Overview
This lecture introduces the concept of electric flux, explores its calculation using symmetry, and leads to a fundamental explanation of Gauss's Law.
Electric Flux Concept
- Electric flux measures how much electric field passes through a given area.
- It is calculated as the product of electric field strength (E), area (A), and the cosine of the angle (θ) between them: Flux = EAcosθ.
- The more perpendicular the electric field is to the surface, the greater the flux.
Mathematical Definition and Interpretation
- Flux can be found via the dot product: Flux = ∫E · dA, though basic problems use Flux = EAcosθ.
- When E is parallel to the area normal, cosθ = 1, making flux maximum; when perpendicular, flux is zero.
Example: Infinite Line of Charge
- The electric field a distance 'a' from an infinite line of charge points radially outward.
- Field strength at distance 'a': E = 2kλ/a, where λ is the charge per unit length.
- The field is symmetric at every point equidistant from the wire.
Calculating Flux with Symmetry
- Surrounding the wire with a cylinder (length L, radius a), the electric field is perpendicular to the surface everywhere.
- The lateral surface area is 2πaL; ends are ignored as field doesn't cross them.
- Total flux through the cylinder: Flux = E × Area = (2kλ/a)(2πaL) = 4πkλL.
- Since k = 1/(4πε₀), flux simplifies to Q/ε₀, where Q is the charge enclosed.
Gauss’s Law
- Gauss's Law states: The total flux through any closed surface equals the enclosed charge divided by ε₀.
- Most useful in cases of high symmetry: cylinders (lines), spheres (point or spherical charges).
Example: Point Charge and Spherical Gaussian Surface
- For a point charge at the center of a sphere (radius r), the electric field is uniform on the surface.
- Flux = E × 4πr² = Q/ε₀, so E = Q/(4πε₀r²).
- This agrees with Coulomb’s law for a point charge.
Spherical Symmetry Implications
- Electric field outside a spherically symmetric charge distribution (all within the Gaussian surface) is the same as for a point charge of equal total charge.
- The field only depends on the total enclosed charge, not its distribution, as long as symmetry is preserved.
Key Terms & Definitions
- Flux — amount of electric field passing through a surface, EAcosθ.
- Gauss's Law — total flux through a closed surface equals enclosed charge divided by ε₀: Φ = Q_enc/ε₀.
- Gaussian Surface — an imaginary closed surface used to exploit symmetry when applying Gauss's Law.
- Spherical Symmetry — same field value at all points equidistant from center; important for Gauss's Law.
Action Items / Next Steps
- Practice applying Gauss’s Law to symmetric cases (line, cylinder, sphere).
- Complete assigned homework problems on calculating electric flux and using Gaussian surfaces.