Gauss's Law Calculator — Calculate the Electric Flux φ = Q/ε₀

Electric Flux

Electric flux is usually calculated for a given surface. It depends on the strength of the electric field going through the surface, the area of that surface, and the orientation of the electric field lines relative to the surface.

In the general case where the surface is not necessarily flat and the electric field is not uniform, the electric flux is calculated in its integral form, where the surface is divided into very small flat surfaces and the electric field lines are orthogonally projected through those surfaces.

When a closed surface surrounds an electric charge, the Gauss's law equation becomes very handy for calculating the total electric flux through that surface and bypassing all the integral-form calculations.

What is Gauss's Law?

Gauss's law stipulates that when we consider a completely closed surface around an electric charge, the total electric flux through that surface is only proportional to the strength of that charge; it is independent of the shape and size of the surface and the exact position and distribution of the electric charge inside that surface.

This proportionality is expressed in the Gauss's law equation:

φ = Q / ε₀

where:

• φ — the electric flux through the closed surface;
• Q — the total electric charge inside the surface; and
• ε₀ — the vacuum permittivity (permittivity of free space). It is equal to 8.854 × 10⁻¹² F/m.

Conversely, if we know the electric flux through a closed surface, we can also use Gauss's law to calculate the total electric charge inside that surface:

Q = φ · ε₀

Using the Gauss's Law Calculator

When using the Gauss's law calculator, you can either input the value of the electric charge Q to receive the electric flux φ, or you can provide the electric flux φ and the calculator will give you the corresponding electric charge Q.

You can also see the exact value of the vacuum permittivity ε₀. Remember, it is constant and shouldn't be changed except in certain special cases.

The unit of electric flux used in this calculator is V·m or, equivalently, N·m²/C. The unit of electric charge is set by default to nC (nanocoulomb) to get flux and charge numbers of similar orders of magnitude. You can, however, select a different unit for the electric charge.

Unit Systems

Metric (SI)

• Charge: coulombs (C), millicoulombs (mC), microcoulombs (μC), nanocoulombs (nC), picocoulombs (pC), elementary charge (e)
• Flux: V·m, kV·m, MV·m, GV·m (equivalently N·m²/C)

American (CGS/Gaussian)

• Charge: statcoulombs (statC = esu), elementary charge (e), microcoulombs (μC), nanocoulombs (nC)
• Flux: V·m and its multiples

Useful conversion: 1 statC = 3.3356 × 10⁻¹⁰ C, and the elementary charge is e = 1.602 × 10⁻¹⁹ C.

Worked Example

Problem: Calculate the magnitude of the electric flux through a closed surface around a 10 nC electric charge.

Solution:

φ = Q / ε₀ = 10 × 10⁻⁹ C / (8.854 × 10⁻¹² F/m)
φ ≈ 1129 V·m

So a charge of 10 nC produces an electric flux of about 1129 V·m (equivalently, 1129 N·m²/C) through any closed surface that encloses it — regardless of the surface's shape or size.

How to Use This Calculator

1. Select your unit system: Metric (SI) or American (CGS).
2. Choose the calculation mode:
   • Charge → Flux — enter the enclosed charge Q to get the electric flux φ.
   • Flux → Charge — enter the electric flux φ to get the enclosed charge Q.
3. Click Calculate. Results are shown in V·m / N·m²/C (flux) or C and nC (charge).
4. Use the Share button to get a direct link to your calculation.

Gauss Law Problems and Related Subjects

Make sure to check out our other electromagnetism calculators if, for example, you need to know the magnitude of the electric field or the electric potential due to a point charge. You can also solve more elementary problems like calculating the electrostatic force between two charged particles, with our Coulomb's law calculator, or finding out how a magnetic field affects these particles, with our handy Lorentz force calculator.

FAQs

What is the unit of electric flux?

The SI unit of electric flux is the volt-meter (V·m), which is equivalent to the newton-meter² per coulomb (N·m²/C).

Does the shape of the closed surface matter?

No. Gauss's law tells us that the total electric flux through a closed surface depends only on the total charge enclosed — not on the shape, size, or the exact position of the charge inside.

What is the value of the vacuum permittivity ε₀?

The vacuum permittivity (permittivity of free space) is ε₀ = 8.854 × 10⁻¹² F/m. It is a fundamental physical constant.

Can I find the charge from a known flux?

Yes. Switch to the Flux → Charge mode and enter the electric flux. The calculator returns the enclosed charge using Q = φ · ε₀.