Understanding Gauss’s Law: The Key to Understanding Electric Fields

Gauss’s Law

Understanding Gauss’s law is a bit different from understanding the electric field. In both cases, you need to know how to calculate the total electric field in a closed system in an electric field. But in the electric field case, you do that by knowing the total charge in the system. In the Gaussian case, you do that by knowing the charge density in the system.

What is Gauss’s Law

The electric flux through an area is calculated by multiplying the electric field by the surface area projected in a plane perpendicular to the field. Gauss’ Law is a general principle that can be applied to any closed surface. It’s a helpful technique since it allows you to estimate how much contained charge there is by mapping the field on a surface outside the charge distribution. It simplifies the computation of the electric field for geometries with adequate symmetry.

Define Gauss’s law

The charge contained divided by the permittivity equals the total electric flux out of a closed surface.

Gauss's Law formula
Gauss’s Law formula. Image source: hyperphysics.phy-astr.gsu.edu

Another approach to visualize this is to imagine a probe with an area of A that can measure the electric field perpendicular to the region. It can get a measure of the net electric charge within any closed surface by stepping over it and measuring the perpendicular field times its area, regardless of how that internal charge is structured.

Gauss’ Law Applications

Gauss’s law is the most important law of electromagnetism because it is the basis of how current is conducted within a conductor. It also provides us with the natural laws that describe the electric field and the electric charge in a vacuum (minus the electric charge on an object outside the field, which is not included in the electric field).

The Gaussian distribution for the electric field is the basis for estimating the magnitude of a photon’s intensity; this is the basis of the law of thermal radiation. Gauss’s law is the fundamental law of electromagnetic fields and is vital to the success of modern radio communication (radio waves and microwaves), radar, GPS, satellite and other related technologies.

When electric fields arise from charge distributions with adequate symmetry, Gauss’ law is a valuable tool for calculating them.

Applications of Gauss' Law
Gauss’ Law Applications: Image source: hyperphysics.phy-astr.gsu.edu

Key Takeaways from Gauss’s Law

This theory is said to be fully valid up to an emf of one trillion voltvoltsper cm2 – approximating a point charge of 10 uF. of one trillion voltvoltsper cm-2 – approximating a point charge of 10 uF. If applied to the surface of a hollow cylindrical metal such as copper, an emf of ~1 wattm – 1 at this emf may be observed. – 1 at this emf may be observed. Since this emf is known, it can be applied to other areas.

The principle of Electric Potential (Principle of Inverse Square Law) is just as powerful as the principle of Gauss’s law, in the sense that it quantifies how any electric field propagates in space. It has the property that the electric field gradient is also the electric potential gradient.

How to Use Gauss’s Law for Different Situations

In cases when Gauss’s law is written as a series, with the surface area enclosed as “r”, and the electric charge formula_3 enclosed by the surface as “p”, the constant constant “k” at each point is the amplitude of the electric field in that point: However, note that the output may still be nonzero, even when “k” is large.

In particular, in some situations, it is useful to construct an expression that allows the output to be shown to depend only on “k”, without any other necessary spatial variables: that is, only on the vector containing “k”. This provides a way of highlighting a different physical property of the electric field: its dependence on the geometry of the surface enclosed by the charge.

Calculating Gauss’s Law

The simplest way to calculate Gauss’s law is to use an empirical, mathematical model that describes how electricity moves in a wire. It is called an approximate model because Gauss’s law is a law of nature. It must be used as an approximate model because the actual electric field is unknown and there is no one way to describe it.

Our approximate model will be a one-dimensional surface called a Gaussian, which has the same general shape as a real conductor and a Gaussian surface which has a spherical shape. This surface is very simple and is based on the type of surface that Gauss used in his work. For example, this surface could have been the surface of a cube, a sphere, or even a flat plane.

Gauss’s law Sphere

Because we already know the electric field in such a circumstance, let’s compute the electric flux through a spherical surface surrounding a positive point charge q. Remember that when a point charge is placed at the origin of a coordinate system, the electric field at a point P that is r away from the charge at the origin is given by

gauss's law equation

The radial vector from the charge at the origin to the point P is denoted by r. As illustrated in Figure, we can utilize this electric field to calculate the flux across a spherical surface of radius r.

A point charge q is surrounded by a closed spherical surface.

gauss law sphere
gauss law sphere

Then we use

gauss's law equation

to replace known values in this system. Because n=r and r=R on the sphere, given an infinitesimal area dA,

gauss's law equation

We can now calculate the net flow by integrating this flux across the sphere’s surface:

gauss's law equation

Where the spherical surface’s entire surface area is 000. The flow through the closed spherical surface at radius r is calculated as

gauss's law equation

The flow is independent of the size of the spherical surface, which is a noteworthy feature of this equation. This is due to the fact that the electric field of a point charge drops with distance as 1/r2, cancelling out the r2 rate of increase of the surface area.


Understanding Gauss’s law is extremely important for creating transistors and other diodes because it must be applied before the voltage regulator’s circuit designer can apply the maximum current to any given area on the circuit. Since Gauss’s law can be derived without knowing any of the details of the circuit being designed, it is a principle that almost always applies even if other properties of the circuit are known.

This is one of the principles of circuit design known as the Zener rule. In physics and electromagnetism, Gauss’s law, also known as Gauss’s flux theorem, (or sometimes simply called Gauss’s theorem) is a law relating the distribution of electric charge to the resulting electric field.


  1. opentextbc.ca
  2. hyperphysics.phy-astr.gsu.edu
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