101 lines
4.9 KiB
TeX
101 lines
4.9 KiB
TeX
\subsection{Gauss's Law and Symmetry Reduction}
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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.
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\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 $q_{\mathrm{enc}}$.}
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\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
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\[
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\oint_S \vec{E}\cdot d\vec{A}=\frac{q_{\mathrm{enc}}}{\varepsilon_0}.
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\]
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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.}
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\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.}
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\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
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\[
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\vec{E}\cdot d\vec{A}=E(r)\,dA
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\]
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everywhere on the surface. Therefore,
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\[
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\oint_S \vec{E}\cdot d\vec{A}=E(r)\oint_S dA=E(r)(4\pi r^2).
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\]
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Since the enclosed charge is $q_{\mathrm{enc}}=Q$, Gauss's law gives
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\[
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E(r)(4\pi r^2)=\frac{Q}{\varepsilon_0},
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\qquad
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E(r)=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}.
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\]
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The law itself is general, but the symmetry is what allowed $E(r)$ to be pulled outside the integral.}
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\qs{Worked AP-style problem}{A very long straight wire carries a uniform positive linear charge density
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\[
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\lambda=3.0\times 10^{-6}\,\mathrm{C/m}.
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\]
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Let point $P$ be at perpendicular distance
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\[
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r=0.20\,\mathrm{m}
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\]
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from the wire. Choose a cylindrical Gaussian surface of radius $r$ and length $L$ coaxial with the wire.
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Find:
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\begin{enumerate}[label=(\alph*)]
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\item the enclosed charge $q_{\mathrm{enc}}$ for that Gaussian surface,
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\item the electric flux through the curved side and through the two flat end caps, and
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\item the magnitude and direction of the electric field at $P$.
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\end{enumerate}}
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\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
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\[
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q_{\mathrm{enc}}=\lambda L.
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\]
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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.
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On the curved side, the area element $d\vec{A}$ also points radially outward, so $\vec{E}$ is parallel to $d\vec{A}$. Thus,
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\[
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\vec{E}\cdot d\vec{A}=E(r)\,dA
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\]
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on the curved surface.
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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,
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\[
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\vec{E}\cdot d\vec{A}=0
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\]
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on both end caps, so the flux through each end cap is zero.
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The total flux is therefore entirely through the curved side:
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\[
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\oint_S \vec{E}\cdot d\vec{A}=E(r)\bigl(2\pi rL\bigr).
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\]
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Apply Gauss's law:
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\[
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E(r)(2\pi rL)=\frac{\lambda L}{\varepsilon_0}.
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\]
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Cancel $L$:
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\[
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E(r)=\frac{\lambda}{2\pi \varepsilon_0 r}.
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\]
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Now substitute $\lambda=3.0\times 10^{-6}\,\mathrm{C/m}$, $r=0.20\,\mathrm{m}$, and $\varepsilon_0=8.85\times 10^{-12}\,\mathrm{C^2/(N\,m^2)}$:
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\[
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E(r)=\frac{3.0\times 10^{-6}}{2\pi(8.85\times 10^{-12})(0.20)}\,\mathrm{N/C}.
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\]
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This gives
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\[
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E(r)=2.7\times 10^5\,\mathrm{N/C}.
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\]
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So the fluxes are
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\[
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\Phi_{\mathrm{curved}}=E(r)(2\pi rL)=\frac{\lambda L}{\varepsilon_0},
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\qquad
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\Phi_{\mathrm{cap\ 1}}=0,
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\qquad
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\Phi_{\mathrm{cap\ 2}}=0,
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\]
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and the electric field at $P$ is
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\[
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\vec{E}(P)=(2.7\times 10^5\,\mathrm{N/C})\hat{s},
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\]
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where $\hat{s}$ points radially away from the positively charged wire.
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