Electric Potential Calculator
Calculate electric potential from one point charge, combine the scalar contributions of several point charges, or find potential difference from work per unit charge.
Inputs
Choose the physical relationship you want to evaluate.
Dimensionless; use 1 for vacuum.
Changes presentation only; the model remains in volts.
Signed source charge; scientific notation is accepted.
Radial distance from the point charge; must be greater than zero.
Enter at least two complete charge – distance pairs. Empty optional rows are ignored.
Signed work or potential-energy change in the selected unit.
Must be nonzero because ΔV = W/q.
Live results
Electric potential
35,950.2071 V
Positive potential relative to the zero-at-infinity convention.
Effective Coulomb constant
8.987551786 × 10⁹
Canonical distance
0.1 m
Calculation detail
| Source | Charge (C) | Distance (m) | Potential contribution (V) |
|---|---|---|---|
| Point charge | 4.000000 × 10⁻⁷ | 0.1 | 35,950.2071 |
How to use this Electric Potential Calculator
What this calculator does
This calculator evaluates electric potential, a scalar quantity measured in volts, using three closely related electrostatic relationships. In Electric potential of a point charge mode, it uses the point-charge expression V = q/(4πε₀εr r). In Electric potential of a system of point charges mode, it applies superposition and adds the signed potential contribution from each source charge. In Potential difference from work and charge mode, it uses ΔV = W/q. The tool is intended for idealized electrostatic calculations: point-like source charges, a uniform relative permittivity for the point-charge modes, and a conventional reference potential of zero at infinity. It does not model extended charge distributions, spatially varying dielectric materials, time-varying electromagnetic fields, or conductor boundary-value problems.
When to use it
Use it when checking homework or lab calculations for the voltage created by a small isolated charge, comparing how distance or dielectric environment changes potential, adding the potential from several point sources at one observation point, or converting a known work-per-charge relationship into potential difference. The underlying point-charge equation and the scalar nature of electric potential are explained in OpenStax's treatment of electrical potential due to a point charge.
How to calculate
- The calculator opens with a ready demonstration: Charge (q) = 4e-7 C, Distance (r) = 10 cm, Relative permittivity (εr) = 1, and output in volts. The result and a validated XLSX workbook are available immediately.
- Choose a Calculation mode. Replace the demonstration values with your own. For a system, enter at least two complete charge and distance rows and choose the shared System charge unit and System distance unit. For potential difference, enter Work done (W) and a nonzero Test charge (q).
- Use Charge unit, Distance unit, Work unit, Test charge unit, and Output unit as needed. Changing an input unit converts the value already present so the represented physical quantity stays the same. The model performs its calculations in SI base units and then converts the displayed result.
- Read the primary result, the two secondary cards, the formula summary, and the Calculation detail table. The table is particularly useful for verifying signed contributions in a multi-charge system.
- Select Download Excel to create a fresh workbook from the current validated inputs and canonical results. Reset clears the demonstration and all user-entered data, restores neutral unit choices, removes calculated results, and may disable Download Excel until you enter another complete valid state.
Input guide
Calculation mode is required and selects the formula family. Relative permittivity (εr) is a required positive dimensionless number in the point-charge and system modes; 1 represents vacuum, and larger values reduce the potential magnitude in inverse proportion. Do not enter zero or a negative value. The vacuum permittivity used to derive Coulomb's constant follows the published NIST CODATA value for ε₀.
Charge (q) is the signed source charge for point-charge mode. It accepts ordinary decimal or scientific notation such as 4e-7, with Charge unit choices C, mC, µC, nC, and pC. Positive q gives positive potential and negative q gives negative potential. Zero is mathematically valid. A common mistake is to type a microcoulomb value as though it were coulombs without changing the unit. Distance (r) is required and must be strictly greater than zero. Its Distance unit can be mm, cm, m, or km; the demonstration 10 cm is 0.1 m.
