Electric Potential Calculator — voltage and energy near a point charge

Electric potential is the electric potential energy a unit of charge would have at a given point, measured in volts. One volt is one joule per coulomb. Near a single point charge it falls off as one over the distance, so doubling how far away you stand halves the potential — not quarters it, which is the behaviour of the field rather than the potential. That single difference in exponent is the source of most of the confusion around this topic, and it is worth getting straight before you calculate anything.

This calculator returns the potential at a chosen point from one or two source charges, the potential at a second point, the difference between them, and the energy consequences for a test charge you place there. Arb Digital publishes it alongside a set of electrostatics tools because potential is the quantity that actually connects charge geometry to energy, and energy is usually the thing you were trying to find out.

What This Electric Potential Calculator Does

You give it up to two source charges and the distance from each of them to two observation points, A and B. It computes the potential at each point by superposition, subtracts them to give the potential difference, and then multiplies by a test charge to convert potential into energy and potential difference into work. The relative permittivity field lets you place the whole arrangement in a dielectric rather than in vacuum.

The headline number is the potential at point A in volts. The supporting grid carries the potential at B, the difference between the two, the potential energy a test charge would have sitting at A, and the work needed to carry that test charge from B to A. Those last two are separate quantities that get conflated constantly: one is an absolute energy relative to infinity, the other is a difference, and only the difference is measurable.

Setting the second charge to zero collapses the tool to the single-charge case and its two distance fields stop mattering. The two presets show the cases where potential behaves in a way that surprises people: a dipole pair, where two equal and opposite charges can give exactly zero potential at a point where the field is emphatically not zero, and the same charge immersed in water, where a relative permittivity near eighty cuts every potential by almost two orders of magnitude.

How to Use It

1. Enter the source charge with its unit. Laboratory electrostatics lives in nanocoulombs and microcoulombs. A whole coulomb is an enormous amount of static charge, so if you find yourself typing single digits with coulombs selected, check the unit before you trust the answer.
2. Keep the sign. A negative charge produces a negative potential, and the sign propagates through every result on the page. Stripping signs to make the numbers look tidier destroys the superposition, which is the one thing potential does better than field.
3. Set the distance unit first, then the four distances. Point A is where you want the answer; point B is the reference you are comparing it against. If you only care about one point, leave B far away and read the hero figure.
4. Choose a test charge. It is a probe, not a source: the tool assumes it is small enough not to disturb the charges producing the potential. It converts volts to joules and nothing more.
5. Adjust the relative permittivity if the charges sit in a medium. Vacuum and air are both close enough to 1. Anything else divides the result, sometimes dramatically. Values come from the same dielectric data tables engineers use for capacitor design.

The Formula: V = kQ ÷ r

For a single point charge the potential at distance r is V = kQ ÷ r, where k is the Coulomb constant, 8.9875 × 10⁹ N m² C⁻². That constant is not independent: it is 1 ÷ (4πε₀), built from the vacuum electric permittivity whose 2022 CODATA value of 8.8541878188 × 10⁻¹² F m⁻¹ is published by NIST. Inside a dielectric, ε₀ becomes ε₀ε_r and every potential shrinks by the factor ε_r.

With more than one source charge you add the individual potentials as ordinary signed numbers: V = ∑ kQ_i ÷ r_i. There is no vector addition, no resolving into components, no angle anywhere in the calculation. Georgia State University's HyperPhysics treatment of electric potential for different charge geometries works through the standard arrangements on this basis.

Work the defaults. A 5 nC charge, point A at 0.10 m, vacuum. V_A = 8.9875 × 10⁹ × 5 × 10⁻⁹ ÷ 0.10 = 449.4 V. At point B, 0.30 m out, the same arithmetic gives 149.8 V — exactly one third, because the distance tripled and potential goes as 1/r. The difference is 299.6 V. A 2 nC test charge at A has energy qV = 2 × 10⁻⁹ × 449.4 = 8.99 × 10⁻⁷ J, and carrying it from B to A takes 2 × 10⁻⁹ × 299.6 = 5.99 × 10⁻⁷ J of work against the repulsion. The volt itself is a derived SI unit, defined in the BIPM SI Brochure as one joule per coulomb.

Potential Is a Scalar, Field Is a Vector — and This Page Is About the Scalar

Arb Digital publishes an electric field calculator, and the boundary between the two pages is the single most useful thing to understand here. That tool computes E, a vector: it has a magnitude in newtons per coulomb and a direction, it falls off as 1/r², and combining two sources means adding arrows. This page computes V, a scalar: it has a value in volts and no direction, it falls off as 1/r, and combining two sources means adding numbers. The field page reports potential as one supporting figure for a single charge; this page makes potential the subject, adds superposition from a second charge, and carries the result through to potential difference and work — which is what you need when the question is about energy rather than force.

