## **Definition**
>Electric flux ($\Phi_E$) refers to the number of electric field lines passing through a given surface placed in an electric field. It represents the flow of electric field lines through the surface and is mathematically expressed as:
$
\Phi_E = \vec{E} \cdot \vec{A}
$
Where:
- $\Phi_E$: Electric flux
- $\vec{E}$: Electric field vector
- $\vec{A}$: Area vector, whose magnitude equals the surface area and direction is perpendicular to the surface.
To learn more about Electric fields. Refer to [[Electric Field]]
and [[Representation of Electric Field Lines]]
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## **Diagram**
![[Pasted image 20241130095047.png]]
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## **Nature**
Electric flux is a scalar quantity obtained from the dot product of the electric field $\vec{E}$ and the area vector $\vec{A}$. The **SI unit** of electric flux is:
$
\text{Unit of } \Phi_E = \text{unit of } \vec{E} \cdot \text{unit of } \vec{A} = N \cdot m^2 \cdot C^{-1}
$
---
## **Flux at Any Angle**
When the area $\vec{A}$ is tilted at an angle $\theta$ with respect to the electric field $\vec{E}$, the electric flux is calculated as:
$
\Phi_E = E A \cos \theta
$
Where $\cos \theta$ accounts for the orientation of the surface relative to the electric field.
### **Diagram**
![[Pasted image 20241130095119.png]]
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## **Dependence**
Electric flux depends on:
1. **Surface Area ($A$)**: Larger surface areas allow more field lines to pass through.
2. **Electric Field Intensity ($E$)**: Stronger fields result in more flux.
3. **Angle ($\theta$)**: The orientation of the surface impacts the flux:
- $\theta = 0^\circ$: Maximum flux
- $\theta = 90^\circ$: Zero flux
---
## **Maximum Flux**
Maximum flux occurs when the surface is **perpendicular** to the electric field, i.e., $\theta = 0^\circ$. In this case:
$
\Phi_E = E A \cos 0^\circ = E A
$
---
## **Zero Flux**
Zero flux occurs when the surface is **parallel** to the electric field, i.e., $\theta = 90^\circ$. In this case:
$
\Phi_E = E A \cos 90^\circ = 0
$
### **Diagram**
![[Pasted image 20241130095156.png]]
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## **Electric Flux Through a Closed Surface**
When considering a closed surface, such as a sphere of radius $r$ with a [[Charge]] $q$ at its center:
1. The surface is divided into small patches $\Delta A$, each contributing a small flux.
2. The total flux is the sum of the contributions from all patches:
$
\Phi_E = \sum \vec{E} \cdot \Delta \vec{A}
$
### Diagram
![[Pasted image 20241130095219.png]]
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### **Specific Cases**
#### a) Zero Flux
Flux through a closed surface is zero when:
- No field lines intercept the surface.
- The number of field lines entering equals the number leaving.
### Diagram
![[Pasted image 20241130095322.png]]
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#### b) Negative Flux
Flux is negative when more field lines enter the surface than leave it.
### Diagram
![[Pasted image 20241130095410.png]]
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#### c) Positive Flux
Flux is positive when more field lines leave the surface than enter it.
### Diagram
![[Pasted image 20241130095425.png]]
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## Summary
### Key Points:
1. **Definition**: Electric flux is the measure of electric field lines passing through a surface.
2. **Key Relationship**: $\Phi_E = \vec{E} \cdot \vec{A} = E A \cos \theta$.
3. **Maximum and Zero Flux**:
- **Maximum**: Surface perpendicular to field ($\theta = 0^\circ$).
- **Zero**: Surface parallel to field ($\theta = 90^\circ$).
| Condition | Flux Value | Orientation |
|-------------------------|----------------------------|-------------------------------|
| Maximum Flux | $\Phi_E = E A$ | Surface perpendicular to field |
| Zero Flux | $\Phi_E = 0$ | Surface parallel to field |
| Positive/Negative Flux | $\Phi_E = E A \cos \theta$ | Depends on orientation |
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### Real-World Application
Electric flux is crucial in understanding **[[Gauss's Law]]**, which relates the flux through a closed surface to the [[charge]] enclosed. This concept has practical applications in electrostatics, circuit design, and field distribution analysis.
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## References
