Electric Potential Calculator: Equipotential Lines and Work Done Moving a Charge

Shuffle across a carpet in winter and reach for a doorknob, and somewhere around the last centimetre a spark jumps the gap on its own. Nothing about your charge changed in that last centimetre — what changed was your electric potential relative to the doorknob, and once the gap got small enough, the difference was enough to break down the air. A 9V battery keeps exactly that kind of difference standing between its two terminals, on purpose, for as long as the battery lasts.

This page is an electric potential calculator built around something you can actually grab and move: point charges on a 2D plane, with the resulting equipotential lines drawn live underneath them, like a topographic map of an electrostatic hill. Drag a test probe between two points and the calculator reads off the potential difference and the work done — no force diagram, no vector addition, just two numbers and a subtraction.

What Is Electric Potential?

Electric potential ($V$) at a point in space is the potential energy per unit charge that a small positive test charge would have if placed there. For a single point charge $q$, at distance $r$:

$V = \frac{kq}{r} = \frac{1}{4\pi\epsilon_0}\frac{q}{r}$

with $k = 8.99 \times 10^9$ N·m²/C² and $\epsilon_0 = 8.85 \times 10^{-12}$ C²/(N·m²), the same constants from Coulomb's Law.

Here's the part that trips people up: potential is a property of the location, not of any charge sitting there. It exists at every point in space around $q$ whether or not you ever park a test charge at that point — exactly the way a hillside has an altitude at every point whether or not a hiker happens to be standing on it. Put a charge there and it picks up potential energy $U = qV$; take the charge away and $V$ is still there, waiting.

One more wrinkle worth knowing before you touch the simulator: only differences in potential are ever physically measurable. The formula above sets $V = 0$ at infinite distance from an isolated charge, which is a convention, not a law of nature — shift every value on the map by the same constant and nothing physical changes. A voltmeter never reads "the potential"; it reads the potential difference between its two probes.

How Is Electric Potential Different from Electric Field?

That last row is worth checking with real numbers: move from 10 cm to 5 cm away from a +5 nC charge and $V$ goes from 449.5 V to 899.0 V — exactly double, because $V \propto 1/r$. The electric field at the same two distances would quadruple, because $E \propto 1/r^2$. Same charge, same move, very different growth rate.

The two are connected by $E = -\dfrac{dV}{dr}$: the field always points in the direction potential drops fastest — the electrostatic equivalent of "downhill." Positive charges are pushed from high potential to low, the same way a ball rolls from high altitude to low.

How Do You Calculate Electric Potential from Multiple Charges?

Because potential is a scalar, combining the effect of several charges is just addition — no angles, no components:

$V_{\text{total}} = \sum_i \frac{kq_i}{r_i}$

Compare that with superposing electric fields, where you have to break each contribution into $x$ and $y$ components before adding — exactly what the electric field simulator has to do at every point on its grid. Here, you just add plain numbers, sign and all.

Interactive Equipotential Simulator: Drag Charges, Measure Work

Every closed loop on the map below is an equipotential line — a curve where $V$ has the same value at every point. Warm colours are high potential, cool colours are low, and the heavy dashed line is $V = 0$. The green dot is a test probe: it reads the potential wherever it sits, and it remembers the path you drag it along.

Try the midpoint experiment before reading on. With the default charges (+6 nC and −6 nC, symmetric about the centre), the probe at the origin reads $V \approx 0.0$ V even though a positive test charge placed there would feel a very real push to the right. Zero potential does not mean zero field; it just means the two contributions happened to cancel at that one spot. The dashed $V=0$ line through the middle is the whole family of such spots.

The trail experiment is the deeper one. Mark A, take the probe on any loop you like — out to a corner, around a charge, anywhere — and mark B. The work readout is identical to what you'd get walking A to B in a straight line, because the electrostatic force is conservative: $W$ depends only on the endpoints, never on the road taken between them. It's the same property that lets you define gravitational potential energy for a ball rolling down an arbitrarily bumpy hill.

What Are Equipotential Lines?

An equipotential line (or, in three dimensions, an equipotential surface) is a curve along which $V$ never changes. Moving a charge along one costs zero work, because $W = q(V_A - V_B)$ and $V_A = V_B$ for any two points A and B on the line, by definition.

Why Are Equipotential Lines Always Perpendicular to the Electric Field?

If the field had any component running along an equipotential line, that component would do nonzero work pushing a charge along the line — contradicting $\Delta V = 0$ there. So the field can only point across equipotential lines, never along them: field lines and equipotential lines always cross at right angles. The simulator's field-arrow overlay lets you audit this claim directly — every arrow crosses every contour at 90°, no matter how you arrange the charges. It's the same geometric fact as a topographic map, where the steepest downhill path always cuts straight across the contour lines, never runs along one.

This is also why the surface of a conductor in electrostatic equilibrium is always an equipotential surface — any leftover field component tangent to the surface would push charge around until it vanished, which is exactly what "equilibrium" means here.

What Is the Formula for Work Done by an Electric Field?

The work done by the electric force moving a charge $q$ from point A to point B is:

$W = q(V_A - V_B)$

This comes straight from the definition of potential energy, $U = qV$: work done by a conservative force equals the drop in potential energy, $W = U_A - U_B = q(V_A - V_B)$.

How Do You Calculate the Work Done Moving a Charge Between Two Points?

1. Find the potential at the starting point, $V_A$, by summing $kq_i/r_i$ over every source charge.
2. Find the potential at the ending point, $V_B$, the same way.
3. Multiply the charge being moved by the difference: $W = q(V_A - V_B)$.
4. A positive $W$ means the field itself pushed the charge there — no outside effort needed. A negative $W$ means an external agent had to do work against the field.

Where Does Electric Potential Matter in Real Life?

• Batteries: a battery's voltage rating is the potential difference it maintains between its terminals — a 9V battery keeps $V_+ - V_- = 9$ V for as long as its chemistry can sustain it.
• Capacitors: the charge stored on a capacitor's plates is directly proportional to the potential difference across them, $Q = CV$ — the whole basis of energy storage in circuits.
• Power lines and birds: a bird perched on a single high-voltage wire is fine, because both its feet sit at (essentially) the same potential — no potential difference across its body means no current. Touch that wire and a grounded pole at the same time, and the enormous $\Delta V$ becomes lethal.
• Defibrillators: deliver a large, brief potential difference across the chest specifically to force current through the heart muscle and reset its rhythm.
• Lightning: charge separation between a thundercloud and the ground can build a potential difference of hundreds of millions of volts — once the difference gets large enough relative to the gap, it breaks down the insulating air, the same effect (at a vastly larger scale) as the doorknob spark this article opened with.