Electric Flux, Gauss's Law, and Divergence Theorem

Michael Faraday's Experiment

An experiment conducted in 1837 using two concentric hollow spheres, where a positively charged inner sphere induced an equal magnitude of negative charge on the grounded outer sphere, demonstrating the concept of electric flux.

Electric Flux (Φ or Ψ)

A measure of the amount of electric field passing through a surface. Quantitatively, it is equal to the total charge enclosed within the surface (Φ = Q) in SI units.

Electric Flux Density (D)

The amount of electric flux per unit area. It is a vector field with units of Coulombs per meter squared (C/m²), and its direction is along the electric flux lines.

Relationship between Electric Flux Density (D) and Electric Field (E) in Free Space

D = ε₀E, where ε₀ is the permittivity of free space.

Gauss's Law

A generalization of Faraday's experiment stating that the total electric flux passing through any closed surface is equal to the total charge enclosed by that surface (∮s D ⋅ dS = Q_enclosed).

Gaussian Surface

An imaginary closed surface chosen to apply Gauss's law for calculating electric flux density, ideally satisfying conditions where D is either normal or tangential to the surface and constant where it's not zero.

Conditions for Applying Gauss's Law Easily

1) The electric flux density (D) is either normal or tangential to the closed surface. 2) On the portion of the closed surface where D is not zero, its magnitude is constant.

Electric Flux Density due to a Point Charge

D = Q / (4πr²) arhat.

Electric Flux Density due to a Uniform Line Charge

D = ρL / (2πρ) aρhat for a cylindrical Gaussian surface, where ρL is the line charge density.

Coaxial