Gauss's Law and Electric Flux - Comprehensive Study Notes

Electric Flux

Electric flux is the scalar product of the electric field and the surface area vector perpendicular to the field
For a flat surface with uniform E and angle θ between E and the surface normal, the flux is Φ_E = E A cosθ
For a differential area element, dΦ_E = E · dA
Maximum flux occurs when the surface is perpendicular to the field; flux is zero when the surface is parallel to the field
If the field varies over the surface, ΦE = ∫S E · dA (surface integral) for the chosen surface
Units of electric flux: N·m^2/C
General relation for a small area element: dΦE = E dA cosφ (alternatively, ΦE = E · dA)

Gauss’s Law relates the total electric flux through a closed surface to the charge enclosed by that surface
Statement: ∮S E · dA = Qenclosed / ε0
The total flux through a closed surface can be due to charges inside or outside; Gauss’s Law counts only enclosed charge for the flux value
Gauss’s Law is most useful when charges have high symmetry (spherical, cylindrical, planar)
Gaussian surface is an imaginary construct you choose to exploit symmetry; it need not coincide with a real surface
Blade of intuition: if Q_enclosed = 0, net flux through the closed surface is zero

For an infinite plane with surface charge density σ, the electric field is uniform and does not depend on distance r from the plane
Magnitude on either side: E = σ / (2 ε0)
Direction: perpendicular to the plane, away from the plane for σ > 0, toward the plane for σ < 0
The field is the same on both sides (magnitude) but directed away from the plane on the side where σ is positive
Note: flux through a pillbox straddling the plane yields ΦE = ∮S E · dA = (σ/ε0) A for the pillbox area A

Consider a line with linear charge density λ
Using a cylindrical Gaussian surface of radius r and length L, symmetry gives E perpendicular to the line and constant on the curved surface
Magnitude: E(r) = λ / (2 π ε0 r)
Direction: radially outward (or inward for negative λ)
Flux through curved surface: Φ_E = E × (2π r L) = (λ L) / ε0 → per unit length, λ / ε0

Outside the sphere (r > R): E(r) = (1 / (4 π ε0)) × Q / r^2
Inside the sphere (r ≤ R): E(r) = (1 / (4 π ε0)) × Q r / R^3
Rationale: ρ = Q / [(4/3) π R^3] is uniform; enclosed charge scales with r^3, so E ∝ r inside

If Q is uniformly distributed in a volume of radius R, what is the ratio of E at r = 2R (outside) to E at r = R/4 (inside)?
Outside at r = 2R: E_out = (1 / (4 π ε0)) × Q / (2R)^2 = Q / (16 π ε0 R^2)
Inside at r = R/4: E_in = (1 / (4 π ε0)) × Q (R/4) / R^3 = Q / (16 π ε0 R^2)
Ratio: Eout / Ein = 1

Electrostatic equilibrium: no net motion of charge within the conductor; charges reside in static positions; charges experience zero net force inside
Inside a conductor, the electric field is zero: E = 0 everywhere
Isolated conductor with net charge qc: the charge resides on the surface; the field inside the conductor is zero
A conductor with a cavity and an internal charge: the inner surface of the cavity carries charge −q to cancel the field in the conductor; the outer surface carries the remainder so that the total charge on the outer surface equals qc + q (depending on the initial charge and cavity charge)

The field just outside a charged conductor is perpendicular to the surface and has magnitude E_out = σ / ε0 when the field inside is zero
Boundary condition at a conductor surface: Eout,⊥ − Ein,⊥ = σ/ε0; with Ein = 0, we get Eout,⊥ = σ/ε0
The surface charge density σ is related to the outward normal field just outside the surface
If the conductor carries no net charge, the outer surface carries zero net charge; with cavities and inner charges, induced charges adjust accordingly

Point charge q at a point: E(r) = q / (4 π ε0 r^2)
Point charge on the surface of a conducting sphere (radius R): outside sphere (r > R): E(r) = q / (4 π ε0 r^2); inside (r < R): E = 0
Infinite line with linear density λ: E(r) = λ / (2 π ε0 r)
Infinite conducting cylinder (radius R) with line density λ: outside (r > R): E(r) = λ / (2 π ε0 r); inside (r < R): E = 0
Solid insulating sphere with uniform charge Q and radius R: outside E = (1 / (4 π ε0)) Q / r^2; inside E = (1 / (4 π ε0)) Q r / R^3
Infinite sheet of charge with surface density σ: E = σ / (2 ε0) on each side; direction perpendicularly away from sheet for positive σ; field does not depend on distance from sheet
Between two large conducting plates with surface charges +σ and −σ: E between plates is E = σ / ε0; outside the field is typically zero for ideal parallel plates
Just outside a conductor: E_⊥ = σ/ε0; field is perpendicular to the surface

Gauss’s Law provides a bridge between electric flux and enclosed charge, and it formalizes how symmetry simplifies E-field calculations
The concept of a Gaussian surface helps translate a 3D field problem into a tractable flux integral using symmetry
The idea that conductors in electrostatic equilibrium have E = 0 inside and charge resides on surfaces is fundamental to shielding, capacitors, and electrostatic design
Charge distribution behavior in conductors with cavities demonstrates how local charges influence induced charges on inner vs outer surfaces

Flux through a surface: oxed{ \PhiE = \ointS \mathbf{E} \cdot d\mathbf{A} }
Gauss’s Law: \boxed{ \ointS \mathbf{E} \cdot d\mathbf{A} = \frac{Q{ ext{enc}}}{\varepsilon_0} }
Volume density: \boxed{ \rho = \frac{Q}{V} }
Surface density: \boxed{ \sigma = \frac{Q}{A} }
Linear density: \boxed{ \lambda = \frac{Q}{L} }
Point charge field: \boxed{ \mathbf{E} = \frac{q}{4\pi\varepsilon_0}\frac{\hat{r}}{r^2} }
Infinite plane: \boxed{ E = \frac{\sigma}{2\varepsilon_0} }\quad \text{(on each side, direction away from plane for } \sigma>0)
Infinite line charge: \boxed{ E(r) = \frac{\lambda}{2\pi\varepsilon_0\,r} }
Outside a uniformly charged sphere: \boxed{ E(r) = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2} } \text{ for } r > R
Inside a uniformly charged sphere: \boxed{ E(r) = \frac{1}{4\pi\varepsilon_0}\frac{Q\,r}{R^3} } \text{ for } r \le R
Field between two plates (capacitor-like): \boxed{ E = \frac{\sigma}{\varepsilon_0} }
Boundary condition at conductor surface: \boxed{ E{\ ext{out},\perp} - E{\text{in},\perp} = \frac{\sigma}{\varepsilon0} } with E{in} = 0 inside

Ex. 22.1: How to measure charge inside a box without opening it (Gaussian surface intuition; outward flux vs inward flux indicates positive vs negative charge inside)
Ex. 22.3: Electric flux through a sphere (using ∮ E · dA) and Gauss’s Law for enclosed charge
Ex. 22.5: Field by a conducting sphere (inside E = 0; outside behaves like a point charge)
Ex. 22.6: Field of an infinite line charge
Ex. 22.7: Field of an infinite plane sheet of charge
Ex. 22.9/22.10: Field of a uniformly charged sphere and other charge configurations
Ex. 22.x: Charges on conductors, cavities, and surface charges