Gauss's Law Calculator — Electric Flux and Field (Φ = Q/ε₀)
Enter the enclosed electric charge and the radius of the Gaussian surface to compute the total electric flux and the electric field strength at that radius using Gauss's law.
Charge unit
m
Total flux through any closed surface enclosing Q (Gauss’s law: Φ = Q / ε₀)
- 1
Enclosed charge Q in coulombs
10 nC × 10⁻⁹ = 0.00000001 - 2
Electric flux Φ = Q ÷ ε₀
0.00000001 ÷ 8.854 × 10⁻¹² = 1,129.4091ε₀ = 8.854 × 10⁻¹² F/m is the vacuum permittivity — flux depends only on enclosed charge.
How does this calculator work?
Gauss's law: Φ = Q / ε₀ (ε₀ = 8.854 × 10⁻¹² F/m). For a spherical Gaussian surface of radius r around a point charge Q, the electric field is E = kQ/r² where k = 8.988 × 10⁹ N·m²/C². Flux depends only on enclosed charge, not surface size or shape. Positive Q → field points outward; negative Q → inward.
Formula
How this is calculated
Gauss's law states that the total electric flux Φ through any closed surface is equal to the net charge enclosed by that surface divided by the vacuum permittivity ε₀: Φ = Q_enc / ε₀. Flux is measured in N·m²/C (equivalently V·m) and is independent of the shape or size of the Gaussian surface — only the enclosed charge matters.
For a spherically symmetric charge distribution (a point charge, uniformly charged sphere, or spherical shell), choosing a concentric spherical Gaussian surface of radius r makes the electric field E constant and perpendicular to the surface everywhere. Applying Gauss's law gives E × 4πr² = Q/ε₀, so E = Q/(4πε₀r²) = kQ/r², where k = 1/(4πε₀) = 8.988 × 10⁹ N·m²/C² is Coulomb's constant. This is identical to the field from a point charge at distance r.
Gauss's law applies to any geometry, not just spheres: for an infinite line charge (linear charge density λ) the field at radius r is λ/(2πε₀r); for an infinite plane (surface charge density σ) it is σ/(2ε₀). This calculator implements the spherical case, which covers point charges and any spherically symmetric charge distribution. Note that E here is the field outside the surface; for a uniformly charged sphere, the field inside depends on the enclosed charge at that radius.
Frequently asked questions
Gauss's law (Φ = Q/ε₀) relates the electric flux through a closed surface to the enclosed charge. It is always true but most useful when the charge distribution has a high symmetry (spherical, cylindrical, or planar), because symmetry lets you factor E out of the surface integral and solve for it algebraically — much faster than Coulomb's law for extended charge distributions.
Electric flux Φ measures how many electric field lines pass through a surface. Mathematically Φ = ∫ E · dA. Its SI unit is N·m²/C (= V·m). For a uniform field perpendicular to a flat surface, Φ = E × A. Positive flux means field lines exit the surface (net positive charge inside); negative flux means they enter (net negative charge inside).
No. By Gauss's law, Φ = Q_enc/ε₀ depends only on the net enclosed charge, not on the shape or size of the surface. A small sphere and a large irregular surface enclosing the same charge have the same total flux. The shape choice only affects how you calculate E — spherical, cylindrical, and planar surfaces simplify E by symmetry.
Also known as
TG we-Calculate Editorial Team. (2026). Gauss's Law Calculator — Electric Flux and Field (Φ = Q/ε₀) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/gauss-law-calculator
TG we-Calculate Editorial Team. "Gauss's Law Calculator — Electric Flux and Field (Φ = Q/ε₀)." TG we-Calculate. 2026. https://we-calculate.com/calculator/gauss-law-calculator.
TG we-Calculate Editorial Team, "Gauss's Law Calculator — Electric Flux and Field (Φ = Q/ε₀)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/gauss-law-calculator
@misc{wecalculate_gauss_law_calculator, title = {Gauss's Law Calculator — Electric Flux and Field (Φ = Q/ε₀)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/gauss-law-calculator}}, year = {2026}, note = {TG we-Calculate} }
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