Overview
This lecture explains how to use Gauss's Law to find the electric field in systems with spherical, cylindrical, and planar symmetry by exploiting the symmetries to simplify calculations.
Symmetry in Charge Distributions
- Spherical symmetry: charge density depends only on distance from a center point.
- Cylindrical symmetry: charge density depends only on distance from the axis, not along or around it.
- Planar symmetry: charge density is uniform in a plane, independent of position in the plane.
Applying Gauss’s Law: General Steps
- Identify the symmetry of the charge distribution.
- Choose an appropriate Gaussian surface matching the symmetry.
- Calculate the electric flux through the surface.
- Find the enclosed charge within the surface.
- Solve for the electric field using Gauss’s Law.
Spherical Symmetry
- Electric field at distance r: always radial, depends on enclosed charge.
- For r ≥ R (outside sphere): (E = \frac{1}{4\pi\varepsilon_0} \frac{q_{tot}}{r^2}).
- For r < R (inside sphere): (E = \frac{1}{4\pi\varepsilon_0} \frac{q_{within\ r}}{r^2}).
- For a uniformly charged sphere, (q_{enc} = \frac{4}{3}\pi r^3 \rho_0) for r < R.
- Electric field inside increases linearly with r, outside falls off as (1/r^2)._
Non-Uniform Spherical Charge
- If (\rho(r) = a r^n): integrate to find (q_{enc}) using (q_{enc} = \int_0^r a r'^n 4\pi r'^2 dr').
- Outside: (E \propto 1/r^2), inside: (E \propto r^{n+1}).
Cylindrical Symmetry
- For r > R (outside cylinder): (E = \frac{\lambda_{enc}}{2\pi\varepsilon_0 r}), where (\lambda) is linear charge density.
- For r < R (inside): (E) determined by enclosed charge up to radius r.
- For an infinite shell: inside field is zero; outside depends on total charge per unit length._
Planar Symmetry
- For an infinite plane: (E = \frac{\sigma}{2\varepsilon_0} n^), where (\sigma) is surface charge density.
- Field is perpendicular to the plane and does not depend on distance from the plane.
Key Terms & Definitions
- Gauss’s Law — The total electric flux through a closed surface equals enclosed charge divided by (\varepsilon_0).
- Gaussian Surface — Imaginary surface used in Gauss’s Law to exploit symmetry.
- Permittivity of Free Space ((\varepsilon_0)) — Constant in Gauss’s Law.
- Electric Flux — Measure of electric field passing through a surface.
- Spherical/Cylindrical/Planar Symmetry — Types of symmetry simplifying Gauss’s Law application.
Action Items / Next Steps
- Practice identifying symmetries and choosing the appropriate Gaussian surface.
- Solve example problems for each symmetry type.
- Review calculations for enclosed charge in uniform and non-uniform distributions.