Electric Potential and Voltage

Definition

Electric potential is defined as electric potential energy per unit charge. If a charge q placed at some point has electric potential energy U there, the electric potential at that point is:

V = U / q

Equivalently, V at a point is the work W an external agent must do, per unit charge, to carry a small positive test charge from a chosen reference point to that point without changing its kinetic energy:

V = W_ref→point / q₀

The SI unit is the volt (V), named after Alessandro Volta. One volt is one joule per coulomb:

1 V = 1 J/C = 1 kg·m²·s⁻³·A⁻¹

Potential is a scalar field — it has a value but no direction — which makes it far easier to work with than the vector electric field. Crucially, potential is a property of the location, set up by surrounding charges; the test charge merely samples it and does not have to be present for V to be defined.

Potential of a point charge

Integrating the Coulomb field of a single point charge q inward from infinity (where we set V = 0) gives the most-used result in the subject:

V(r) = k·q / r        with  k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²

Note that potential falls off as 1/r, whereas the field magnitude E = kq/r² falls off as 1/r². The sign of V follows the sign of q: near a positive charge V > 0 (a "hill" for positive charges), near a negative charge V < 0 (a "valley").

Because potential is a scalar, the potential of several charges is a simple algebraic sum — no vector components, no angles:

V = Σᵢ k·qᵢ / rᵢ

This superposition-by-addition is why physicists often compute V first and then extract the field from it.

Field from potential: E = -∇V

The electric field is the negative gradient of the potential:

E = -∇V = -(∂V/∂x, ∂V/∂y, ∂V/∂z)

In one dimension this reduces to E_x = -dV/dx. The minus sign encodes a physical fact: the field points from high to low potential, so a positive charge accelerates "downhill," converting potential energy into kinetic energy. The steeper the potential slope, the stronger the field. Consequently the field's unit, volts per metre (V/m), is identical to newtons per coulomb (N/C).

The inverse relationship recovers potential difference by a line integral of the field:

V_B - V_A = -∫_A^B E · dl

Because the electrostatic field is conservative (∇ × E = 0), this integral is path-independent — the same ΔV results no matter which route you take between A and B. That is exactly the condition that lets us define a single-valued potential at all.

Equipotential surfaces

An equipotential surface is the set of all points at the same potential V. Two properties follow directly:

• No work along the surface. Moving a charge between two points on the same equipotential costs zero work, since W = qΔV and ΔV = 0.
• Perpendicular to the field. The field is everywhere normal to equipotentials. If E had any component along the surface, moving along it would change V — contradicting "equal potential." So field lines and equipotentials always cross at 90°.

For a point charge the equipotentials are concentric spheres; near a large flat charged plate they are parallel planes; the surface of any conductor in electrostatic equilibrium is itself an equipotential (which is why the field just outside a conductor is perpendicular to it).

Worked example: work to move a charge

Suppose point A sits at 12 V and point B at 4 V. How much work does the field do on a +2 μC charge carried from A to B?

ΔV = V_B - V_A = 4 V - 12 V = -8 V
W_field = -q·ΔV = -(2 × 10⁻⁶ C)(-8 V) = +1.6 × 10⁻⁵ J

The field does positive work (+16 μJ): a positive charge moving from high to low potential loses potential energy, which appears as kinetic energy. Equivalently, the external agent that instead holds it back does -16 μJ. Had the charge been negative, every sign would flip and the field would do negative work.

A convenient energy unit follows from W = qΔV: accelerate one electron (q = 1.602 × 10⁻¹⁹ C) through 1 volt and it gains one electron-volt, 1 eV = 1.602 × 10⁻¹⁹ J — the working currency of atomic and particle physics.

Reference points and why voltage is a difference

Only differences in potential produce measurable work, force, or field. The zero of potential is therefore an arbitrary choice, like sea level for altitude:

• Electrostatics. Set V = 0 at infinity — this makes V = kq/r come out cleanly.
• Circuits. Set V = 0 at the earth (ground) or the battery's negative terminal, and measure everything relative to it.

Shifting the reference adds the same constant to V everywhere, which never changes E = -∇V or any voltage you can measure with a meter. A "9-volt battery" does not put its terminals at 9 V absolutely; it maintains a 9 V difference between them. This is why "voltage" in engineering almost always means potential difference ΔV, and why a voltmeter always has two probes.

