## **Statement**
*Gauss's Law states*:
>*"The net [[Electric Flux]] through a closed surface is equal to the total [[Charge]] $q$ enclosed by the surface divided by the permittivity of free space $\varepsilon_0$."*
Mathematically,
$
\Phi_E = \frac{q}{\varepsilon_0}
$
---
## **Explanation**
To understand Gauss's law, consider a **spherical closed surface** of radius $r$ that encloses a point Charge $q$ at its center.
### Flux Through Small Areas
The entire surface is divided into $n$ small patches with areas:
$
\Delta A_1, \Delta A_2, \Delta A_3, \ldots, \Delta A_n
$
Since the electric field intensity is constant over the sphere (as all points are equidistant from the charge), the [[Electric Flux]] through each patch is:
$
\phi_i = \overrightarrow{E} \cdot \Delta \overrightarrow{A_i} = E \Delta A_i \cos \theta
$
For a spherical surface, $\theta = 0$, so $\cos \theta = 1$, giving:
$
\phi_i = E \Delta A_i
$
The total Electric flux through the surface is the sum of fluxes through all the patches:
$
\Phi_E = \phi_1 + \phi_2 + \ldots + \phi_n = \sum E \Delta A
$
Learn more about Electric field at [[Electric Field]]
---
### Total Flux for a Sphere
For a sphere, the electric field at any point is:
$
E = \frac{1}{4 \pi \varepsilon_0} \frac{q}{r^2}
$
Substituting into the total flux:
$
\Phi_E = \frac{1}{4 \pi \varepsilon_0} \frac{q}{r^2} \sum \Delta A
$
Since the total area of a sphere is $4 \pi r^2$, we have:
$
\sum \Delta A = 4 \pi r^2
$
Thus:
$
\Phi_E = \frac{1}{4 \pi \varepsilon_0} \frac{q}{r^2}(4 \pi r^2) = \frac{q}{\varepsilon_0}
$
This shows that the flux depends only on the enclosed charge $q$ and the medium's permittivity $\varepsilon_0$, **not on the surface's shape**.
---
## **Electric Flux Through Irregular Surfaces**
For an irregular closed surface enclosing multiple charges:
$
q_1, q_2, q_3, \ldots, q_n
$
The total flux is:
$
\Phi_E = \frac{q_1}{\varepsilon_0} + \frac{q_2}{\varepsilon_0} + \ldots + \frac{q_n}{\varepsilon_0}
$
Combining charges:
$
Q = q_1 + q_2 + \ldots + q_n
$
The total flux becomes:
$
\Phi_E = \frac{Q}{\varepsilon_0}
$
---
## Conclusion
Gauss's law establishes that:
1. The electric flux through any closed surface is proportional to the net charge enclosed.
2. The shape or geometry of the surface does not affect the flux, only the enclosed charge matters.
This law simplifies electric field calculations, especially for symmetric charge distributions.
---
# Summary
| **Concept** | **Explanation** |
|-----------------------------------|---------------------------------------------------------------------------------|
| **Electric Flux** ($\Phi_E$) | The total electric field passing through a closed surface. |
| **Gauss's Law** | $\Phi_E = \frac{q}{\varepsilon_0}$, relating flux to enclosed charge. |
| **Surface Independence** | Flux depends only on the charge enclosed, not the shape of the closed surface. |
**Key Insight**:
Gauss's law provides a powerful tool for solving problems involving electric fields, particularly in symmetrical setups like spheres, cylinders, or planes. Its real-world applications include understanding electric fields around conductors, capacitors, and charge distributions.
---
# References
