Overview
This lecture introduces the concept of electric flux, covering its definition, calculation, conceptual analogies, and applications, particularly through Gauss's law, with examples involving spheres and planes.
Electric Flux Concept
- Electric flux measures the flow of the electric field through a surface.
- For a small surface element dA with normal vector, flux dφ = E • dA (dot product).
- For the entire surface, total flux is the surface integral ∫ E • dA.
- Flux is a scalar and can be positive, negative, or zero.
- SI unit of electric flux is newton meter squared per coulomb (N·m²/C).
Flux Analogy and Calculation
- Analogy: Airflow through a surface depends on the angle between airflow direction and surface normal—similar for electric fields.
- When the field is parallel to the normal, flux is maximized (E·A); if perpendicular, flux is zero.
Flux Through Closed and Open Surfaces
- For open surfaces, the direction of the normal vector is arbitrary.
- For closed surfaces (like a sphere), the normal always points outward by convention.
- Total flux through a closed surface is ∮ E • dA, indicating net flow in or out.
Gauss's Law
- For a point charge inside a sphere, total flux = Q/ε₀, independent of the sphere’s size.
- Gauss's law: Net electric flux through any closed surface equals the net enclosed charge divided by ε₀ (∮ E • dA = Q_enclosed/ε₀).
- Gauss's law holds for any closed surface and any enclosed charge configuration.
- Practical calculation of E using Gauss's law requires symmetry: spherical, cylindrical, or planar.
Applications of Gauss's Law
Spherical Symmetry: Thin Shell
- For a uniformly charged thin sphere (shell):
- Electric field inside (r < R): E = 0 (due to symmetry).
- Electric field outside (r > R): E = Q/(4πε₀r²), directed radially.
- Result matches a point charge at the center for external points.
Planar Symmetry: Infinite Plane
- For an infinite, uniformly charged plane (surface charge density σ):
- Electric field magnitude: E = σ/(2ε₀), constant and independent of distance from the plane.
- Direction: Away from the plane if σ > 0, toward for σ < 0.
Parallel Plates (Capacitor Model)
- For two oppositely charged, parallel planes:
- Outside both plates: E = 0.
- Between plates: E = σ/ε₀, uniform and directed from positive to negative plate.
- Field near edges (fringe effect) is not uniform or zero.
Demonstrations
- The electric field near a large charged plane is nearly constant close to the surface, dropping off only when far away.
- The electric field from a sphere falls off as 1/r², shown by a rapid decrease in force with distance.
- No measurable electric field exists inside a uniformly charged closed conductor (hollow sphere).
- Between two parallel plates, the electric field is strong and nearly uniform; fringe fields appear at the edges.
Key Terms & Definitions
- Electric Flux (φ) — Measure of the electric field passing through a surface (∫ E • dA).
- dA — Infinitesimal area element vector with direction normal to the surface.
- Gauss’s Law — The total electric flux through a closed surface is Q_enclosed/ε₀.
- Surface Charge Density (σ) — Charge per unit area (C/m²).
- Symmetry — Uniformity in a system allowing simplification with Gauss’s law (spherical, cylindrical, planar).
Action Items / Next Steps
- Work out the electric field near a uniformly charged infinite plane using Gauss's law and symmetry arguments.
- Review the concepts of electric field for spherical and planar symmetry.
- Complete the assigned homework problems on Gauss’s law before the next class.