About the Gauss's Law Calculator
Gauss's Law is one of the four Maxwell equations that govern electricity and magnetism. It states that the total electric flux ΦE passing through any closed ("Gaussian") surface is proportional to the electric charge enclosed by that surface: ΦE = ∮E·dA = Q_enc / ε, where ε is the permittivity of the surrounding medium. This calculator computes that flux from an enclosed charge and permittivity, and — assuming the charge distribution is spherically symmetric — the resulting electric field at a chosen radius from the center.
How the calculation works
Enter the enclosed charge Q_enc and its unit, then the relative permittivity εr of the surrounding medium (εr = 1 for vacuum or air; higher for dielectrics such as glass, mica, or water). The calculator first finds the absolute permittivity ε = εr × ε₀, where ε₀ ≈ 8.854 × 10⁻¹² C²/(N·m²) is the permittivity of free space. It then applies Gauss's Law directly to get the flux, ΦE = Q_enc / ε. Finally, treating the Gaussian surface as a sphere of radius r centered on a spherically symmetric charge (a point charge or a uniformly charged sphere), it divides the flux by the sphere's surface area to get the electric field: E = ΦE / (4πr²), which is algebraically identical to the Coulomb's-law field E = Q_enc / (4πεr²).
Choosing the Gaussian surface and symmetry
Gauss's Law is true for any closed surface — the flux never depends on the surface's shape, only on the charge it encloses. But you can only solve for E by itself when the surface is chosen to match the symmetry of the charge distribution, so that E has the same magnitude everywhere on the surface and points either parallel or perpendicular to it. A sphere works for a point charge or any spherically symmetric charge; an infinite cylinder works for a uniform line charge; a "pillbox" works for an infinite charged plane. This calculator assumes the common spherical case for its electric-field result.
Working with charge, permittivity, and units
- Enter Q_enc in whichever unit is most natural (µC and nC are common in lab settings); the calculator converts to coulombs internally.
- Relative permittivity εr is dimensionless: 1.000 for vacuum, about 1.0006 for air, roughly 2–4 for common plastics, about 4–7 for glass, and about 80 for liquid water at room temperature.
- Radius r must be measured from the center of symmetry of the charge to the Gaussian surface, not from an edge — and it must be positive and outside (or on) the charge distribution for the point-charge-style field formula to apply.
- Negative enclosed charge is valid and gives a negative flux, meaning the net field points inward through the surface.