Fields · potential and force

Electric Field

Shape a dipole or a quadrupole. Move a probe across the potential landscape, scan the field, and test how its direction follows the downhill slope.

Probe potential $\text{—}$

Probe field strength $\text{—}$

Horizontal field $\text{—}$

Vertical field $\text{—}$

Net source charge $\text{—}$

Closed-loop circulation $\text{—}$

Normalized softened Coulomb model. Height is a graph of potential, not a physical surface. Flow beads show field-line direction; they are not dynamical particles. Read the scan and 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 real dynamical 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: Treat potential as terrain: the electric field is its steepest downhill direction, the negative gradient.

Core mathematical model

Normalized softened model in this Lab

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 section of a three-dimensional field. Softening makes the source core finite; the field is the negative gradient of this same potential.

Superposed point-charge potential

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

Positive charges raise potential, negative charges lower it, and scalar contributions 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 modelPotential depends on the chosen reference; force depends on nearby change, the gradient.

Field lines are not particle paths

Typical misconceptionA released charge must trace one field line.

Better mental modelField lines give instantaneous force direction; inertia means velocity need not stay parallel to the field.

Run the experiment

01
Read a dipole
Choose Dipole and show field lines with equipotentials.
What to observe: The line families are nearly orthogonal and field lines run positive to negative.
02
Compare the midpoint
Switch to Like charges and inspect the center.
What to observe: Potential adds while symmetric field vectors can cancel.
03
Change the geometry
Drag a source, or edit it with the source sliders.
What to observe: Potential surface, contours, and field lines rebuild together.

Electric Field

A scan through the field

Electric fields: from potential terrain to force