Representative potentials and voltages

Situation	Potential difference / potential
Neuron resting membrane potential	≈ -70 mV
Single AA / alkaline cell	1.5 V
Car battery	12 V
Household mains (RMS, US / EU)	120 V / 230 V
Van de Graaff generator dome	up to ~10⁶ V
High-voltage transmission line	≈ 10⁵–10⁶ V
Lightning leader-to-ground	≈ 10⁸–10⁹ V
Potential 1 cm from a +1 nC charge	V = kq/r ≈ 900 V

Key relations at a glance

Quantity	Relation	Notes / units
Potential (definition)	V = U / q	volts; U in J, q in C
Point charge	V = kq / r	V = 0 at r → ∞; scalar sum for many charges
Potential energy of a charge	U = qV	joules; depends on the charge placed
Work by the field	W = -qΔV	positive charge falls from high to low V
Field from potential	E = -∇V	V/m = N/C
Potential from field	V_B - V_A = -∫ E·dl	path-independent (conservative field)
Pair of point charges	U = kq₁q₂ / r	interaction energy, joules
Energy unit	1 eV = 1.602 × 10⁻¹⁹ J	one electron through 1 V

JavaScript — potential calculations

const k = 8.9875517923e9;   // Coulomb constant, N·m²/C²
const e = 1.602176634e-19;  // elementary charge, C

// Potential of a single point charge at distance r (V=0 at infinity)
function pointPotential(q, r) { return k * q / r; }

console.log(`+1 nC at 1 cm: ${pointPotential(1e-9, 0.01).toFixed(0)} V`); // ~899 V

// Potential from many charges — SCALAR sum (no vectors!)
function totalPotential(charges, x, y) {
  return charges.reduce((V, c) => {
    const r = Math.hypot(x - c.x, y - c.y);
    return V + k * c.q / r;
  }, 0);
}

// A dipole: +1 nC and -1 nC, 2 cm apart, evaluated at the midpoint-off-axis
const dipole = [{ q: 1e-9, x: -0.01, y: 0 }, { q: -1e-9, x: 0.01, y: 0 }];
console.log(`Dipole V at (0, 0.02): ${totalPotential(dipole, 0, 0.02).toFixed(3)} V`); // ~0 by symmetry

// Work done BY the field moving charge q from A to B
function workByField(q, V_A, V_B) { return -q * (V_B - V_A); }

console.log(`+2 µC from 12 V to 4 V: ${(workByField(2e-6, 12, 4) * 1e6).toFixed(1)} µJ`); // +16.0 µJ

// Convert a kinetic energy in joules to electron-volts
function toEV(joules) { return joules / e; }

// Electron accelerated through 100 V gains 100 eV
console.log(`Electron through 100 V: ${toEV(e * 100).toFixed(0)} eV`); // 100 eV

// Interaction potential energy of two point charges
function pairEnergy(q1, q2, r) { return k * q1 * q2 / r; }

console.log(`Two protons 1 fm apart: ${pairEnergy(e, e, 1e-15).toExponential(2)} J`); // ~2.3e-13 J

Where electric potential shows up

• Circuits. Voltage drives current: Ohm's law V = IR, power P = IV, and Kirchhoff's voltage law (Σ ΔV = 0 around a loop) are all statements about potential.
• Capacitors. Stored charge Q = CV and energy U = ½CV² depend directly on the potential difference across the plates.
• Particle accelerators. Charged particles gain kinetic energy qΔV; the electron-volt is the natural unit — the LHC runs to 6.8 TeV per beam.
• Electrochemistry. Cell EMF and electrode potentials (measured in volts) determine which redox reactions run spontaneously.
• Neuroscience. Membrane potentials (~-70 mV rest, spiking to +40 mV) encode and transmit every nerve signal.
• Electronics design. "Ground," logic-level thresholds, and signal levels are all defined relative to a chosen reference potential.
• Field mapping. Equipotential/field-line plots make electrostatic geometry (conductors, shielding, corona) intuitive.

Common mistakes

• Confusing potential V with potential energy U. V is energy per charge (volts, a field property); U = qV is energy (joules, tied to a specific charge). Doubling the test charge doubles U but leaves V unchanged.
• Treating potential as a vector. V is a scalar — you add the numbers. It is the field E = -∇V that has direction. Never resolve potential into components.
• Forgetting the reference point. A bare statement "V = 20 V" is meaningless without saying relative to what. Only ΔV between two points is measurable.
• Assuming zero field means zero potential (or vice versa). At the center of a uniformly charged sphere or midway between equal charges, E = 0 but V ≠ 0. Potential can be flat (large V, zero slope) or zero-crossing with strong slope.
• Dropping the minus sign in E = -∇V. The field points toward decreasing potential; forgetting the sign reverses the force direction.
• Confusing the 1/r of potential with the 1/r² of the field. V = kq/r but E = kq/r² for a point charge — they scale differently with distance.