Rucete ✏ AP Physics C In a Nutshell
11. Gauss’s Law
This chapter introduces electric flux, Gauss’s law, applications of Gauss’s law to systems with planar, cylindrical, and spherical symmetry, and properties of conductors.
The Concept of Flux
• Flux is the amount of a field passing through a surface, calculated as Φ = F ⋅ A.
• For electric fields, flux represents the number of field lines through a surface.
• Over complex surfaces or varying fields, flux is computed by integrating dΦ = F ⋅ dA.
Gauss’s Law
• Gauss’s law states that the electric flux through a closed surface equals the enclosed charge divided by ε₀.
• It is a reformulation of Coulomb’s law and remains valid even for moving charges.
• Charges outside the surface contribute no net flux.
Solving Gauss’s Law Problems
• Gauss’s law is most useful for planar, cylindrical, or spherical symmetry.
• Select a gaussian surface matching the symmetry of the charge distribution.
• Symmetry ensures constant E over parts of the surface, simplifying the flux integral.
Examples of Gauss’s Law Applications
• For an infinite plane of charge, E = σ/2ε₀ and points perpendicular to the surface.
• For an infinite line of charge, E = λ/2πε₀r and points radially outward.
• For a spherical shell, E outside is like a point charge and E inside is zero.
Applications to Conductors
• The electric field inside a conductor in electrostatic equilibrium is zero.
• Any excess charge on a conductor resides on its surface.
• Gaussian surfaces entirely within a conductor enclose zero net charge.
Variation Examples with Conductors
• A charged spherical shell distributes its charge such that the inside field is zero, and outside behaves like a point charge.
• For hollow conductors with internal charges, induced charges arrange to maintain zero field inside the metal.
• Conductors are always at constant potential in electrostatic equilibrium.
Calculating Potential by Integration
• Potential difference is the negative integral of the electric field along a path.
• Standard reference: potential is taken to be zero at infinity unless otherwise specified.
• When crossing regions where E changes form, break the integral into parts accordingly.
Problem-Solving Techniques for Gauss’s Law
• Identify the symmetry: planar, cylindrical, or spherical.
• Choose a gaussian surface aligned with the symmetry.
• Evaluate the electric flux and enclosed charge to find the electric field.
• Check if E is continuous across boundaries unless infinite charge densities exist.
In a Nutshell
Gauss’s law provides a powerful alternative to Coulomb’s law for calculating electric fields in systems with high symmetry. By relating the electric flux through a surface to the enclosed charge, it simplifies complex problems, especially for spheres, cylinders, and planes. It also offers deep insights into the behavior of conductors and supports the connection between electric fields and potentials.