its center at r greater than its radius) — it does not apply inside a charged conductor or to extended charge distributions like plates or lines without integrating over them.
- Potential from multiple charges adds algebraically (superposition): V_total = V₁ + V₂ + ... — simply sum the potential from each source charge, keeping signs, unlike forces which must be added as vectors.
- As r → 0 the point-charge model predicts an unbounded potential, which is a mathematical idealization; real charge carriers have finite size, so treat very small r values as illustrative rather than physical.