Gauss Law Calculator - Flux and E-field from Q

A gauss law calculator is an electromagnetism tool that turns the charge Q_enclosed inside any closed surface into the total electric flux Phi_E and the surface electric field E, using the integral form Phi_E = Q_enclosed / epsilon_0. Use it to solve the standard intro-physics problem of finding E from Q for a sphere, cylinder, or infinite sheet without doing the surface integral by hand.

• • Intro electromagnetism homework: Solve the textbook problem set where you are given Q and a Gaussian surface and asked for the flux or surface field, in SI units.
• • Symmetry problems in physics labs: Check the closed-form answers for spherical, cylindrical, and planar charge distributions against a worked example from a teaching lab.
• • Engineering back-of-envelope checks: Estimate the electric field outside a charged sphere, cylinder, or plane patch for an electrostatics or corona-discharge design problem.
• • Capacitance of simple geometries: Use the surface field of a charged sphere or plane with a parallel-plate formula to back out the capacitance of the geometry you are studying.

Unlike Coulomb's law, which gives the force between two point charges, Gauss law gives the total flux through any closed surface around a known charge distribution, which turns symmetry problems into one-line algebra.

It shines when the charge distribution has spherical, cylindrical, or planar symmetry, because only then can E be pulled outside the integral and solved directly.

For the two-charge point-force case that Gauss law generalises, a Coulomb's law calculator computes F = k_e q_1 q_2 / r^2 with the same NIST CODATA constant.

The gauss law calculator takes your enclosed charge Q, the Gaussian surface geometry, and the matching dimension, then solves Phi_E = Q / epsilon_0 for the flux and the geometry-specific closed-form surface field.

• Q_enclosed: Total charge inside the Gaussian surface, in coulombs. The calculator converts from mC, muC, nC, pC, or elementary charge units.
• epsilon_0: Vacuum permittivity, listed by NIST CODATA 2018 as 8.854 187 8128(13) x 10^-12 F/m. The 2019 SI redefinition left it as a derived constant (relative uncertainty near 1.5 x 10^-10), so the calculator embeds the CODATA 2018 digits rather than a redefinition figure.
• k_e = 1/(4 pi epsilon_0): Coulomb constant, equal to 8.988e9 N m^2 / C^2. Used in the sphere and cylinder closed-form surface fields.
• Geometry and dimension: Sphere uses radius r, cylinder uses r and L (lambda = Q / L), plane uses area A (sigma = Q / A).

The integral form Phi_E = E . dA = Q_enclosed / epsilon_0 is the master equation, and the calculator evaluates the closed-form surface field E for sphere, cylinder, and plane directly.

Because Phi_E depends only on Q and epsilon_0, it is the same number for every Gaussian surface that encloses the same charge; only E changes with geometry.

According to the NIST CODATA 2018 Fundamental Constants, epsilon_0 = 8.854 187 8128(13) x 10^-12 F/m with a relative uncertainty near 1.5 x 10^-10, because the 2019 SI redefinition left it as a derived constant rather than an exact definition; the gauss law calculator embeds those digits so flux answers stay aligned with modern metrology.

According to the NIST 2019 SI redefinition page, the integral form of Gauss law is Phi_E = Q_enclosed / epsilon_0, with closed-form surface fields E = k_e Q / r^2 (sphere), E = 2 k_e lambda / r (cylinder), and E = sigma / (2 epsilon_0) (infinite sheet); the calculator pulls E outside the integral only for those three symmetries.

When the next problem uses Gauss law to derive the capacitance of a parallel-plate or spherical capacitor, the capacitor charge calculator computes Q = C V for the chosen geometry and stored charge.

Four ideas drive every gauss law calculator answer. Once you understand them, the calculator's outputs map onto the textbook steps you are expected to show on paper.

These are the same steps you would write out by hand, which is why the calculator's intermediate conversions are worth tracing through.

