Electric Potential Calculator
Enter a charge and a distance to instantly calculate the electric potential (voltage) at that point in space using Coulombs law constant. Switch between coulombs, microcoulombs or nanocoulombs, pick metre or centimetre distances, and see the full step-by-step working. Use the Mode selector to reverse-solve for unknown charge or unknown distance, calculate potential difference between two points, or add a second charge and apply the superposition principle.
Formula
Worked example
A point charge of +2 µC in air at a distance of 10 cm: V = (8.9876e9 * 2e-6) / (1 * 0.1) = 17975 / 0.1 = 179,751 V (about 180 kV). The electric field at the same point is E = V / r = 179,751 / 0.1 = 1,797,510 V/m (about 1.8 MV/m).
What is electric potential?
Electric potential (symbol V, unit: volt) is the amount of electric potential energy a unit positive charge would possess if placed at a given point in space. It is a scalar quantity, meaning it has magnitude but no direction, which makes it easier to work with than the electric field vector when analysing complex charge distributions. The potential created by a single point charge falls off as 1/r (inversely with distance), while the electric field falls off more steeply as 1/r^2. Electric potential is always measured relative to a reference - in the point-charge formula, that reference is infinity (zero potential at an infinite distance).
The formula V = kq / r explained
For a single point charge q in a medium with relative permittivity er, the electric potential at a distance r is V = k_e * q / (er * r), where k_e = 8.9876 * 10^9 N*m^2/C^2 is Coulombs constant (equal to 1 / (4 * pi * epsilon_0)). A positive charge produces a positive (repulsive) potential; a negative charge produces a negative (attractive) potential. In a medium other than vacuum the potential is reduced by the factor er, because the medium partially screens the charge. For a system of several point charges, the superposition principle says the total potential is the algebraic (signed) sum of each individual contribution, V = sum(k_e * q_i / r_i), because potential is a scalar.
Electric potential difference and work
Potential difference (deltaV = V_A - V_B) is the work done per unit charge moving a positive test charge from point B to point A. If you move a charge Q from point B to point A, the work done on the charge is W = Q * deltaV. This is the basis of how capacitors store energy, how batteries drive current, and how particle accelerators impart kinetic energy. Between two distances r1 and r2 from the same charge, deltaV = k_e * q * (1/r1 - 1/r2). Moving toward the charge (decreasing r) does positive work on a positive test charge; moving away does negative work.
Reverse-solving: finding charge or distance
The formula V = kq / r can be rearranged in two useful ways. To find the charge that produces a known potential at a known distance: q = V * r / k_e. To find the distance at which a known charge produces a given potential: r = k_e * q / V. These reverse-solve modes are useful in sensor design, where the detector voltage is known and the source charge (or the placement distance) must be inferred. They also appear in problems involving Geiger counters, Van de Graaff generators, and coaxial cable electrostatics.
Relative permittivity of common materials
| Material | Relative permittivity er | Effect on V |
|---|---|---|
| Vacuum / air | 1.0 | Baseline |
| Teflon (PTFE) | 2.1-2.5 | About 50% weaker |
| Silicon dioxide | 3.9 | About 74% weaker |
| Glass | 4-10 | About 75-90% weaker |
| Mica | 5-8 | About 80-88% weaker |
| Nylon | 3.5 | About 71% weaker |
| Distilled water | 80 | About 99% weaker |
| Ethanol | 24.3 | About 96% weaker |
The relative permittivity (dielectric constant) reduces the effective Coulomb constant by the factor er. Higher er means weaker potential at the same distance.
Frequently asked questions
What is the difference between electric potential and electric field?
Electric potential V is a scalar (a single number, in volts) that tells you the potential energy per unit charge at a point. Electric field E is a vector (it has magnitude and direction, in V/m or N/C) that tells you the force per unit charge. They are related by E = -dV/dr: the electric field points in the direction of steepest decrease in potential. For a point charge, V = kq/r and E = kq/r^2, so E = V/r for that geometry.
Why is electric potential a scalar while the electric field is a vector?
The electric field is derived from Coulombs force law, which is a force (inherently directional). The electric potential is derived from energy, which is a scalar quantity. When you sum contributions from multiple charges, scalar addition is much simpler: you just add the signed magnitudes. Vector addition requires resolving components in each direction. This is one reason potential is often more convenient for multi-charge problems.
What happens to the potential inside a conducting sphere?
Inside a spherical conductor, all excess charge resides on the surface, and the electric field inside is exactly zero. Because E = -dV/dr = 0 everywhere inside, the potential is constant throughout the interior and equals the potential at the surface. This is why the interior of a Faraday cage is shielded from external electric fields, and why high-voltage Van de Graaff generator spheres can store large charges without an internal field.
Does the medium (material) affect electric potential?
Yes. In a material with relative permittivity er (the dielectric constant), the effective Coulomb constant becomes k_e / er, so the potential at the same charge and distance is reduced by the factor er. Water (er about 80) is highly effective at screening charges, which is why ionic compounds dissolve so readily in it: the attractive potential between ions is reduced by a factor of 80 compared with vacuum.
How is electric potential used in everyday technology?
Electric potential is fundamental to capacitors (energy stored = C * V^2 / 2), batteries (cell potential is the voltage difference between terminals), semiconductors (the built-in potential at a p-n junction governs diode behavior), and medical devices (an ECG measures the potential difference across the heart). High-voltage transmission lines carry electrical energy long distances at very high potential (hundreds of kilovolts) to keep current low and minimise resistive losses.