Q.

Gauss's Law states that the total electric flux through a closed surface is proportional to the net charge enclosed within that surface. Mathematically, it is expressed as:

 ΦE = ∮S E · dA = Qenc / ε0
 

Where:

• Φ_E is the total electric flux through the surface S,
• E is the electric field vector,
• dA is an infinitesimal area vector on the surface S,
• Q_enc is the total charge enclosed within the surface S,
• ε₀ is the permittivity of free space.

Explanation

• Electric Flux (Φ_E): It is a measure of the number of electric field lines passing through a given area.
• Closed Surface: A surface that completely encloses a region of space, such as a sphere or cube, with no gaps.
• Proportionality: Gauss's law states that the electric flux through the surface is directly proportional to the total charge inside the surface. This means that the flux does not depend on the shape or size of the surface but only on the amount of charge enclosed within the surface.

Importance of Gauss's Law in Electrostatics

1. Simplifies Complex Electric Fields: Gauss's law simplifies the calculation of electric fields for systems with symmetry (e.g., spherical, cylindrical, or planar symmetry). By choosing an appropriate Gaussian surface, we can easily calculate the electric field without having to solve complicated integrals.
2. Helps in Understanding Charge Distribution: Gauss's law allows us to understand how the electric field is distributed around charges. It helps in visualizing the electric flux and how charges interact with each other.
3. Foundation of Electrostatics: Gauss's law is one of the fundamental principles in electrostatics and is derived from Coulomb's law. It is the basis for the derivation of many other electrostatic laws like the electric field of a point charge, electric field inside conductors, and the potential due to a charge distribution.
4. Gauss's Law and Conductors: It helps in understanding the behavior of electric fields inside and outside conductors. For example, inside a conductor in electrostatic equilibrium, the electric field is zero, which is a direct consequence of Gauss's law.
5. Used in Maxwell's Equations: Gauss's law is one of the four Maxwell equations, which are the foundation of classical electromagnetism. These equations describe how electric and magnetic fields propagate and interact with matter.