Electric Field Guide & Visualizer

What is an electric field?

The electric field E at a point in space describes the force per unit charge that a positive test charge would experience. For a point charge qat the origin, Coulomb's law gives:

E = (1 / 4πε₀) · q / r² · r̂

k = 1 / 4πε₀ ≈ 8.99 × 10⁹ N·m²/C²

Field lines point away from positive charges and toward negative charges. The superposition principle states that the total field from multiple charges is the vector sum of individual contributions.

Key formulas for introductory & GRE physics

Point charge field:     E = k q / r²
Superposition:            E_total = Σ E_i
Force on test charge:     F = q_test E
Potential (scalar):       V = k q / r
Field from potential:     E = −∇V
Uniform field:            E = σ / ε₀  (parallel plates, ideal)
Dipole field (far):       E ∝ p / r³

Gauss's law connection

Gauss's law relates the flux of the electric field through a closed surface to enclosed charge:

∮ E · dA = Q_enc / ε₀

For highly symmetric charge distributions (spheres, cylinders, infinite planes), Gauss's law provides the fastest route to the field magnitude without integrating Coulomb's law directly.

Common exam scenarios

• Two point charges: find null points on the line connecting them where E = 0.
• Dipole: field falls off as 1/r³ at large distances; torque τ = p × E.
• Conductors in equilibrium: interior E = 0, excess charge resides on the surface.
• Energy: potential energy U = qV; stored energy density u = ½ ε₀ E².

Units & constants

Study workflow

1. Read the field definition and superposition principle above.
2. Drag charges in the visualizer to see how field arrows respond.
3. Practice null-point and dipole-limit problems from GRE-style sets.
4. Cross-reference E&M formulas in the Physics GRE Guide.
5. Plot field-related functions (e.g. Coulomb potential V(r) ∝ 1/r) with the graphing calculator to check scaling and asymptotic behavior.

Use the formulas above with the interactive canvas to build intuition — then try sketching field lines by hand before checking the visualizer.