Home / Learn / Physics / Potential mapTool #L257 (formerly #1402)

Potential map

Drag charges around a board and see their electric potential as a coloured map with equipotential lines and as hills and wells in 3D. Probe any point for V = Σ kq/r and the field, and let a positive test charge roll downhill along the field, crossing every equipotential at right angles.

Keep it one tap away for everyday use.

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Equipotential map

high potential (hill)low potential (well)field at the probe

Drag a charge to move it; tap a charge to change it; tap anywhere else to put the probe there.

Potential hills in 3D

Height = potential, cut off at ±1 kV so the peaks fit. Each equipotential line on the map is a contour line on these hills.

Try this

  • Put the probe on the middle line of the dipole: V = 0 there, yet the field is not zero. Potential and field are different things.
  • Let a test charge roll: it runs downhill, crossing every equipotential at right angles.
  • Where the lines crowd together the hill is steep and the field is strong.
  • Choose Two like charges: between them is a saddle, where the field is zero but the potential is not.
  • Turn a charge to 0: its hill disappears and the other charges' hills are unchanged. Potentials simply add.

Check yourself

What it does

The electric potential V at a point is the work done per unit charge to bring a small positive charge there from far away. For point charges it is V = Σ kq/r, with k = 1/(4πε₀). Unlike forces, potentials are plain numbers (scalars), so the potentials of several charges simply add.

A good way to picture potential is as a landscape. A positive charge makes a hill and a negative charge makes a well. Lines joining points at the same height are equipotentials, like contour lines on a map. The electric field is the downhill slope: it points straight down the steepest way, so it is always at right angles to the equipotentials, and it is strongest where the lines crowd together.

This page shows both pictures at once. The map is shaded red where the potential is high and blue where it is low, with equipotential lines every ±50, 100, 200, 400 and 800 V (dashed for 0 V). The 3D view draws the same potential as a surface you can turn. Tap anywhere to put a probe there and read V and the field. Press Let a + test charge roll and a positive charge rolls downhill from the probe along the field.

It follows the electrostatic potential and capacitance chapter of Class 12 Physics (potential due to a point charge, a dipole and a system of charges, equipotential surfaces, relation between field and potential). Start with forces in coulomb forces and flux in gauss's law; for conductors at fixed voltages see the 2D electrostatics solver, and for stored charge the parallel-plate capacitor lab.

How to use it

  1. Pick a starting layout: one charge, a dipole, two like charges or three charges. Charges are in nanocoulombs (nC) on a 36 cm by 28 cm board.
  2. Drag a charge to move it. Tap a charge to choose it and change it with the slider (−5 to +5 nC).
  3. Tap an empty spot to move the probe (the small ring). Read V there, each charge’s share, and the field’s size and direction.
  4. Press Let a + test charge roll to watch it run downhill (it stops at a negative charge or the edge of the board).
  5. Turn the 3D view with its slider. Heights are cut off at ±1000 V so the peaks fit.

Worked example

The page opens with a dipole: A = +2 nC at (12, 14) cm and B = −2 nC at (24, 14) cm, with the probe at (18, 20) cm.

  • Each charge is √(6² + 6²) = 8.49 cm from the probe. A gives 8.99 × 10⁹ × 2 × 10⁻⁹ / 0.0849 = 212 V and B gives −212 V.
  • So V = 0: the probe sits on the 0 V line that runs straight up the middle.
  • But the field is not zero: it is 3530 V/m, pointing to the right (0°), from the hill towards the well.

Change the layout to two like charges and the same probe reads 424 V, while the field there points straight up.

Tips

  • E = −dV/dr: in a uniform field, E = ΔV/Δd between two equipotentials.
  • No work is done moving a charge along an equipotential; moving it between two of them takes W = q ΔV whatever the path.
  • Around a single charge the equipotentials are circles, closer together near the charge, because V falls off as 1/r.

FAQ

Why are the heights cut off? Right at a point charge V goes to infinity. The cut-off just keeps the picture on the screen; the numbers are not cut off.

Is this the real 3D field? The board is a flat slice through space with point charges in it, and the potential is worked out in 3D, so the numbers are real; only the slice is shown.

Is anything saved? No. Everything is worked out on this page and nothing leaves your device.

Where the numbers come from

  • Potential of point charges V = Σ q/(4πε₀r), field E = −grad V: standard physics. ε₀ = 8.8541878188 × 10⁻¹² F/m, the CODATA 2022 recommended value (NIST).
  • Checked against an independent NumPy script, the field compared with a numerical gradient of the potential, and the equipotential lines checked by evaluating V at their ends.

Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.

Last updated 2026-11-09.

Change log for Potential map

  1. Add #1402 Equipotentials and potential hills Details on GitHub

Every change to fibub

Related tools

2D electrostatics solver #T650 Type or draw electrodes at any voltages inside a box, and the page solves Laplace's equation on a grid by over-relaxation. See the potential map, equipotentials and field arrows, probe any point, and get the charge on each electrode and the capacitance per metre. Parallel-plate capacitor lab #L22 Change the plate area and the gap of a parallel-plate capacitor, slide a dielectric slab in and out, and connect or disconnect the battery. Watch the capacitance, charge, voltage, stored energy and field change, and see two capacitors combined in series and in parallel. Coulomb forces #L254 Drag up to four point charges around a board, choose one, and see the force from each of the others as an arrow, with Coulomb's law worked out with your numbers. The arrows add head to tail into the net force, with its size, direction and x and y parts. Gauss's law #L255 Drag charges and a closed surface around a board and count field lines across it: only the charge inside changes the flux, Φ = q/ε₀. Then use a Gaussian surface to find the field of a long charged wire, a large flat sheet and a thin spherical shell, with your numbers and a graph of E against distance. Dipole torque #L256 Turn an electric dipole in a uniform field and see the two equal and opposite forces, the torque τ = pE sin θ and the potential energy U = −pE cos θ, with your numbers and a graph of both against angle. Let it go and watch it swing about the field direction. Capacitor network reducer #L258 Pick a capacitor network, set each capacitor and the battery, and reduce it one series or parallel group at a time to a single equivalent capacitance, with the working for every step. Then see the charge, voltage and energy on each capacitor, worked back out from the battery. Faraday cage #L259 Put a metal tube in an electric field and watch the free charges move to its surface so that the field inside is exactly zero. Probe the field anywhere, turn the outside field, take the tube away, or put a charge inside and see why a cage must be earthed to hide it.