Electric Flux & Gauss’s Law Explained | Chapter 22 of University Physics

Electric flux and Gauss’s Law provide a powerful framework for calculating electric fields in systems exhibiting symmetry. In this chapter summary, we explore how flux through a surface relates to enclosed charge, apply Gauss’s Law to key geometries, and examine electrostatic behavior in conductors.

Watch the full video summary here to see step-by-step derivations and visualizations of Gaussian surfaces.

Electric Flux

Electric flux (Φ_E) measures the “flow” of electric field through a surface. For a flat surface of area A in a uniform field:

Φ_E = E · A · cos(φ) or Φ_E = E ⋅ A. In general, for a curved or nonuniform field, flux is:

Φ_E = ∮ E ⋅ dA,

where positive flux indicates field lines exiting the surface and negative flux indicates entry.

Gauss’s Law

Gauss’s Law links electric flux through a closed surface to the net charge enclosed:

∮ E ⋅ dA = Q_enclosed / ε₀.

This law is equivalent to Coulomb’s Law but significantly simplifies field calculations when high symmetry allows choosing an appropriate Gaussian surface.

Applications of Gauss’s Law

Charged Conducting Sphere

Inside a conducting sphere (r < R), E = 0. Outside, the field behaves like a point charge:

E = (1 / 4πε₀) · (q / r²).

Infinite Line of Charge

Using a cylindrical Gaussian surface of radius r around a line with linear charge density λ:

E = (1 / 2πε₀) · (λ / r).

Infinite Sheet of Charge

For a uniformly charged infinite sheet with surface density σ:

E = σ / (2ε₀), perpendicular to the sheet on both sides.

Parallel Conducting Plates

Between two oppositely charged parallel plates:

E = σ / ε₀. Outside the plates, the fields from each plate cancel, yielding a near-zero net field.

Uniformly Charged Insulating Sphere

For an insulating sphere of radius R and total charge Q:

• Inside (r < R): E = (1 / 4πε₀) · (Q · r / R³)
• Outside (r ≥ R): E = (1 / 4πε₀) · (Q / r²)

Electrostatics in Conductors

In electrostatic equilibrium, the field inside a conductor is zero and any excess charge resides on its surface. For a hollow cavity, induced charges on inner and outer surfaces ensure zero field within the conductor’s material. Conductors act as Faraday cages, shielding interiors from external fields. The field just outside a charged conductor is:

E_⊥ = σ / ε₀, perpendicular to the surface.

Conclusion

Gauss’s Law provides an elegant method to determine electric fields for symmetric charge distributions and explains key electrostatic properties of conductors. Mastery of flux integrals and Gaussian surfaces is essential for tackling advanced problems in electromagnetism.

For visual demonstrations and guided problem solving, watch the video summary. Practice applying Gauss’s Law to diverse geometries to build intuition and proficiency.

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