Gauss's Law Calculator

Created by Salam Moubarak, PhD
Reviewed by Hanna Pamuła, PhD and Jack Bowater
Last updated: Oct 10, 2022

Our Gauss law calculator allows you to compute the magnitude of the electric flux generated by the electric field of an electric charge.

In the following sections, we will first explain to you the concept of electric flux, then you will learn about the Gauss law equation and see how the total electric flux around an electric charge relates to the magnitude of that charge.

We'll finish by showing you how to use our Gauss law calculator and proposing some other similar calculators.

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 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=Qε0\phi = \frac{Q}{\varepsilon_0}

where:

  • ϕ\phi — The electric flux through the closed surface;
  • QQ — The total electric charge inside the surface; and
  • ε0\varepsilon_0 — The vacuum permittivity or permittivity of free space. It is equal to 8.854×1012 F/m8.854\times10^{-12}\ \mathrm{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=ϕε0Q =\phi\cdot\varepsilon_0.

Using the Gauss's law calculator

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

You can also click on Advanced mode to see the exact value of the vacuum permittivity ε0\varepsilon_0. Remember, it is constant and shouldn't be changed except in certain special cases.

The unit of electric flux used in this calculator is Vm\mathrm{V\cdot m} or, equivalently, Nm2/C\mathrm{N\cdot m^2/C}. The unit of electric charge is set by default to nC\mathrm{nC} (nanocoulomb) to get flux and charge numbers of similar orders of magnitude. You can however select a different unit for the electric charge.

For example, let's say you want to calculate the magnitude of the electric flux through a closed surface around a 10 nC10\ \mathrm{nC} electric charge. You can do so using our Gauss law calculator with two very simple steps:

  1. Enter the value 10 nC10\ \mathrm{nC}** in the field "Electric charge Q".
  2. The Gauss law calculator gives you the value of the electric flux in the field "Electric flux ϕ": In this case, ϕ=1129 Vm\phi = 1129\ \mathrm{V\cdot m}**.

Make sure to check out our other calculators if, for example, you need to know the magnitude of the electric field or the electric potential due to a point charge: the electric potential calculator snd the electric field calculator

. 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.

Salam Moubarak, PhD
Electric charge (Q)
nC
Electric flux (ϕ)
V·m
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