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Sandbox Physics

Fields · electrostatics

Electric Field

Place positive and negative charges on the plane. Surface height and color encode electric potential, streamlines run from positive to negative, and equipotentials wrap around equal heights.

Interactive modelElectric Field
Potential VV+0.0
Field strength ∣E∣\lvert\mathbf E\rvert0.00
Charges2
Net charge0

Physics tutorial

Electric fields: from potential terrain to force

BackgroundElectric potential is potential energy per unit charge at each position. Point-charge potentials add directly, and their spatial variation gives the field.

Why it mattersSolving for a scalar potential is often easier than summing vector fields and immediately explains voltage, equipotentials, and shielding.

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

Superposed point-charge potential

V(r)=14πε0∑iqi∣r−ri∣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

E=−∇V\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=0V=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

  1. 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.
  2. 02

    Compare the midpoint

    Switch to Like charges and inspect the center.

    What to observe: Potential adds while symmetric field vectors can cancel.
  3. 03

    Change the geometry

    Drag one charge.

    What to observe: Potential surface, contours, and field lines rebuild together.