Electric Potential Calculator

Calculate electric potential (voltage) from a point charge using V = kQ/r.

⚡ Electricity📐 V = kQ/r🔌 Potential
Charge (Q) Coulombs
Distance (r) m
⚠️ Enter valid numbers. Q and r must be non-zero.

What Is Electric Potential?

Electric potential (V) at a point in space is the work done per unit positive charge to bring a test charge from infinity to that point: V = kQ/r, where k = 8.99×10⁹ N·m²/C² is Coulomb's constant, Q is the source charge (C), and r is the distance (m). V is a scalar quantity (not a vector) — positive near positive charges, negative near negative charges.

The electric field E and potential V are related: E = −dV/dr (field is the negative gradient of potential). High-potential regions have strong outward-pointing fields; low-potential (negative) regions have inward-pointing fields. Electric potential difference ΔV = V_A − V_B is the voltage between two points — the quantity measured by a voltmeter.

Work done moving charge q between two points: W = q·ΔV. This is why potential is called 'voltage' — it's the energy per unit charge. A 9 V battery maintains 9 V difference between terminals; moving 1 coulomb from negative to positive terminal takes 9 joules of chemical energy and delivers it to the circuit.

Superposition applies: total potential from multiple charges = Σ kQᵢ/rᵢ (algebraic sum, since V is a scalar). This is simpler than vector addition required for electric field. Equipotential surfaces (V = constant) are always perpendicular to field lines. The surface of a conductor at equilibrium is an equipotential.

Formula Reference Table

Solve ForFormulaNotes
Electric potentialV = kQ/rk = 8.99×10⁹ N·m²/C²
Charge from VQ = Vr/kFrom known V and r
Distance from Vr = kQ/VFrom known V and Q
Potential energyU = qV = kQq/rPE of charge q in field of Q
Work doneW = q·ΔVJ; move charge q through ΔV
SuperpositionV_total = Σ kQᵢ/rᵢAlgebraic sum (scalar)

3 Worked Examples

Example 1
Proton at 1 nm

Find electric potential 1 nm from a proton (Q = 1.6×10⁻¹⁹ C).

  • V = kQ/r = 8.99×10⁹ × 1.6×10⁻¹⁹ / 10⁻⁹
  • V = 1.438×10⁻⁹ / 10⁻⁹ = 1.438 V
  • Energy for another proton to reach 1 nm: U = qV = 1.6×10⁻¹⁹ × 1.438 = 2.3×10⁻¹⁹ J = 1.44 eV
✓ V = 1.44 V at 1 nm from proton
Example 2
Van de Graaff Sphere

Sphere radius 0.3 m, carries Q = 2×10⁻⁶ C. Surface potential?

  • V = kQ/r = 8.99×10⁹ × 2×10⁻⁶ / 0.3
  • V = 17,980 / 0.3 = 59,933 V ≈ 60 kV
  • Hair-raising demo: hairs repel because all charged to same V
✓ V = 60 kV on 0.3 m Van de Graaff sphere
Example 3
Find Charge from Potential

A point in space at r = 0.5 m has V = 180,000 V. What is the source charge?

  • Q = Vr/k = 180,000 × 0.5 / (8.99×10⁹)
  • Q = 90,000 / 8.99×10⁹ = 1.0×10⁻⁵ C = 10 μC
✓ Q = 10 μC (10 microcoulombs)

Real-World Applications

⚡
Van de Graaff Generators
Research Van de Graaff generators reach V = 20+ MV. At high V, the electric field at the sphere surface (E = V/r) approaches air breakdown (3 MV/m), limiting maximum V for a given sphere size.
🔬
Particle Accelerators
Linear accelerators use electric potential differences to accelerate charged particles. A proton accelerated through 1 MV gains 1 MeV of kinetic energy. The LHC uses 6.5 MV per turn × thousands of turns to reach 7 TeV.
🧬
DNA Electrophoresis
Gel electrophoresis uses V ≈ 50–200 V across the gel. The electric field E = V/L drives charged DNA fragments through the gel, separating by size. Potential, not field, is the directly applied quantity.
🔋
Battery Terminals
A 9V battery maintains V_positive − V_negative = 9 V between terminals. Current flows because charges move from low to high potential inside the battery (chemical energy), and high to low in the external circuit.
💡
Electric Potential Energy
A 1 μC charge at V = 10,000 V has electric PE = qV = 10⁻⁶ × 10⁴ = 0.01 J. Moving it to V = 0 releases this as kinetic energy (particle accelerator principle).

