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.

What is Electric Flux?

Electric flux is the estimation of the total number of electric field lines (imaginary lines considered around a charged particle, these are thought to originate from the positive electric charges and thought to sink in negative electric charges), passing through a given closed surface, it can also be defined as the total charge contained in a closed surface (Gauss’s Law). It is a vector quantity and has direction. The electric flux is considered to have a sign associated with it which denotes whether the electric field lines are entering into the surface or coming out of the surface. Thus,

Definition of Electric Flux

Electric flux is the measure of electric lines of force (or electric field lines) passing through a given closed surface. It is a scalar quantity, representing the total number of electric field lines passing through a given surface.

Electric Flux Symbol

Electric flux is denoted by a Greek letter Φ, which is pronounced as phi i.e., Φ.

Electric Flux Formula

The electric flux depends on the different parameters namely, the strength of the electric lines of forces, the area of the surfaces and it also depends on the orientation between the surface area and the electric lines of forces. These quantities together yields electric fields through the surface and they are related as:

Φ = E • A

OR

Φ = E A cos θ

Where,

• Φ denotes the electric flux,
• E denotes the electric field strength,
• A denotes the area of the closed surface, and
• θ denotes the angle between the electric field lines and the area vector

Factors Affecting Electric Flux

Some of the factors affecting electric flux are:

• Electric Field Strength (E): Φ ∝ E, thus it increase with the increase in electric filed lines.
• Area of the Surface (A): Φ ∝ A, thus its magnitude increases with increase of the area.
• cos θ : It attains the maximum value of 1 for θ = 0, and minimum value as -1 for θ = 180.

To take a deep dive into the topic how electric flux through continuous charge distribution works and what are surface, refer the articles:

• Gauss’ Law
• Application of Gauss’ Law

SI Unit of Electric Flux

The unit of electric flux can be derived from putting the units of different values in the formula for calculating electric flux.

Φ = E × A × cos θ

Therefore,

Unit of (Φ) = Unit of (E) × Unit of (A) × Unit of cos θ

OR Unit of (Φ) = (V/m) × (m²) × 1

OR Unit of (Φ) = V-m.

Thus, the SI unit of electric flux is V-m (Volt-meter).

Electric Flux Dimensional Formula

Since electric flux depends on the some parameters, therefore dimensional formula of electric flux can be derived by putting the dimensional formula of the quantities together in the formula of electric flux.

[Φ] = [E] • [A]

[Φ] = [MLT^-3A^-1] • [L²] = [ML³T^-3A^-1]

Therefore, dimensional formula of electric flux is [ML³T^-3A^-1].

Types of Electric Flux

Since the electric flux also depends on the angle between the field lines and the area vector, it can have a negative or positive value.

• Positive Electric Flux: When the electric field lines pass outward through a closed surface, the electric flux is considered positive. This occurs when the electric field lines are in the same direction as the outward-pointing normal vector to the surface.
• Negative Electric Flux: When the electric field lines pass inward through a closed surface, the electric flux is considered negative. This happens when the electric field lines are in the opposite direction to the outward-pointing normal vector to the surface.

Properties of Electric Flux

Electric flux has several key properties that helps in understanding and analyzing electric fields. Some of the significant properties of electric flux are mentioned below:

• Electric flux is a scalar quantity, it has only magnitude with no direction. It quantifies the total number of electric field lines passing through a given surface, irrespective of its direction.
• The electric flux through a surface is directly proportional to the strength of the electric field passing through the surface (E). A stronger electric field will result in higher electric flux through the surface.
• It also depend on the angle between the area vector and the field lines.
• Electric flux follows the principle of superposition, i.e the total flux through a surface, is the sum of the individual fluxes through different parts of the surface.

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:

What is Electric Flux Density?

Electric field density is yet another important concept in electromagnetism, which is allows us to understand and predict how electric fields interact within substances, including insulators, conductors, and dielectrics. It signifies the amount of electric flux passing through a specific area within the material. It is also defined as the sum of the free charge effect (expressed through the electric field, E) and the impact of the material’s polarization (P) due to an external electric field. It is also referred to as electric displacement.

Electric field density is defined as, the electric flux passing through a unit area perpendicular to the direction of electric flux.

