Lec-Jan23

Gauss's Law

• Definition: Gauss's Law relates the net charge enclosed by a Gaussian surface to the electric flux through that surface.

• Key Equation: $ ext{Net charge inside g.s.} = ext{Net flux passing through g.s.}$

• Term Definition:

  • g.s. = Gaussian surface (a closed, imaginary mathematical surface).

Implications of Charge

• Charge Example: In a 2D E-field line diagram, if there's a charge of +1 C, consider what the net charge could be in the hidden region.

  • Clicker options:

    • A. +1C

    • B. +2C

    • C. -1C

    • D. -2C

Electric Flux Calculation

• Example: Verify Gauss’s Law using a Gaussian surface around a negative point charge.

• Consider geometrical simplicity based on the shape of g.s.

Agenda Overview (January 23, 2025)

• Important topics to cover include:

  • Review Gauss's Law

  • Milestones in using Gauss's Law and symmetry for different geometries:

    1. E-field outside spherical charge distribution (3D)

    2. E-field outside an infinite line of charge (1D)

    3. E-field outside an infinite plane of charge (2D)

Gauss’s Law Summary

• Gaussian Surface: Imaginary closed surface used in calculations.

• Applications within the chapter include deriving the E-field for symmetrical charge distributions and properties regarding conductors.

Key Concepts in Gauss’s Law

• Clicker Question Analysis:

  • A closed surface with no charges inside leads to:

    • A. Electric field is zero everywhere at surface.

    • B. No E-field lines crossing the surface.

    • C. Indeterminate conclusions.

Flux Calculation with a Uniform Electric Field

• Exercise: Determine the flux passing through a Gaussian surface subjected to a uniform electric field with magnitude $E_0$.

General Applications of Gauss’s Law

• Limitations: While Gauss's Law holds true universally, it may not provide practicality when calculating E-fields in non-symmetric charge distributions.

• Complex Example: Discuss a scenario yielding negative net flux, indicating complexity in application but reliability of the law for symmetric cases.

Example 1a: Electric Field Outside a Sphere

• Scenario: Calculate the electric field for a spherical shell with uniformly distributed charge Q, using Gauss's law.

  • Direction: Radially outward from the shell's center.

  • Formula: ( \oint g.s. E \cdot dA = \frac{q_{enc}}{\varepsilon_0} )
    - Integral evaluates to ( E = \frac{Q}{4\pi\varepsilon_0 r^2} ) for r > R.

Example 1b: Electric Field Inside a Sphere

• Inside Spherical Shell: No enclosed charge within a Gaussian surface very much inside the shell results in zero electric field.

• Simplification leads to using Gauss's law for a spherical shell with all charge on the surface.

Cylindrical Symmetry

• Application: Evaluating electric field due to an infinite line charge.

  • Select cylindrical Gaussian surface of radius r.

  • No flux through the endcaps; constant E-field along the walls.

  • Employs λ for charge density.

Useful Symmetries for Gauss's Law

• Types of Symmetries:

  1. 1D charge distribution (infinite)

  2. 2D charge distribution (infinite)

  3. 3D charge distribution (finite or infinite)

• Resulting E-Fields: Specific forms of electric field equations for each symmetry noted.

Conductors and Electric Fields

• Inside a conductor at equilibrium, the electric field (E) equals 0, ensuring charge resides on outer surfaces.

• This conclusion leads to implications regarding induced charges in cavities within conductors and highlights applications such as Faraday cages.