APPLICATIONS OF GAUSS’S LAW

Gauss’s law is valid for any distribution of charges and for any closed surface. Gauss’s law can be used in two ways. If we know the charge distribution, and if it has enough symmetry to let us evaluate the integral in Gauss’s law, we can find the field. Or if we know the field, we can use Gauss’s law to find the charge distribution, such as charges on conducting surfaces.

In this section we present examples of both kinds of applications. As you study them, watch for the role played by the symmetry properties of each system. We will use Gauss’s law to calculate the electric fields caused by several simple charge distributions; the results are collected in a table in the chapter summary. 

In practical problems we often encounter situations in which we want to know the electric field caused by a charge distribution on a conductor. These calculations are aided by the following remarkable fact: When excess charge is placed on a solid conductor and is at rest, it resides entirely on the surface, not in the interior of the material. (By excess we mean charges other than the ions and free electrons that make up the neutral conductor.) Here’s the proof. We know from Section 21.4 that in an electrostatic situation (with all charges at rest) the electric field E at every point in the interior of a conducting material is zero. If were not zero, the excess charges would move. Suppose we construct a Gaussian surface inside the conductor, such as surface in Fig. 1. Because E = 0 every where on this surface, Gauss’s law requires that the net charge inside the surface is zero. Now imagine shrinking the surface like a collapsing balloon until it encloses a region so small that we may consider it as a point then the charge at that point must be zero. We can do this anywhere inside the conductor, so there can be no excess charge at any point within a solid conductor; any excess charge must reside on the conductor’s surface. (This result is for a solid conductor. In the next section we’ll discuss what can happen if the conductor has cavities in its interior.) We will make use of this fact frequently in the examples that follow.

Fig.1: Under electrostatic conditions (charges not in motion), any excess charge on a solid conductor resides entirely on the conductor’s surface.

For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law. This is best understood by some examples.

1. Field of a charged conducting sphere
2. Field due to an infinitely long straight uniformly charged wire
3. Field of an infinite plane sheet of charge
4. Field between oppositely charged parallel conducting plates
5. Field of a uniformly charged sphere

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