In system mode, Charge q₁ through Charge q₄ and Distance r₁ through Distance r₄ form up to four source rows. At least two complete rows are required. An entirely blank optional row is ignored, but a half-complete row is rejected because every included source needs both qᵢ and rᵢ. System charge unit applies to all system charges, and System distance unit applies to all system distances. Mixed positive and negative charges are allowed, so contributions can reinforce or cancel.
In potential-difference mode, Work done (W) is a signed energy/work quantity with J, mJ, µJ, or nJ choices under Work unit. Test charge (q) is the signed nonzero charge used in ΔV = W/q, with its own Test charge unit. A zero test charge is invalid because division by zero has no finite result. The sign convention must be interpreted consistently with your definition of work or change in potential energy; the relation between work, potential energy, and voltage is reviewed by OpenStax's potential-energy and potential-difference section. Output unit chooses mV, V, kV, or MV and does not change the underlying canonical result.
Output guide
The header pills report Mode, the current εr when relevant, the chosen Output unit, and calculation Status. The main Electric potential or Potential difference result is a signed voltage estimate under the selected idealized formula. Positive and negative values represent opposite potential signs relative to the chosen reference; zero can result from zero source charge, zero work, or exact cancellation in a multi-charge system. In point and system modes, Effective Coulomb constant is k/εr in N·m²/C². The second card shows Canonical distance for a single charge or Active sources for a system. In work mode the cards report Canonical work and Canonical test charge. The Formula summary shows the active identity with current canonical values.
The Calculation detail table uses canonical SI values. In point and system modes its columns are Source, Charge (C), Distance (m), and Potential contribution (V). Each contribution is exact for the stated point-charge model before display rounding. In potential-difference mode the table changes to Quantity, Canonical value, Unit, and Role, so the work and test charge used in the division remain auditable.
Worked example
For the startup values q = 4 × 10⁻⁷ C, r = 10 cm = 0.1 m, and εr = 1, the calculator uses k = 1/(4πε₀) ≈ 8.987551786 × 10⁹ N·m²/C². Therefore V = kq/r = (8.987551786 × 10⁹)(4 × 10⁻⁷)/0.1 = 35,950.2071 V, or about 35.95 kV. That same canonical value feeds the first-open result, the detail table, and the Excel workbook.
How the model behaves
Electric potential from a point charge falls as 1/r, not 1/r². Doubling the observation distance halves the potential magnitude, while doubling the source charge doubles it. Relative permittivity scales the result inversely: at the same q and r, a medium modeled with εr = 2 gives half the vacuum potential. For multiple point charges, potential is especially convenient because it is a scalar. You add signed values directly rather than resolving vector directions as you would for electric field. This means opposite-sign sources can cancel at a point even when the local electric field is not zero.
The zero of potential is a reference choice. For the isolated point-charge model, the standard convention is V → 0 as r → ∞. Equipotential surfaces around an isolated point charge are spheres centered on the charge; OpenStax discusses this geometry in its equipotential lines and surfaces section.
Interpretation and common mistakes
Keep signs rather than using absolute values unless your problem explicitly asks only for magnitude. In a system, a small net potential can represent genuine cancellation between large positive and negative contributions. Do not interpret that as evidence that no electric field exists. Also distinguish electric potential, measured in volts or joules per coulomb, from electric potential energy, measured in joules. The former belongs to the field at a location; the latter depends on the charge placed there.
Distance must be measured from the source point to the observation point. For a spherically symmetric charged conductor, the point-charge relation can describe the exterior potential when the distance is measured from the center, but this calculator does not model the interior conductor solution. Relative permittivity is treated as one uniform scalar constant; real materials can be dispersive, anisotropic, nonlinear, or spatially nonuniform, so advanced dielectric problems need a field model rather than a single εr entry.
For input syntax, use a dot as the decimal separator. Standard thousands grouping such as 1,000 is accepted, and scientific notation such as 2.5e-6 is accepted. Ambiguous decimal-comma text such as 1,5 is rejected instead of being silently reinterpreted. All calculations must remain finite; values that overflow the browser's numeric range are rejected before results or workbook cells are produced.