Electric flux - Gauss’s law

Gauss’s law is an other way to express Coulomb’s law that quantifies the amount of force between two stationary electrically charged particles or the electric field due to a point charge. It is used to calculate the electric field due to a continuous distribution of charge in particular when there is some symmetry in the problem.

Gauss’s law is based on the concept of flux of field lines Φ, defined as the number of lines passing through a certain surface.

The question is therefore, how can we count the field lines passing through a surface?

The next figure represents in green any electric field. As you can see, the number of field lines passing through a surface (it is a plane in this example), is dependent upon the orientation of the field lines with respect to the surface. In the figure, no lines pass though the surface in (a), but the number of lines passing through it increases as the surface is inclined, with a maximum in (d).

Furthermore, to count the net number of field lines passing through a surface, we need to consider their direction. The net flux through the surface represented in the next figure is zero:

The mathematical operator used to “count” the number of field lines passing through a certain surface is the dot product. If we represent the surface using a vector dS perpendicular to it at any point and with its magnitude equal to the surface area, the flux of the electric field is defined as:

In the last two examples, the perpendicular vector to the (plane) surface always had the same orientation. But in the general case, vector dS will be different at each point of the surface. To calculate the flux passing through any surface, we decompose it into surface elements, we calculate the flux for each one of them and then we sum all these contributions. This explains why the flux definition includes an integral (or, and it is the same, a sum).

With the definition of the dot product we get:

As you can see in the first figure (a), when the angle between E and dS is 900, its cosine is zero and therefore the flux passing through the surface is zero. As the angle between the two decreases, its cosine increases until the situation represented in (d) where the cosine is 1 and therefore the flux passing through the plane is maximum.

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Gauss’s law

The net electric flux through any hypothetical closed surface is equal to the net electric charge within that closed surface divided by the vacuum permittivity ε0.

The closed surface through which the flux is calculated is called a Gaussian surface.

We are going to demonstrate Gauss’s law for a point charge. Let’s consider a positive point charge and the electric field lines due to it as the ones that are represented in the next figure.

We have represented in red a sphere of radius r that we will use as the Gaussian surface through which we will calculate the flux of the electric field. As you can see in the figure, the number of field lines passing through the sphere (flux) is independent of its position. We are going to calculate the flux in the situation (a) represented in the figure, with the sphere’s center is co-located with the charge. In this case, the flux is given by:

Furthermore, as you can see in the figure, the vector dS is parallel to ur at any point of the sphere. We can therefore write it as:

And after substituting in the flux expression and integrating, we obtain:

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Gauss’s law is fulfilled independently of the shape of the Gaussian surface. What is important is that it be a closed surface. As you can see in the next figure, if the sphere is enclosed in an arbitrary Gaussian surface, the number of field lines passing through both surfaces is the same.

If there is more than a single charge within the Gaussian surface, Gauss’s law is expressed as:

Where Σqi is the sum with their sign of the charges enclosed within the Gaussian surface.

You can see how Gauss’s law is used to calculate the electric field of different charge distributions using the links at the end of this page.

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