Electric Flux Density Formula

Electric flux is denoted by the symbol D. The formula for electric flux can be given as,

D = ε₀E + P

Where,

• D is the electric flux density vector,
• ε₀ is the permittivity of free space,
• E is the electric field strength and,
• P is the polarization vector, representing the dipole moment induced in the material per unit volume due to an external electric field.

SI Unit of Electric Flux Density

The SI unit of electric flux density can be derived from putting the units of different values in the formula of electric flux density.

D = ε₀E + P

Putting,

• farads per meter (F/m) as unit of ε₀
• volts per meter (V/m) as unit of E and,
• Polarization has a unit coulombs per square meter (C/m²).

The expression will yield the unit of Electric Flux density as C/m².

The SI unit of electric flux density is coulombs per square meter (C/m²).

Electric Flux Density Dimensional Formula

The formula for Electric Flux Density can also be represented as,

D = Φ․A

Putting the dimensions of each quantities together we can get the dimensional formula of Electric Flux Density,

[D] = [Φ]․[A]

[D] = [ML³T^-3A^-1]/[L²]

Hence the dimensional formula of the Electric Flux Density Dimensional as [ML¹T^-3A^-1].

Applications of Electric Flux

Electric flux is the basic behind various concepts in physics, including:

• Electric flux plays a key role in Gauss’s law, which relates the total electric flux through a closed surface and the total charge enclosed inside the surface.
• Electric flux is also used to understand the behavior of capacitors, which store electrical energy.
• Electric flux is used in the study of electromagnetic induction. When a magnetic field passing through a closed loop changes, it induces an electric field, and the concept of flux helps in understanding the induced electromotive force.
• Electric flux plays a crucial role in the study of dielectric materials.

Read More,

• Magnetic Flux
• Lorentz Force
• Electric Field Lines

Solved Problems on Electric Flux

1. The surface of area 5 m² when an electric field of 2 N/C makes an angle of 180 degrees with the surface. What is the flux passing through the surface?

Given, A = 5 m², E = 2 N/C and θ = 180

putting everything in the formula, Φ = E A cos θ

Φ = 2 × 5 × cos(180) = 10 × -1 = -10,

where negative sign indicates that the electric field lines are leaving the surface.

2. Derive the unit of electric flux.

Since,

Φ = E • A

Putting the unit of E as Volt per meter (V/m) and the unit of A as m²,

The unit of Φ = V/m × m²

Unit of Φ= V-m.

3. Derive the dimensional formula for electric flux.

We know the unit of Φ as V-m, putting the dimensions of the quantity in the formula the we get

Dimensional formula of electric flux = [MLT^-3A^-1] × [L²]

Dimensional formula of electric flux = [ML³T^-3A^-1].

Practice Problems on Electric Flux

Problem 1: Calculate the electric flux through a surface of area 1.414 m² when an electric field of 5 N/C makes an angle of 45 degrees with the surface.

Problem 2: A plane surface has an electric field of 100 N/C directed perpendicular to it. Calculate the electric flux through the surface if the area is 10 m².

Problem 3: Given a surface of area 5 m² and the electric filed in the region as 10 N/C. The flux passing through the surface is 0. What is the angle between the area vector and electric field vector?

Problem 4: Can an object having a considerable area when placed in a considerable electric field having 0 electric flux passing through it? If yes explain.

Electric Flux: FAQs

1. Explain Electric Flux.

Electric flux is a measure of the total number of electric field lines passing through a given area. It is the measure of the strength of an electric field passing through a surface.

2. How to Calculate Electric Flux?

Electric flux (Φ) through a surface can be calculated as the dot product of the electric field (E) passing through the surface and the surface area (A) i.e.,

Φ = E • A = E A cos θ

3. What is the Unit of Electric Flux?

The SI unit of electric flux is the Volt-meter (V⋅m).

4. Define Electric Flux Density.

Electric Flux Density (D) is a measure of electric flux per unit area. It represents the electric field strength in a material, indicating how much electric flux passes through a given area.

5. What is the Dimensional Formula for Electric Flux?

The dimensional formula for electric flux is [ML³T^-3A^-1].

6. Can Electric Flux be Zero?

Yes, electric flux can be negative. It depends on the orientation of the surface with respect to the electric field.

7. What Does a Positive or Negative Sign in Electric Flux Indicate?

A positive electric flux indicates that the electric field is penetrating the surface in the direction of the normal to the surface, while a negative electric flux implies that the electric field is leaving the surface or penetrating in the opposite direction.