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6: Gauss's Law

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    Flux is a general and broadly applicable concept in physics. However, in this chapter, we concentrate on the flux of the electric field. This allows us to introduce Gauss’s law, which is particularly useful for finding the electric fields of charge distributions exhibiting spatial symmetry. The main topics discussed here are

    1. Electric flux. We define electric flux for both open and closed surfaces.
    2. Gauss’s law. We derive Gauss’s law for an arbitrary charge distribution and examine the role of electric flux in Gauss’s law.
    3. Calculating electric fields with Gauss’s law. The main focus of this chapter is to explain how to use Gauss’s law to find the electric fields of spatially symmetrical charge distributions. We discuss the importance of choosing a Gaussian surface and provide examples involving the applications of Gauss’s law.
    4. Electric fields in conductors. Gauss’s law provides useful insight into the absence of electric fields in conducting materials.

    Gauss’s law gives us an elegantly simple way of finding the electric field, and, as you will see, it can be much easier to use than the integration method described in the previous chapter. However, there is a catch—Gauss’s law has a limitation in that, while always true, it can be readily applied only for charge distributions with certain symmetries.

    • 6.1: Prelude to Gauss's Law
      So far, we have found that the electrostatic field begins and ends at point charges and that the field of a point charge varies inversely with the square of the distance from that charge. These characteristics of the electrostatic field lead to an important mathematical relationship known as Gauss’s law. This law is named in honor of the extraordinary German mathematician and scientist Karl Friedrich Gauss.
    • 6.2: Electric Flux
      The electric flux through a surface is proportional to the number of field lines crossing that surface. Note that this means the magnitude is proportional to the portion of the field perpendicular to the area. The electric flux is obtained by evaluating the surface integral \[\Phi = \oint_S \vec{E} \cdot \hat{n} dA = \oint_S \vec{E} \cdot d\vec{A},\]  where the notation used here is for a closed surface S.
    • 6.3: Explaining Gauss’s Law
      if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. Now, what happens to the electric flux if there are some charges inside the enclosed volume? Gauss’s law gives a quantitative answer to this question. Gauss’s law relates the electric flux through a closed surface to the net charge within that surface.
    • 6.4: Applying Gauss’s Law
      For a charge distribution with certain spatial symmetries (spherical, cylindrical, and planar), we can find a Gaussian surface over which \(\vec{E} \cdot \hat{n} = E\),  where E is constant over the surface. The electric field is then determined with Gauss’s law.
    • 6.5: Conductors in Electrostatic Equilibrium
      The electric field inside a conductor vanishes. 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 given by \(E = \frac{\sigma}{\epsilon_0}\).
    • 6.A: Gauss's Law (Answers)
    • 6.E: Gauss's Law (Exercises)
    • 6.S: Gauss's Law (Summary)

    Thumbnail: Karl Friedrich Gauss (1777–1855) was a legendary mathematician of the nineteenth century. Although his major contributions were to the field of mathematics, he also did important work in physics and astronomy. (Public Domain; Christian Albrecht Jensen).

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