Common Mistakes to Avoid

⚠️
Confusing V (scalar) with E (vector)

V = kQ/r is a scalar; electric field E = kQ/r² is a vector. V is J/C; E is N/C = V/m. They are related by E = −dV/dr.

⚠️
Forgetting sign for negative charges

V = kQ/r is negative when Q < 0 (negative charge). The potential is negative near negative charges — electrons flow from high to low V; positive charges from low to high.

⚠️
Confusing electric potential with potential energy

V = kQ/r is potential (J/C) at a point — a property of the field, not of a specific charge. Potential energy U = qV depends on which test charge q is placed there.

⚠️
Using radius of the sphere instead of distance for external potential

For a uniformly charged sphere, V outside = kQ/r where r is from the center. On the surface: V = kQ/R. Inside a conductor: V = constant = kQ/R everywhere inside.

⚠️
Not applying superposition correctly

Multiple charges: V_total = Σ kQᵢ/rᵢ (algebraic sum — signs matter!). Two equal opposite charges (+Q and −Q) at ±d/2: V at midpoint = kQ/(d/2) + k(−Q)/(d/2) = 0. The field is not zero there, but the potential is.

Frequently Asked Questions

What is electric potential physically? ▾
V (volts = J/C) is the electric potential energy per unit charge at a point. V = kQ/r: the work done per unit positive charge to bring a test charge from infinity to distance r from charge Q. It's a scalar — much easier to calculate than the vector field E.
What is the difference between potential and potential energy? ▾
V (volts) = work per unit charge = J/C — a property of the field at a point. U (joules) = qV = potential energy of a specific charge q in the field. V is like the 'height' of water; U depends on how much water you have (the charge q).
How does E relate to V? ▾
E = −∇V (field is the negative gradient of potential). In 1D: E = −dV/dx. V decreases in the direction of E. Between parallel plates (separation d, voltage V): E = V/d (uniform field). Equipotential surfaces (V = const) are always perpendicular to E field lines.
What is an equipotential surface? ▾
Surfaces where V = constant. No work is done moving a charge along an equipotential. E field lines are always perpendicular to equipotentials. Concentric spheres are equipotentials for a point charge. Conducting surfaces at equilibrium are equipotentials (E_tangential = 0 at conductor surface).
What happens at the surface of a conductor? ▾
Inside a conductor at equilibrium: E = 0 (free charges redistribute until net E = 0). The entire conductor is at the same V (equipotential). Charges distribute on the surface; more dense at sharper points (higher E at points). This is why lightning rods work — sharp point concentrates E until discharge occurs.
How does a Van de Graaff generator work? ▾
A motor-driven belt carries charge from ground to the sphere. Charge is deposited on the inside of the sphere, then redistributes to the outside (conductor). V = kQ/R increases as Q accumulates. Sparks occur when V/R = E_breakdown ≈ 3 MV/m. Large sphere = higher V before breakdown.
What is potential in quantum mechanics? ▾
Schrödinger equation: iħ∂ψ/∂t = [−ħ²/2m ∇² + V(r)]ψ. The classical electric potential V(r) directly enters the quantum mechanical energy operator (Hamiltonian). The hydrogen atom energy levels arise from V(r) = −ke²/r (Coulomb potential). Quantum tunneling occurs when a particle crosses a potential barrier classically impossible.
What is the Millikan oil drop experiment? ▾
Millikan (1909) balanced gravity and electric force on oil droplets in a known E field: qE = mg → q = mg/E. By adjusting V (and thus E = V/d), he found q was always a multiple of 1.6×10⁻¹⁹ C — the first measurement of the elementary charge. This confirmed the quantization of charge.

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Interpretation: This formula links charge, voltage, current, resistance, capacitance, power or circuit time response. Assumption: Confirm DC versus AC conditions, RMS versus peak values, component topology and steady-state versus transient behavior. Ideal components may be assumed.

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