The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Gauss’s Law – Lecture 3 (University Physics II)

Overview

This lecture introduces electric flux and Gauss's law, the first of Maxwell's four equations and one of the central results of the course. After briefly closing out the topic of electric dipoles, the focus shifts to thinking of an electric field as something that "flows" through a surface, and to counting how much field passes through a chosen surface. Gauss's law states that the net flux through any closed surface depends only on the charge enclosed — not on the surface's shape or the charge's position inside it. Because we are free to pick the surface, this turns hard field calculations into easy ones whenever the charge distribution has symmetry. We use it to recover Coulomb's law, to find the fields of charged spheres, infinite lines, and infinite planes, and to explain why excess charge on a conductor lives entirely on its surface.

Key Concepts

Electric dipole (chapter wrap-up). A dipole is two equal and opposite charges held a fixed distance apart (e.g. a water molecule). Placed in a uniform field, the two opposite forces produce a torque that tends to align the dipole with the field.

• Dipole moment $\vec p$: a vector pointing from the negative to the positive charge, with magnitude $p = qd$ ($d$ = separation).
• Torque: $\vec\tau = \vec p \times \vec E$, magnitude $\tau = pE\sin\theta$.
• Potential energy: $U = -\vec p \cdot \vec E = -pE\cos\theta$ (minimum when aligned). The force comes from a cross product, the energy from the negative dot product.
• Field of a dipole on its axis: $E = \dfrac{2kpx}{(x^2-a^2)^2}$, which for $x \gg a$ falls off as $1/x^3$ — faster than a point charge's $1/r^2$.

Electric flux $\Phi_E$ — a measure of how much field passes through a surface.

• For a flat area in a uniform field: $\Phi_E = EA\cos\theta$, where $\theta$ is the angle between $\vec E$ and the area vector $\vec A$ (which points perpendicular to the surface). Maximum when the surface faces the field ($\theta = 0$); zero when the surface is parallel to the field ($\theta = 90^\circ$).
• For a general surface: $\Phi_E = \displaystyle\int \vec E \cdot d\vec A$, summing $E\cos\theta\,dA$ over infinitely many small flat patches.
• "Flow" is only an intuition — a static field does not actually move; the word flux (Latin for "flow") borrows the language of moving fluids.

Closed surfaces and sign of flux. For a closed surface the area vector always points outward.

• Positive enclosed charge → net outward flux (positive).
• Negative enclosed charge → net inward flux (negative).
• No enclosed charge → zero net flux (every field line entering also leaves).

Gauss's law. The net flux through any closed surface equals the enclosed charge over $\varepsilon_0$: $\oint \vec E \cdot d\vec A = \frac{Q_{\text{enc}}}{\varepsilon_0}$

• The circle on the integral signals a closed surface. $Q_{\text{enc}}$ is the algebraic sum of all charges inside.
• The result is independent of the surface's shape and of where the charge sits inside it — so we choose a clever imaginary Gaussian surface matching the symmetry. ($\varepsilon_0 \approx 8.85\times10^{-12}$; $k = \frac{1}{4\pi\varepsilon_0}$.)

Key results from Gauss's law.

• Coulomb's law follows directly: a Gaussian sphere of radius $r$ around a point charge gives $E = \dfrac{Q}{4\pi\varepsilon_0 r^2}$.
• Infinite line of charge (linear density $\lambda$), using a coaxial cylinder: $E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$ (falls off as $1/r$).
• Infinite charged plane (surface density $\sigma$), using a pillbox cylinder: $E = \dfrac{\sigma}{2\varepsilon_0}$ — constant everywhere.
• Conductor surface (field exits one side only): $E = \dfrac{\sigma}{\varepsilon_0}$ — twice the isolated-plane value.

Conductors in electrostatics.

• Inside a conductor the field is zero (otherwise charges would keep moving), so all excess charge sits on the surface.
• Outside a uniformly charged sphere (conducting or insulating), the field is identical to that of a point charge $Q$ at the center.
• Inside a conducting sphere: $E = 0$. Inside a uniformly charged insulating sphere (volume density $\rho$): $E = \dfrac{Qr}{4\pi\varepsilon_0 R^3}$, rising linearly from zero at the center.
• A cavity in a conductor stays field-free; placing a charge $+Q_i$ in the cavity induces $-Q_i$ on the cavity wall and an extra $+Q_i$ on the outer surface (the principle behind the Van de Graaff generator).
• A Faraday cage: a conducting enclosure shields its interior from external fields, since the field cannot penetrate the conductor.

Worked Example

Find the field a distance $r$ from an infinitely long line of charge with linear charge density $\lambda$.

Choose a coaxial cylinder of radius $r$ and length $L$ as the Gaussian surface. By symmetry $\vec E$ points radially outward, so no flux passes through the end caps — all flux exits the curved side, whose area is $2\pi r L$. The enclosed charge is $\lambda L$. Apply Gauss's law:

The length $L$ cancels, and the field falls off as $1/r$ — the same answer the Coulomb's-law integral gave, but with far less work.

Summary

• Electric flux measures field through a surface: $\Phi_E = \int \vec E \cdot d\vec A$, equal to $EA\cos\theta$ for a flat area in a uniform field.
• Gauss's law — $\oint \vec E \cdot d\vec A = Q_{\text{enc}}/\varepsilon_0$ — makes the net flux through a closed surface depend only on the enclosed charge, not the surface's shape or the charge's location.
• Choosing a Gaussian surface that matches the symmetry quickly yields the fields of point charges (recovering Coulomb's law), infinite lines ($\lambda/2\pi\varepsilon_0 r$), and infinite planes ($\sigma/2\varepsilon_0$).
• In electrostatics the field inside a conductor is zero, so all excess charge resides on the surface; outside, a uniform sphere acts like a point charge at its center.
• Cavities, the Van de Graaff generator, and Faraday cages all follow from "no field, hence no net charge, inside a conductor."
• Gauss's law is the first of Maxwell's four equations and a major milestone of the course.