6.3 Applying Gauss’s Law

Electric fields exhibit fascinating symmetries that simplify complex calculations. By recognizing spherical, cylindrical, or planar symmetry in charge distributions, we can apply Gauss's law more efficiently to determine electric field strengths.

Gauss's law connects electric flux through a closed surface to the enclosed charge. This powerful tool, combined with symmetry, allows us to solve electric field problems that would otherwise be challenging using Coulomb's law alone.

Symmetry in Electric Field Systems

Symmetry types in electric fields

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• Spherical symmetry occurs when charge distribution is uniform and radially symmetric around a central point (point charge, uniformly charged sphere)
  • Electric field magnitude depends only on distance from the center
  • Electric field direction always radial, pointing towards or away from the center
• Cylindrical symmetry occurs when charge distribution is uniform and radially symmetric along the axis of a cylinder (infinite line of charge, uniformly charged cylinder)
  • Electric field magnitude depends only on distance from the cylinder's axis
  • Electric field direction always perpendicular to the cylinder's axis, pointing towards or away from the axis
• Planar symmetry occurs when charge distribution is uniform and symmetric across a plane (infinite sheet of charge, uniformly charged parallel plates)
  • Electric field magnitude constant at any given distance from the plane
  • Electric field direction always perpendicular to the plane, pointing towards or away from the plane

Identifying electric field symmetry

• Identify shape and distribution of charge(s) in the system
• Determine if charge distribution is uniform and symmetric
  • Spherical: Radially symmetric around a central point (point charge, uniformly charged sphere)
  • Cylindrical: Radially symmetric along the axis of a cylinder (infinite line of charge, uniformly charged cylinder)
  • Planar: Symmetric across a plane (infinite sheet of charge, uniformly charged parallel plates)

Applying Gauss's Law

Gauss's law for symmetrical charges

• Gauss's law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$
  • $\vec{E}$: Electric field (a vector field representing the force per unit charge)
  • $d\vec{A}$: Infinitesimal area element
  • $Q_{enc}$: Total charge enclosed by the Gaussian surface
  • $\epsilon_0$: Permittivity of free space ($8.85 \times 10^{-12} \frac{C^2}{N \cdot m^2}$)
• Choose Gaussian surface that exploits symmetry of charge distribution
  1. Spherical: Use concentric spherical surface
  2. Cylindrical: Use coaxial cylindrical surface
  3. Planar: Use parallel planar surface
• Simplify integral by taking advantage of symmetry
  • Spherical: $\vec{E}$ constant in magnitude and perpendicular to surface at all points
  • Cylindrical: $\vec{E}$ constant in magnitude and perpendicular to curved surface
  • Planar: $\vec{E}$ constant in magnitude and perpendicular to surface
• Solve for electric field magnitude using simplified integral
  • Spherical: $E = \frac{Q_{enc}}{4\pi r^2 \epsilon_0}$
  • Cylindrical: $E = \frac{\lambda}{2\pi r \epsilon_0}$ ($\lambda$ is linear charge density)
  • Planar: $E = \frac{\sigma}{2\epsilon_0}$ ($\sigma$ is surface charge density)

Electric Flux and Closed Surfaces

• Electric flux is the measure of the electric field passing through a given surface
• Gauss's law relates the electric flux through a closed surface to the enclosed charge
• The closed surface used in Gauss's law is called a Gaussian surface
• In electrostatics, the electric field and flux are time-independent
• The superposition principle allows for the addition of electric fields from multiple charges