Electric Flux Through Different Surfaces

From the discussion so far, we have got to know the relation between flux (Φ), Field Strength (E), and net area in the direction of field (A cosθ) as Φ = E A cos θ. So it far clear that electric flux through the surfaces depends on the area of the surface. Also according to the Gauss law, the total flux passing through a closed surface, depends on the net enclosed charge,

Φ = q_enclosed / ε₀

Where q_enclosed is the total charge enclosed in the surface.

Now, we will discuss the electric flux through closed and open surface, and look what they are:

• Flux through Closed Surfaces
• Flux through Open Surfaces
• Electric Flux Through Special Geometries

Let’s discuss these in detail.

Flux through Closed Surfaces

Any surface that completely encloses a three-dimensional region, is a closed surface, examples of closed surfaces include cubes, spheres, cylinders etc. Closed surfaces, according to Gauss’s Law, are critical in understanding the relationship between the total electric flux passing through a surface and the charge enclosed within it.

According to Gauss’s Law that the total electric flux through a closed surface is proportional to the total charge enclosed by that surface, divided by the permittivity of the medium. The symmetric nature of closed surfaces simplifies the calculation of electric flux, enabling straightforward application of Gauss’s Law.

Φ = q_enclosed/ε

where,

• q is the total charged enclosed inside a closed surface, and
• ε is the permittivity of the medium.

Flux through Open Surfaces

Unlike the closed surfaces the open surface doesn’t have a closed boundary and thus doesn’t encloses a volume. The direct application of Gauss’s Law become difficult in case of open surfaces, and thus determining the flux through open surface require integration of dot product of the electric field and the surface area vector over the entire surface.

These calculations are more complex than that of the closed surfaces due to the lack of symmetry, and they involve integrating over irregularly shaped surfaces. Open surfaces includes planes, sheets, rings etc.

The flux Φ through an open surface can be determined using the integral calculation:

Φ = ∲ E ․ dA

Where

• E is the electric field,
• dA is the small area element from the surface, and
• The dot product of the electric field and the differential area vector is integrated over the entire open surface to calculate the total flux.

Electric Flux Through Special Geometries

Electric Flux thorough various special geometries are listed in the following table:

Electric flux is a fundamental concept in physics that helps us understand and quantify the electric field passing through a given surface. It provides a means to describe the flow of electric field lines through an area. Electric flux forms the basis of Gauss’s Law, to calculate the net charge enclosed inside a given Gaussian surface, which says that the flux through a surface will be the result of the total (or net) charge enclosed inside it.

In this article, we will learn about the basics of electric flux, its types, electric flux density, and electric flux through various types of surfaces.