Gauss's Law Calculator

Gauss's law relates the net electric flux through a closed surface to the total electric charge enclosed by that surface. In vacuum, the law is Φ = Q/ε₀, where Φ is net electric flux, Q is enclosed charge, and ε₀ is vacuum permittivity. This calculator uses ε₀ = 8.8541878128 × 10⁻¹² farads per meter and converts the entered charge from nanocoulombs to coulombs. Electric flux measures how much electric field passes normally through a surface. Its SI unit is newton meter squared per coulomb. The total depends only on net enclosed charge, not on the surface's precise shape or the location of charges outside it. External charges can create fields on the surface, but their field lines enter and leave in equal net amounts, contributing zero to the closed-surface integral. The calculator also divides total flux by entered area to report a uniform normal electric field. That second result is valid only when symmetry makes the field magnitude constant across the Gaussian surface and the field is perpendicular to it. Common examples include a spherical surface centered on an isolated point charge, a cylindrical surface around a sufficiently long line charge when end effects are handled correctly, and a pillbox crossing an ideal infinite charged sheet. For arbitrary charge distributions or surfaces, Gauss's law remains true, but field strength cannot generally be found by simple division. Positive enclosed charge produces outward positive flux, while negative charge produces inward negative flux. Surface area must be positive even when charge is negative. Use this tool for electrostatics lessons, symmetric-field checks, and quick order-of-magnitude estimates. It assumes vacuum or air approximated as vacuum. Dielectric materials require care with permittivity, polarization, and whether the problem is written in terms of electric field or electric displacement. Complex geometries typically require direct integration or numerical field analysis rather than the uniform-field estimate.