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.

Probe potential $\text{—}$

Probe field strength $\text{—}$

Horizontal field $\text{—}$

Vertical field $\text{—}$

Net source charge $\text{—}$

Closed-loop circulation $\text{—}$

Move the probe and the source charges

Shape the field

Potential landscapeSurface height shows the potential on a compressed scale.Field linesThe moving beads only show direction; they are not particles.Equipotential contoursThe contours come from the same potential as the landscape.Probe horizontal position

x_p

0

Probe vertical position

y_p

0

Selected source horizontal position

x_s

-1.2

Selected source vertical position

y_s

0

Selected source charge

q_s

1

The map is a flat slice through a three-dimensional Coulomb field, in normalized units. Each charge has a soft core of radius 0.1. Click the plane or use the probe sliders to move the probe; the source sliders do the same job as dragging a charge.

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.

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 →MIT: electrostatics and potentials

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 misconceptionA test charge feels no force wherever $V=0$ .

Better mental modelThe 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 misconceptionA released charge must trace one field line.

Better mental modelA 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

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.
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.
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.

Electric Field

A scan through the field

Electric fields: from potential terrain to force