1. Concept Overview
Electric flux gives a measure of how much electric field passes through a given surface. It acts like counting the number of electric field lines crossing that surface. This idea becomes very important in Gauss’s law, where flux helps relate electric fields to enclosed charge.
2. Definition
Electric flux through a surface is defined as the product of the electric field and the area projected in the direction of the field.
Mathematically:
\( \Phi_E = \vec{E} \cdot \vec{A} = EA \cos \theta \)
Here, \( A \) is the area vector, and \( \theta \) is the angle between \( \vec{E} \) and the normal to the surface.
3. Understanding the Area Vector
Every surface has an associated direction, called the area vector. For a flat surface, the direction is perpendicular to (normal to) the surface.
3.1. Area Vector Definition
The area vector \( \vec{A} \) for a flat surface of area \( A \) is a vector of magnitude \( A \), pointing normal to the surface.
3.2. Interpretation of \( \theta \)
The angle \( \theta \) measures how the field is oriented relative to the surface:
- \( \theta = 0^\circ \): Field is perpendicular → flux is maximum.
- \( \theta = 90^\circ \): Field is parallel → flux is zero.
4. General Expression for Flux
For non-uniform electric fields or curved surfaces, the surface is divided into tiny elements, and flux is summed over them:
\( \Phi_E = \int \vec{E} \cdot d\vec{A} \)
4.1. Meaning of the Integral
Each small patch contributes its own small flux value, and the integral adds them up over the whole surface.
5. Flux Through Different Orientations
The amount of flux depends on how the surface is tilted relative to the field.
5.1. Surface Perpendicular to Field
\( \theta = 0^\circ \), so flux is:
\( \Phi_E = EA \)
This gives the highest flux.
5.2. Surface Parallel to Field
\( \theta = 90^\circ \), so flux is:
\( \Phi_E = 0 \)
No field lines pass through the surface.
5.3. Tilted Surface
Flux becomes:
\( \Phi_E = EA \cos \theta \)
Only the component of the field perpendicular to the surface contributes.
6. Worked Examples
6.1. Example 1: Simple Flat Surface
An electric field of \( 200 \text{ N/C} \) passes through a square sheet of area \( 0.5 \text{ m}^2 \). The field is perpendicular to the sheet.
Flux: \( \Phi_E = EA = 200 \times 0.5 = 100 \text{ N m}^2/\text{C} \).
6.2. Example 2: Inclined Surface
Electric field \( E = 300 \text{ N/C} \) makes an angle of \( 60^\circ \) with the normal of an area \( 0.2 \text{ m}^2 \).
Flux: \( \Phi_E = EA \cos 60^\circ = 300 \times 0.2 \times 0.5 = 30 \text{ N m}^2/\text{C} \).
7. Physical Interpretation
Electric flux gives a sense of how many field lines pass through a surface. More lines crossing means greater flux. This concept helps relate electric fields to charge distribution, forming the basis of Gauss’s law, which uses flux to determine electric fields around symmetric charge configurations.