Phi_E depends only on Q_enclosed, so the flux number is the same for any Gaussian surface that encloses the same charge.

When the worksheet moves from static electric fields to circuits, an Ohm's law calculator gives the V = I R relationship between voltage, current, and resistance using the same SI base units.

Pick a Gaussian surface geometry, enter the enclosed charge and matching dimension, and read the flux and surface field at once.

1. 1 Choose the geometry: Use the geometry dropdown to pick sphere, cylinder, or plane. The right dimension field applies for each choice.
2. 2 Enter the enclosed charge: Type the total charge Q_enclosed in the chosen unit. The calculator converts to coulombs automatically.
3. 3 Pick the matching dimension: Sphere uses radius r; cylinder uses r and L; plane uses area A. Unused dimensions are ignored.
4. 4 Read the flux and field: Phi_E is the same for any geometry that encloses the same charge; E changes because it is Phi_E divided by the geometry-specific area.
5. 5 Check the implied charge density: Sphere shows total Q, cylinder shows line density lambda = Q / L, plane shows surface density sigma = Q / A.

Try a 1 muC charge on a 10 cm by 10 cm plate: pick plane, enter 1.0 muC, set area = 0.01 m^2, and read sigma = 100 muC/m^2 with E = 5.65e6 N/C at the surface.

When the next step is the work done by the surface field on a test charge q, the work energy power calculator turns W = q E d and P = V I into the same SI answer you already used for the field.

The calculator handles the unit conversions and the closed-form geometry branching, so you can spend more time on physics and less on algebra.

• • Geometry-independent flux output: Phi_E = Q / epsilon_0 is reported in N m^2 / C, so the same answer works for any Gaussian surface that encloses the same charge.
• • Closed-form surface field for three symmetries: Get E = k_e Q / r^2, E = 2 k_e lambda / r, and E = sigma / (2 epsilon_0) by switching the geometry dropdown.
• • Six charge unit selectors: Type Q in C, mC, muC, nC, pC, or elementary charge units; the calculator converts to coulombs internally.
• • NIST CODATA 2018 vacuum permittivity: epsilon_0 = 8.854 187 8128(13) x 10^-12 F/m is embedded directly, and the 2019 SI redefinition left it as a derived constant (relative uncertainty near 1.5 x 10^-10), so flux and surface field answers stay aligned with the modern metrology roundings.
• • Real-time recalculation on phones: Outputs update as you type, handy on a tablet in the lab or at the physics help desk.

Use the flux value when comparing different Gaussian surfaces that enclose the same charge, because Phi_E is the same for all of them.

Use the surface field value when you want the magnitude of E at the surface, which is the answer most intro-E&M problems ask for.

When the question turns to F = q E on a charge inside the field, the forces and Newton's laws calculator converts the surface electric field from this calculator into a net force using Newton's second law.

The arithmetic is exact, but the choice of geometry and the assumption of vacuum decide how close the calculator's answer is to a real measurement.

• • Closed-form surface fields only work for spherical, cylindrical, or planar symmetry; for other shapes fall back on the integral form or superposed Coulomb's law.
• • The fields assume a vacuum; inside a dielectric with relative permittivity epsilon_r, multiply epsilon_0 by epsilon_r to scale E down.

Treat the calculator as a working tool for physics homework, intro E&M lab reports, and engineering back-of-envelope checks; for Maxwell-FEM work, switch to a numerical solver.

According to OpenStax University Physics Volume 2, Gauss law reduces to E A = Q_enclosed / epsilon_0 for spherical, cylindrical, and planar symmetry, with Q_enclosed = Q for a sphere, Q_enclosed = lambda L for a cylinder, and Q_enclosed = sigma A for an infinite sheet.

When Gauss law is combined with Ampere law to derive the speed of electromagnetic waves in vacuum, the wave speed calculator reports c from the permittivity and permeability values the calculator already uses for the surface field.