Electric Flux and Gauss's Law Physics Lesson 23 by Owen Borville 12.22.2025
The electric flux through a surface is proportional to the number of field lines crossing that surface, and the magnitude is proportional to the portion of the field perpendicular to the area. The electric flux is calculated using the surface integral. Φ = ∫E*ndA = ∫E*dA. Electric flux for an open field is Φ = E * A = θ.
For a uniform electric field, electric flux is Φ = E * A = EA cos θ
For an open surface, flux depends on the field and area, requiring integrals. For a closed surface, flux is simplified to Gauss's Law.
Gauss's law (1777-1855) shows the relationship of electric flux through a closed surface to the net charge within that surface: Φ = ∫sE*ndA = qenc/ε0, where qenc is the total charge inside the Gaussian surface S. All surfaces that include the same amount of charge have the same number of field lines crossing it, regardless of the shape or size of the surface, as long as the surfaces enclose the same amount of charge.
For a charge distribution with certain spatial symmetries (spherical, cylindrical, and planar), we can find a Gaussian surface over which E*n = E, where E is constant over the surface. The electric field is then determined by Gauss's law.
For spherical symmetry, the Gaussian surface is also a sphere, and Gauss's law simplifies to 4πr^2E = qenc/ε0.
For cylindrical symmetry, we use a cylindrical Gaussian surface where Gauss's law is 2πrLE = qenc/ε0.
For planar symmetry, the Gaussian surface is a box penetrating the plane, with two faces parallel to the plane and the remainder perpendicular, resulting in Gauss's law = 2AE = qenc/ε0.
The electric field inside a conductor disappears or becomes zero in electrostatic equilibrium, because any internal field would cause free charge electrons to move until they create an opposing field that cancels the original one. Therefore, conductors act as shields for electric fields. Any excess charge placed on a conductor resides entirely on the surface of the conductor. The electric field is perpendicular to the surface of a conductor everywhere on that surface. The magnitude of the electric field just above the surface of a conductor is E = σ/ε0.
The electric flux through a surface is proportional to the number of field lines crossing that surface, and the magnitude is proportional to the portion of the field perpendicular to the area. The electric flux is calculated using the surface integral. Φ = ∫E*ndA = ∫E*dA. Electric flux for an open field is Φ = E * A = θ.
For a uniform electric field, electric flux is Φ = E * A = EA cos θ
For an open surface, flux depends on the field and area, requiring integrals. For a closed surface, flux is simplified to Gauss's Law.
Gauss's law (1777-1855) shows the relationship of electric flux through a closed surface to the net charge within that surface: Φ = ∫sE*ndA = qenc/ε0, where qenc is the total charge inside the Gaussian surface S. All surfaces that include the same amount of charge have the same number of field lines crossing it, regardless of the shape or size of the surface, as long as the surfaces enclose the same amount of charge.
For a charge distribution with certain spatial symmetries (spherical, cylindrical, and planar), we can find a Gaussian surface over which E*n = E, where E is constant over the surface. The electric field is then determined by Gauss's law.
For spherical symmetry, the Gaussian surface is also a sphere, and Gauss's law simplifies to 4πr^2E = qenc/ε0.
For cylindrical symmetry, we use a cylindrical Gaussian surface where Gauss's law is 2πrLE = qenc/ε0.
For planar symmetry, the Gaussian surface is a box penetrating the plane, with two faces parallel to the plane and the remainder perpendicular, resulting in Gauss's law = 2AE = qenc/ε0.
The electric field inside a conductor disappears or becomes zero in electrostatic equilibrium, because any internal field would cause free charge electrons to move until they create an opposing field that cancels the original one. Therefore, conductors act as shields for electric fields. Any excess charge placed on a conductor resides entirely on the surface of the conductor. The electric field is perpendicular to the surface of a conductor everywhere on that surface. The magnitude of the electric field just above the surface of a conductor is E = σ/ε0.