Gauss's Law Calculator

Created by Kenneth Alambra
Last updated: Sep 14, 2022

If you're curious about how to calculate the electric flux passing through a surface given the electric charge it has, this Gauss law calculator is for you. Performing Gauss's law calculations is surprisingly easy, and this tool will show you exactly how. Keep on reading to learn:

  • What Gauss's Law is;
  • How to calculate electric flux; and
  • How to use this Gauss law calculator.

What is Gauss's law?

Gauss's law states that the electric flux coming out of a closed surface of any size is proportional to the electric charge around it and varies inversely due to the permittivity of free space constant.

Electric flux is a representation of the electric field passing through a surface. The more electric field lines we can observe radiating a surface, the stronger the electric flux is. We use Gauss's law to quantify the strength of that electric field by determining its equivalent electric flux.

We usually find the total electric flux through an object by integrating the electric flux through infinitesimally small flat areas on a surface. In this tool, we consider a closed flat surface enclosed by a uniform electric field from a single electric charge.

In the next section of this text, let us learn about the Gauss law equation we can use in calculating electric flux.

How to calculate electric flux using the Gauss law formula

When determining the electric flux through a closed flat surface, we use the Gauss law formula, as shown below:

ϕ=Qε0\phi = \frac{Q}{\varepsilon_0}

Where:

  • ϕ\phi is the electric flux through the closed surface in volt-meters (Vm\small{\text{V}\cdot\text{m}});
  • QQ is the total electric charge inside the surface in coulombs (C\small\text{C}); and
  • ε0\varepsilon_0 is permittivity of free space electric constant with the value of 8.854×1012 C2Nm2\small{8.854\times 10^{-12}\ \tfrac{\text{C}^2}{\text{N}\cdot\text{m}^2}}.

Let's say we have a closed surface inside an electric charge of 20 nanocoulombs or 2.0×108 coulombs\small{2.0\times 10^{-8}\ \text{coulombs}}, substituting that to the Gauss's law equation gives us:

ϕ=Qε0=2.0×108 C8.854×1012 C2Nm2=2258.866049 Nm2C2259 Vm\begin{align*} \phi &= \frac{Q}{\varepsilon_0}\\[1.0em] &=\frac{2.0\times 10^{-8}\ \text{C}}{8.854\times 10^{-12}\ \tfrac{\text{C}^2}{\text{N}\cdot\text{m}^2}}\\[1.5em] &= 2258.866049\ \tfrac{\text{N}\cdot\text{m}^2}{\text{C}}\\[0.8em] &\approx 2259\ \text{V}\cdot\text{m} \end{align*}

Note that we can use Vm\small{\text{V}\cdot\text{m}} to replace Nm2C\small{\tfrac{\text{N}\cdot\text{m}^2}{\text{C}}} since 1 NmC\small{1\ \tfrac{\text{N}\cdot\text{m}}{\text{C}}} is equivalent to 1 V\small{1\ \text{V}}. You can learn more about the unit of volts (V) by checking out our electric potential calculator.

💡 Newtons per coulomb or NC\tfrac{\text{N}}{\text{C}}, as shown occurring in the units we use above, is the unit of electric field. You can expound your knowledge of the topic by exploring our electric field calculator.

As additional knowledge, we can also find the electric charge around a surface given the electric flux passing through it by rearranging the Gauss law equation, expressed in equation form as shown below:

Q=ϕ×ε0Q = \phi\times\varepsilon_0

How to use this Gauss law calculator

Our Gauss law calculator is very simple and works both ways. That means you can either enter a value for the electric charge to find the electric flux, or vice versa. That's how easy-to-use our Gauss law calculator is!

Suppose you need to experiment on other dielectric materials and, therefore, need to enter different values for permittivity. In that case, you can do that by clicking on the Advanced mode button below our Gauss's law calculator. Doing so will display the permittivity variable where you can enter other values.

Want to learn more?

If you're still craving more electromagnetic stuff, perhaps you'll find our Coulomb's law calculator interesting. There you'll learn how to determine the electrostatic force acting between two charges.

Kenneth Alambra
Electric charge (Q)
nC
Electric flux (ϕ)
Vm
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