Electric Flux: Definition, Formula, Derivation, Applications, & Class 12 Notes

When we visualise an electric field as field lines, it is understandable that the higher the number of lines passing through a region, the stronger the electric field is in that region. So, we measure the electric flux as the number of lines passing through that region. Since electric field lines are an imaginary concept. We can say that electric flux is a measure of the amount of electric field passing through a surface.

Electric flux is defined as the total electric field passing through a given surface. It is measured as the dot product of the electric field and the area of the surface it is passing through.  It also helps to visualize the interaction between the electric field and its surface area. SI Unit of electric flux is Volt·meter (V·m) or Newton·meter² per Coulomb (N·m²/C).

The  Electric Flux in mathematical terms: ${\mathrm{\Phi }}_E=\int \vec{E}\cdot d\vec{A}$

where:

• ${\mathrm{\Phi }}_E$ electric flux (in $\mathrm{N}{\mathrm{m}}^2{\mathrm{C}}^{-1}$  ),
• $\vec{E}$ electric field (in ${\mathrm{N}\mathrm{C}}^{-1}$  ),
• $d\vec{A}$ : infinitesimal area vector (in ${\mathrm{m}}^2$  ), with direction normal to the surface.

When the total area is A and the angle between the electric field $\vec{E}$ and the surface area A is $\theta$ . The electric flux will be:

${\mathrm{\Phi }}_E=EA\mathrm{c}\mathrm{o}\mathrm{s}\theta$ , where $\theta$  is the angle between $\vec{E}$  and the surface normal.

The dimensional formula of electric flux is 

$(M{L}^3{T}^{-\; <!--\; -\; -->3}{A}^{-\; <!--\; -\; -->1})$

Where, 

• M is mass
• L is length
• T denotes time
• A denotes electric current

Gauss's law states that the total electric flux passing through a closed surface is directly proportional to the electric charge enclosed within that surface. In simple words," the electric flux through a closed surface will be 1/ $\epsilon_0$

Mathematically, ${\mathrm{\Phi }}_E=\text{∮}\mathrm{}\mathrm{}\vec{E}\cdot d\vec{A}=\frac{q_{\mathrm{e}\mathrm{n}\mathrm{c}}}{{ϵ}_0}$

 

where:

${\mathrm{\Phi }}_E$ electric flux through the closed surface (in  ),

$\vec{E}$ electric field (in  ),

$d\vec{A}$ infinitesimal area vector on the surface (in  ),

$q_{\text{enc}}$ net charge enclosed (in C),

Electric flux, along with Gauss's law, has very important applications in calculating the electric field for difficult symmetries. Here are a few important applications of electric flux.

1. For any uniform electric field, it can help to find the enclosed charge on the basis of Gauss' law.
2. It helps in calculating the electric field in a region in a way easier method than vector addition.
3. Gauss's Law uses electric flux to derive the formula for the electric field for various charge distributions.

Important Link: NCERT Solutions 

 

The electric flux mathematical form is given as;

ϕ = EA = EAcosθ

As per the formula, the value of electric flux depends on the 3 quantities;

• Electric flux is directly proportional to the intensity of the electric field.
• Electric flux is directly proportional to the surface area, which means the larger the area is, the larger the value of electric flux will be.
• It is directly proportional to the cosine of the angle between the electric field and the area vector of the surface.

 

The flux density is the amount of flux passing through the surface. It is measured as the multiplication of the electric field (E) by the permittivity of the medium (ε).

Mathematical Definition

The electric flux density is expressed as 
$D=ϵE$

Where,

• D is electric flux density (C/m²)
• ε is the permittivity of the medium (F/m)
• E is the electric field (V/m)

 

Key Characteristics of Electric Flux Density

• Flux density has both magnitude and direction
• The unit of flux density is Coulombs per square (C/m²)
• Electric flux density accounts for free charges.
• The flux density depends on the permittivity of a dielectric material

The key properties of electric flux are as follows:

1. Electric flux is a scalar quantity, not a vector. Hence, it is calculated using the vector, and the result is a single number (can be either positive or negative). 

2. The flux depends on the angle of the electric field.

• When θ = 0° (field lines are perpendicular to the surface), then the electric flux is maximum.
• Electric flux is zero (0) when θ = 90° (field lines are parallel to the surface).

3. The flux is proportional to the electric field or surface area.

4. According to Gauss's law, the charge inside the closed surface affects the net flux, while the external charge does not affect the flux.

5. The electric flux moves outward from positive to inward to negative charge.

6. Electric flux is not dependent on the surface shape. 

7. Flux is zero if the field line is parallel to the surface.