Overview
This lecture explains Gauss's Law, focusing on how symmetry allows for simplified calculations of electric fields, especially with spherical, cylindrical, and infinite planar charge distributions.
Gauss's Law and Symmetry
- Gauss's Law states that the electric flux out of a closed surface equals the enclosed charge divided by epsilon naught (ε₀).
- Symmetry in the charge distribution allows for easier calculations using Gauss's Law.
- Spherical symmetry applies to point charges or spheres of charge, allowing use of a spherical Gaussian surface.
- Cylindrical symmetry works for long, straight charged rods, using a cylindrical Gaussian surface.
- Planar symmetry is used for an infinite plane of charge, where a "can-shaped" Gaussian surface is effective.
Example: Infinite Plane of Charge
- Assume an infinite plane with surface charge density sigma (σ).
- Gaussian surface is a cylinder ("can") with its flat faces parallel to the plane, half above and half below.
- Electric field lines are perpendicular and uniform on each side due to infinite extent.
- Flux through the top face: E_z × area = E_z × πr².
- Flux through the bottom face is equal in magnitude, opposite due to surface orientation.
- Total flux through both faces: 2 × E_z × πr².
Flux and Enclosed Charge Calculation
- Enclosed charge: charge density (σ) times area (πr²), so Q_enclosed = σ × πr².
- By Gauss's Law: total flux = Q_enclosed / ε₀.
- 2 × E_z × πr² = σ × πr² / ε₀.
- Solving for the electric field: E = σ / (2ε₀).
- Electric field is constant and does not depend on distance from the plane.
Direction of Electric Field
- Field points away from the plane on both sides if σ is positive.
- If σ is negative, the field points toward the plane.
Key Terms & Definitions
- Gauss's Law — The total electric flux out of a closed surface equals enclosed charge divided by ε₀.
- Gaussian Surface — An imaginary closed surface chosen to exploit symmetry in electric field calculations.
- Surface charge density (σ) — Charge per unit area on a surface.
- Electric flux — Measure of the electric field passing through a surface.
Action Items / Next Steps
- Review homework on applying Gauss's Law to symmetric charge distributions.
- Read textbook section on electric fields from infinite planes and cylinders.