Gauss’s Law Calculator

Electric flux (Φ) measures the number of electric field lines passing through a closed surface. Gauss’s law states that the net flux through a closed surface (Gaussian surface) equals the total charge enclosed by that surface divided by the vacuum permittivity: Φ = Q/ε₀. Its unit is the volt-meter (V·m), identical to N·m²/C.

Gauss’s law is most useful when the charge distribution has high symmetry: point/spherical charges (spherical symmetry), an infinite straight wire (cylindrical symmetry), or an infinite flat plane (planar symmetry). In these symmetric cases, E can be pulled out of the flux integral so the field is obtained directly without complicated integration. For irregular distributions without symmetry, Coulomb’s law (direct integration) remains the choice.

Because flux only counts the enclosed charge, not the surface’s geometry. As long as the same charge stays inside the surface — whether a small sphere, a large sphere, or a cube — the value Φ = Q/ε₀ stays the same. What changes with distance is the field strength E, not the total flux.

The unit of electric flux is the volt-meter (V·m), equivalent to newton-square-meters per coulomb (N·m²/C). For electric field strength, V/m and N/C are identical units — 1 V/m = 1 N/C — so this calculator displays both with the same value.

The electric field points away from positive charges (radially outward) and toward negative charges (radially inward). For a positively charged wire, the direction is radially away from the wire; for a positively charged plane, the direction is perpendicular to the plane, away from it on both sides. Negative charges reverse all these directions.

Gauss’s law is always true for flux, but the shortcut Φ = E·A only applies when E is uniform and perpendicular to the surface over the entire Gaussian surface — a condition only met by highly symmetric distributions. Without that symmetry, you still know the flux (Q/ε₀) but cannot extract E without full integration.

In electrostatic equilibrium, free charges inside a conductor move until the internal fields cancel each other; if E were nonzero, charges would keep moving. Gauss’s law supports this: a Gaussian surface inside the conductor material encloses no net charge (all net charge is pushed to the outer surface), so the flux — and E — are zero.

Choose a surface that follows the charge distribution’s symmetry: concentric spheres for point/spherical charges, a coaxial cylinder for a straight wire, and a pillbox (flat cylinder) for a flat plane. The goal is for E to have constant magnitude and perpendicular direction over the whole surface, so ∮E·dA becomes the simple E×A.

No. Only the enclosed charge (Q_enclosed) enters Φ = Q/ε₀. Charges outside the surface do affect the electric field E at each point on the surface, but their flux contributions cancel each other — the net incoming flux equals the outgoing flux — so the total is zero.

This calculator computes electric field strength E (V/m) and flux Φ (V·m) from symmetric charge distributions. Electric potential V (volts) is energy per unit charge — a different scalar quantity. To compute V = kQ/r, use the Electric Potential Calculator.