Electric Potential & Potential Energy Calculator

Calculate electric potential V=kQ/r and potential energy U=qV for point charges.

For potential energy U=qV
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Electric Potential Versus Electric Potential Energy

Electric potential V describes the electric energy available per unit charge at a location, while electric potential energy U belongs to a specific pair or system of charges. For a single point source Q with zero potential chosen at infinity, V = kQ/r. Once a test charge q is placed there, its potential energy is U = qV = kQq/r.

This distinction explains why potential is measured in volts, where 1 V = 1 J/C, while potential energy is measured in joules. Potential is a scalar, so potentials from multiple sources add algebraically. The sign is physically meaningful: a positive source creates positive V, a negative source creates negative V, and U depends on the product Qq. Like charges give positive pair potential energy; opposite charges give negative potential energy when infinity is the zero reference.

V = kQ/r     U = qV = kQq/r
SymbolMeaningWhy it appears / units
QSource chargeCoulombs; determines sign and magnitude of V
qCharge placed in the potentialCoulombs; converts V into potential energy U
rSeparation from point sourceMeters; potential falls as 1/r rather than 1/r2
V, UPotential and potential energyVolts and joules respectively

Electric field and potential are related but not interchangeable. For a point charge, |E| falls as 1/r2 while |V| falls as 1/r. Moving a charge through a potential difference changes its potential energy by ΔU = qΔV. The electric force can then convert that potential-energy change into kinetic energy.

Worked Examples

Example 1: 1μC charge, 10cm away
V=8.99e9×1e-6/0.1
Result: 89,900 V = 89.9 kV
Large voltage from small charge
Example 2: Proton at Bohr radius: Q=1.6e-19, r=5.29e-11
V=8.99e9×1.6e-19/5.29e-11
Result: 27.2 V
Hydrogen atom ground state potential
Example 3: Potential from a positive source
Q = +2.0 μC, r = 0.50 m → V = kQ/r
Result: V ≈ +35,950 V
The large voltage reflects energy per coulomb; it does not by itself specify how much energy a particular test charge has.
Example 4: Negative test charge in that potential
q = −3.0 nC, V ≈ +35,950 V → U = qV
Result: U ≈ −1.08 × 10−4 J
The negative sign reflects attraction between opposite charges when zero potential energy is defined at infinite separation.

Common Mistakes

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Using V and U as if they were the same quantity

Potential V is energy per charge in J/C. Potential energy U is the energy for a particular charge and equals qV.

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Dropping charge signs

Signs determine whether potential and pair potential energy are positive or negative. Use magnitudes only when the question explicitly asks for magnitude.

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Using an inverse-square dependence for potential

Point-charge potential varies as 1/r. Electric-field magnitude varies as 1/r2.

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Measuring separation from the wrong point

For the point-charge model, r is the distance from the source charge location, or from the center for an exterior spherically symmetric source.

Frequently Asked Questions

Potential vs field?
Electric field E=-dV/dr (gradient of potential). Field points from high to low potential. V is scalar (easier to add); E is vector. Work to move charge: W=qΔV.
Equipotential surfaces?
Surfaces where V=constant. Work=0 to move charge along equipotential. E field is always perpendicular to equipotential surfaces. Conductors in equilibrium are equipotentials.
Why can electric potential be negative?
Potential is defined relative to a chosen zero. For a point charge with V = 0 at infinity, a negative source charge gives negative potential at every finite distance. Negative potential means a positive test charge would have lower potential energy there than at the chosen zero.
Why does potential vary as 1/r while field varies as 1/r2?
The electric field is related to the spatial rate of change of potential. For the point-charge function V = kQ/r, differentiating with respect to r produces a 1/r2 dependence for the radial field magnitude. The quantities describe related but different aspects of the same interaction.
What does one volt mean physically?
One volt equals one joule per coulomb. A potential difference of 1 V changes the electric potential energy of a +1 C charge by 1 J. For an electron, moving through 1 V corresponds to an energy change with magnitude 1 electronvolt.
How are potentials from multiple point charges combined?
Because electric potential is a scalar, calculate Vi = kQi/ri for each source and add the signed values. This is often easier than adding electric-field vectors. After finding total V, multiply by the test charge q to obtain its potential energy.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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