---
title: "Make an Electric Field Visible Interactive Lab | Sandbox Physics"
description: "How can the shape of the electric potential tell you the force on a charge? Find the steepest downhill direction of the potential."
canonical_url: "https://sandboxphysics.com/labs/electric-field/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/electric-field/"
last_modified: "2026-10-27"
---

Fields · potential and force

# Electric Field

**How can the shape of the electric potential tell you the force on a charge?** Find the steepest downhill direction of the potential. The field points that way, and so does the force on a positive charge. The steeper the slope, the stronger the field, and a spot where the potential is zero can still sit in a strong field.

Set up a dipole or a quadrupole. Move a probe across the potential landscape, scan the field, and check how its direction follows the downhill slope of the potential.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model uses a normalized, softened Coulomb law. The height is a graph of the potential, not a physical surface. The flowing beads mark the direction of the field lines; they are not particles moving under the field. Use the scan and the closed-loop integral to check the field.

Model data and checks

## A scan through the field

The scan follows the probe’s vertical position. The field and potential use one softened kernel. Dots estimate the negative slope of the sampled potential; they provide a separate numerical check.

$$
\mathbf E=-\nabla V,\qquad \oint\mathbf E\cdot\mathrm d\boldsymbol\ell=0
$$

Height and colour visualize potential; they are not a physical surface. The moving beads only show field-line direction. The amber rectangle defines the circulation readout. Coordinates, charge, potential and field use normalized units; the source core has radius 0.1.

[Next: launch a particle and follow its trajectory →](https://sandboxphysics.com/labs/charged-particle/) [MIT: electrostatics and potentials](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter28/section04.html)

Physics tutorial

## Electric fields: from potential terrain to force

**Background** Electric potential is the potential energy per unit charge at a point. The potentials of point charges simply add, and the way the total changes across space gives the field.

**Why it matters** Finding the potential, a single number at each point, is usually easier than adding field vectors. It also explains voltage, equipotentials and shielding right away.

Start with the essentials

- **Focus question**: Why are field lines perpendicular to equipotentials, and why is the field stronger where color changes faster?
- **One-sentence intuition**: Think of potential as terrain: the electric field points down the steepest slope, along the negative gradient.

### Core mathematical model

#### Softened potential used here, in normalized units

$$
V(\mathbf r)=\sum_i\frac{q_i}{\sqrt{|\mathbf r-\mathbf r_i|^2+a^2}},\quad a=0.1
$$

The plane is a slice through a three-dimensional field. Softening keeps the potential finite at each charge, and the field is the negative gradient of this same potential.

#### Potentials of point charges add

$$
V(\mathbf r)=\frac{1}{4\pi\varepsilon_0}\sum_i\frac{q_i}{|\mathbf r-\mathbf r_i|}
$$

Positive charges raise the potential and negative charges lower it. Each contribution is a plain number, so they simply add.

#### Field from potential

$$
\mathbf E=-\nabla V
$$

The gradient points uphill; the minus sign makes the field point toward decreasing potential.

### Common difficulties

#### Zero potential does not mean zero field

**Typical misconception** A test charge feels no force wherever $V=0$.

**Better mental model** The value of the potential depends on where you put zero. The force depends on how fast the potential changes nearby, which is the gradient.

#### Field lines are not particle paths

**Typical misconception** A released charge must trace one field line.

**Better mental model** A field line shows only the direction of the force at that moment. Because the charge has inertia, its velocity need not stay parallel to the field.

### Run the experiment

1. 01

   #### Look at a dipole

   Choose Dipole and turn on both Field lines and Equipotential contours.

   **What to observe:** The two sets of lines cross at nearly right angles, and the field lines run from the positive charge to the negative one.

2. 02

   #### Check the midpoint

   Switch to Like charges and look at the center.

   **What to observe:** The two potentials add there, but the fields from the two sides can cancel.

3. 03

   #### Move a charge

   Drag a charge, or select it and use the source sliders.

   **What to observe:** The potential landscape, the contours and the field lines all update together.

Physical conclusion

Understand the shape of potential first, then read the field from its gradient.

Starter path · Step 1 of 6

## From a field to an electron microscope

[Next step → **A Charge Bending through Fields** Launch with only a magnetic field, then add an electric field. Which one changes the kinetic energy? Model · 10 min](https://sandboxphysics.com/labs/charged-particle/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)