The practical consequence is that potential is far easier to compute. Three charges at awkward angles produce a field that requires trigonometry and a diagram. The same three charges produce a potential you can work out on a phone: three divisions and two additions. Physicists compute V first and, when they need E, differentiate it, because the derivative of a scalar field is easier than the vector sum.

For the force between two specific charges rather than the potential of an arrangement, the Coulomb's law calculator is the right page — that is an inverse-square vector relation between a pair of charges, not a property of a point in space.

Potential, Potential Difference and Potential Energy Are Three Different Things

These three share a word and are not the same quantity. Potential V is measured in volts and belongs to a point in space; it exists whether or not anything is there to experience it. Potential difference is V_A − V_B, also in volts, and belongs to a pair of points. Potential energy U = qV is measured in joules and belongs to a specific charge at a specific point.

A voltmeter measures the second of these, never the first. There is no instrument that reads absolute potential, because the number depends on where you decided zero was. When a datasheet says a node sits at 3.3 V it means 3.3 V above the circuit's chosen reference, and moving that reference moves every voltage in the circuit together without changing any physics.

The energy relation is where the volt earns its keep. A charge of one coulomb falling through one volt releases one joule. That is the definition, and it makes the electron-volt trivially convertible: an electron falling through one volt gains 1.602 × 10⁻¹⁹ J, the elementary charge in coulombs multiplied by one. Every accelerator energy quoted in MeV is a statement about the potential difference the particle crossed.

Where the Zero of Potential Sits, and Why It Matters

The formula V = kQ/r quietly contains a choice: it sets potential to zero at infinite distance. That is the standard convention for isolated charges and it is what this calculator uses, which is why the absolute potentials it reports are meaningful only in that frame.

Circuit work uses a completely different zero — the ground node, chosen for convenience and often physically arbitrary. Neither convention is more correct; both are references. The trouble comes from mixing them, which is how a student ends up trying to add a 449 V electrostatic potential referenced to infinity to a 5 V rail referenced to a circuit ground, as though the two numbers lived on the same scale.

Potential differences are immune to this. V_A − V_B is the same number whatever zero you adopt, because the reference cancels in the subtraction. This is why every measurable electrical quantity is ultimately a difference, and why the second and fourth figures in the results grid — the difference and the work — are the ones you can compare against an experiment.

The Dipole Case: Zero Potential Where the Field Is Strong

Load the dipole preset and something instructive happens. Two charges of equal magnitude and opposite sign, with an observation point equidistant from both, give a potential of exactly zero: the positive contribution and the negative contribution cancel numerically. Yet the electric field at that point is not zero at all. Both charges push a positive test charge in the same direction — away from the positive charge and toward the negative one — so the field vectors reinforce rather than cancel.

Zero potential does not mean zero field, and zero field does not mean zero potential. They are independent conditions. Potential is zero where the signed contributions sum to nothing; field is zero where the vector contributions sum to nothing, which for two opposite charges happens nowhere in the finite region at all.

The useful reading of the zero-potential surface is energetic: a charge can be moved anywhere along it without any net work being done. That surface is an equipotential, field lines cross it at right angles, and the fact that no work is required to move along it is exactly why conductor surfaces in electrostatic equilibrium are equipotentials — if they were not, charge would flow until they were.

Common Mistakes to Avoid

• Using 1/r² for potential — that is the field. Potential goes as 1/r. Three times the distance means one third the potential but one ninth the field.
• Dropping the sign on a negative charge — potential is signed and the signs are what make superposition work. Taking magnitudes turns a cancelling pair into a reinforcing one.
• Adding potentials as vectors — there is nothing to resolve. Potential has no direction, so angles between charges never enter the calculation.
• Treating absolute potential as measurable — only differences are. An absolute figure is always relative to some declared zero, here taken at infinity.
• Confusing energy at a point with work between points — qV is the energy of a charge sitting somewhere; qΔV is the work to move it. The grid reports both because they answer different questions.

Related Free Tools From Arb Digital

For the vector field and the force on a test charge, use the electric field calculator; for the force between a specific pair of charges, the Coulomb's law calculator. Energy stored in a dielectric-filled component is handled by the capacitor energy calculator and the capacitance calculator. The mechanical analogue of this page is the potential energy calculator, which does for height and gravity what this one does for distance and charge. For dividing a supply voltage across a resistor chain, see the voltage divider calculator, and for converting between charge units the electric charge converter. Everything Arb Digital publishes is indexed on the free online tools hub.

Frequently Asked Questions

This tool is provided for educational and study use. It models ideal point charges in a uniform medium and does not account for conductor geometry, induced surface charge, breakdown, or any real high-voltage safety consideration.