Overview
This lecture explains how to calculate the electric field of a point charge using Gauss's law, emphasizing the process and logic behind the symmetry and surface selection.
Gauss's Law for a Point Charge
- Gauss's law states: ∮ E · dA = Q_enclosed / ε₀.
- A point charge (+q) creates an electric field that points radially outward in all directions.
- To use Gauss's law, choose a spherical Gaussian surface centered on the point charge.
Applying Symmetry and Surface Choice
- A sphere is chosen due to spherical symmetry; its radius is labeled "r."
- The area element dA on the sphere points radially outward and is perpendicular to the surface.
- Electric field E at every point on the sphere has the same magnitude due to symmetry (E is constant on the surface).
- E and dA are parallel, so the angle between them is 0°, meaning cos(0°) = 1.
Calculating the Flux Integral
- The left side of Gauss's law simplifies to E × (surface area of the sphere): E × 4πr².
- The right side is Q_enclosed / ε₀, which for a point charge is just q / ε₀.
- Setting these equal: E × 4πr² = q / ε₀.
- Solving for E gives: E = q / (4πε₀ r²).
Direction of the Electric Field
- The direction of E is radial, outward for a positive charge and inward for a negative charge.
- The final result: E = (1 / 4πε₀) × (q / r²) r̂, where r̂ is the radial unit vector.
Key Terms & Definitions
- Gauss's Law — The total electric flux out of a closed surface equals the charge enclosed divided by ε₀.
- Gaussian Surface — An imaginary closed surface used to apply Gauss's law.
- ε₀ (epsilon naught) — The permittivity of free space, a physical constant.
- Surface Area of a Sphere — 4πr² for a sphere of radius r.
- Radial Unit Vector (r̂) — A vector pointing directly outward from the center of the sphere.
Action Items / Next Steps
- Practice applying Gauss's law to different charge distributions.
- Review the concept of symmetry in choosing Gaussian surfaces.