1. Concept Overview
Gauss’s law gives a powerful connection between electric flux and electric charge. Instead of calculating electric fields using direct force formulas, Gauss’s law allows us to use symmetry to find fields quickly and cleanly. It becomes extremely helpful for spherical, cylindrical and planar charge distributions.
2. Definition
Gauss’s law states that the total electric flux through a closed surface is equal to the net charge enclosed by that surface divided by the permittivity of free space.
Mathematically:
\( \Phi_E = \oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{\text{enclosed}}}{\varepsilon_0} \)
3. Key Terms
Understanding the important terms makes the law easier to apply.
3.1. Closed Surface
A closed surface is one that completely encloses a volume—like a sphere, cube, or cylinder. Such a surface is often called a Gaussian surface.
3.2. Electric Flux
Flux measures how many electric field lines pass through a surface. Gauss’s law relates flux through a closed surface to enclosed charge.
3.3. Permittivity
\( \varepsilon_0 \) is the permittivity of free space, which determines how the electric field interacts with the medium. Its value is:
\( \varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 \text{N}^{-1} \text{m}^{-2} \).
4. Understanding the Law
Gauss’s law tells us something deeper than just a formula:
- If net flux through a closed surface is non-zero, there must be net charge inside.
- If net flux is zero, either there is no charge inside or equal positive and negative charges cancel.
- The electric field depends only on the charge inside the Gaussian surface, not on charges outside it.
5. Why Gauss’s Law is Useful
The true power of Gauss’s law appears when the electric field has symmetry. If the charge distribution has spherical, cylindrical or planar symmetry, we can pick a Gaussian surface where:
- The electric field has the same magnitude everywhere on the surface.
- The field is either parallel or perpendicular to the surface.
This simplifies the flux integral drastically and gives the field almost instantly.
6. Mathematical Application
The general form of the law is:
\( \oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{\text{enclosed}}}{\varepsilon_0} \).
6.1. Simplifying for Symmetry
When symmetry allows \( E \) to be constant over the surface, the integral reduces to:
\( E \oint dA = E \cdot A \).
So,
\( E A = \dfrac{Q_{\text{enclosed}}}{\varepsilon_0} \).
7. Examples (Conceptual)
7.1. Example 1: Charge Inside a Spherical Surface
A charge \( Q \) is placed at the centre of a spherical Gaussian surface of radius \( r \). The field is the same everywhere on the surface and radial. Using Gauss’s law:
\( E (4 \pi r^2) = \dfrac{Q}{\varepsilon_0} \).
This gives the familiar point-charge field:
\( E = \dfrac{1}{4 \pi \varepsilon_0} \dfrac{Q}{r^2} \).
7.2. Example 2: Flux Through a Closed Surface
If a closed surface encloses equal positive and negative charges, the net enclosed charge is zero. Gauss’s law then gives:
\( \Phi_E = 0 \).
This does not mean the field is zero everywhere, only that total flux is zero.
8. Physical Interpretation
Gauss’s law essentially says that electric field lines originate from positive charges and terminate on negative charges. A closed surface counts how many field lines enter versus leave it. If more lines leave than enter, positive charge is inside; if more lines enter than leave, negative charge is inside. This gives a deep geometric way to understand electric fields without relying on force calculations.