\subsection{Gauss's Law and Symmetry Reduction}

This subsection states Gauss's law and shows how symmetry can reduce a difficult flux integral to simple algebra when the charge distribution is highly symmetric.

\dfn{Gaussian surface and enclosed charge}{Let $S$ be any closed imaginary surface in space, and let $d\vec{A}$ denote an outward-pointing area element on that surface. The surface $S$ is called a \emph{Gaussian surface}. The \emph{enclosed charge} $q_{\mathrm{enc}}$ is the algebraic sum of all charges contained inside $S$. Charges outside $S$ can affect the electric field on the surface, but they do not contribute to $q_{\mathrm{enc}}$.}

\thm{Gauss's law and when symmetry makes it useful}{Let $S$ be any closed surface with outward area element $d\vec{A}$, and let $q_{\mathrm{enc}}$ be the net charge enclosed by $S$. Then Gauss's law states
\[
\oint_S \vec{E}\cdot d\vec{A}=\frac{q_{\mathrm{enc}}}{\varepsilon_0}.
\]
This law is always true. It becomes a practical method for solving for the electric field when the charge distribution has enough symmetry that one can choose a Gaussian surface for which the magnitude $E=|\vec{E}|$ is constant on each flux-contributing part of the surface and the angle between $\vec{E}$ and $d\vec{A}$ is everywhere $0^\circ$, $180^\circ$, or $90^\circ$. Then the flux integral reduces to algebraic terms such as $EA$, $-EA$, or $0$. Common useful cases are spherical, cylindrical, and planar symmetry.}

Gauss's law is one of Maxwell's four equations (in electrostatic form). Its analogue for magnetism is Gauss's law for magnetic fields, $\oint\vec{B}\cdot d\vec{A}=0$ (Section 12.5), which reflects the absence of magnetic monopoles.

\nt{Gauss's law is always true, but it is not always useful for finding $\vec{E}$. In a general asymmetric charge distribution, knowing only the total flux through a closed surface does not tell you the field at each point on that surface. Also, zero net enclosed charge implies zero \emph{net flux}, not necessarily zero field everywhere. The main strategy is therefore: first identify strong symmetry, then choose a Gaussian surface matched to that symmetry.}

\wc{Gauss's law is always true, not just for symmetric cases}{Gauss's law $\oint\vec{E}\cdot d\vec{A}=q_{\mathrm{enc}}/\varepsilon_0$ holds for \emph{any} charge distribution and \emph{any} closed surface. Symmetry only makes it \emph{useful} for calculating $\vec{E}$. Without symmetry, you know the total flux but cannot solve for the field at each point on the surface.}

\wc{Zero net enclosed charge does not mean zero field}{A Gaussian surface enclosing zero net charge has zero \emph{net flux}, but the electric field on the surface need not be zero. External charges can produce nonzero field on the surface, with field lines entering on one side and leaving on the other, yielding zero net flux. $\oint\vec{E}\cdot d\vec{A}=0$ does \emph{not} imply $\vec{E}=0$.}

\pf{Why symmetry reduces the integral}{Let a point charge $Q$ be at the center of a spherical Gaussian surface of radius $r$. By spherical symmetry, the electric field is radial and has the same magnitude $E(r)$ at every point on the sphere. The outward area element $d\vec{A}$ is also radial, so
\[
\vec{E}\cdot d\vec{A}=E(r)\,dA
\]
everywhere on the surface. Therefore,
\[
\oint_S \vec{E}\cdot d\vec{A}=E(r)\oint_S dA=E(r)(4\pi r^2).
\]
Since the enclosed charge is $q_{\mathrm{enc}}=Q$, Gauss's law gives
\[
E(r)(4\pi r^2)=\frac{Q}{\varepsilon_0},
\qquad
E(r)=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}.
\]
The law itself is general, but the symmetry is what allowed $E(r)$ to be pulled outside the integral.}

\qs{Worked AP-style problem}{A very long straight wire carries a uniform positive linear charge density
\[
\lambda=3.0\times 10^{-6}\,\mathrm{C/m}.
\]
Let point $P$ be at perpendicular distance
\[
r=0.20\,\mathrm{m}
\]
from the wire. Choose a cylindrical Gaussian surface of radius $r$ and length $L$ coaxial with the wire.

Find:
\begin{enumerate}[label=(\alph*)]
\item the enclosed charge $q_{\mathrm{enc}}$ for that Gaussian surface,
\item the electric flux through the curved side and through the two flat end caps, and
\item the magnitude and direction of the electric field at $P$.
\end{enumerate}}

\sol Let the cylinder have radius $r$ and length $L$. Because the wire has uniform linear charge density $\lambda$, the charge enclosed by the Gaussian surface is
\[
q_{\mathrm{enc}}=\lambda L.
\]

By cylindrical symmetry, the electric field due to the long wire points radially outward from the wire and has the same magnitude $E(r)$ everywhere on the curved side of the Gaussian cylinder. Let $\hat{s}$ denote the outward radial unit vector from the wire.

On the curved side, the area element $d\vec{A}$ also points radially outward, so $\vec{E}$ is parallel to $d\vec{A}$. Thus,
\[
\vec{E}\cdot d\vec{A}=E(r)\,dA
\]
on the curved surface.

On each flat end cap, the area element $d\vec{A}$ points along the axis of the wire, while $\vec{E}$ points perpendicular to that axis. Therefore,
\[
\vec{E}\cdot d\vec{A}=0
\]
on both end caps, so the flux through each end cap is zero.

The total flux is therefore entirely through the curved side:
\[
\oint_S \vec{E}\cdot d\vec{A}=E(r)\bigl(2\pi rL\bigr).
\]
Apply Gauss's law:
\[
E(r)(2\pi rL)=\frac{\lambda L}{\varepsilon_0}.
\]
Cancel $L$:
\[
E(r)=\frac{\lambda}{2\pi \varepsilon_0 r}.
\]
Now substitute $\lambda=3.0\times 10^{-6}\,\mathrm{C/m}$, $r=0.20\,\mathrm{m}$, and $\varepsilon_0=8.85\times 10^{-12}\,\mathrm{C^2/(N\,m^2)}$:
\[
E(r)=\frac{3.0\times 10^{-6}}{2\pi(8.85\times 10^{-12})(0.20)}\,\mathrm{N/C}.
\]
This gives
\[
E(r)=2.7\times 10^5\,\mathrm{N/C}.
\]

So the fluxes are
\[
\Phi_{\mathrm{curved}}=E(r)(2\pi rL)=\frac{\lambda L}{\varepsilon_0},
\qquad
\Phi_{\mathrm{cap\ 1}}=0,
\qquad
\Phi_{\mathrm{cap\ 2}}=0,
\]
and the electric field at $P$ is
\[
\vec{E}(P)=(2.7\times 10^5\,\mathrm{N/C})\hat{s},
\]
where $\hat{s}$ points radially away from the positively charged wire.
