HW #3 - Gauss' Law

# HW #3 - Gauss' Law - MasteringPhysics Assignment Print View...

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MasteringPhysics: Assignment Print View Gauss's Law in 3, 2, and 1 Dimension Gauss's law relates the electric flux through a closed surface to the total charge enclosed by the surface: . You can use Gauss's law to determine the charge enclosed inside a closed surface on which the electric field is known. However, Gauss's law is most frequently used to determine the electric field from a symmetric charge distribution. The simplest case in which Gauss's law can be used to determine the electric field is that in which the charge is localized at a point, a line, or a plane. When the charge is localized at a point, so that the electric field radiates in three-dimensional space, the Gaussian surface is a sphere, and computations can be done in spherical coordinates. Now consider extending all elements of the problem (charge, Gaussian surface, boundary conditions) infinitely along some direction, say along the z axis. In this case, the point has been extended to a line, namely, the z axis, and the resulting electric field has cylindrical symmetry. Consequently, the problem reduces to two dimensions, since the field varies only with x and y , or with and in cylindrical coordinates. A one-dimensional problem may be achieved by extending the problem uniformly in two directions. In this case, the point is extended to a plane, and consequently, it has planar symmetry. Consider a point charge in three-dimensional space. Symmetry requires the electric field to point directly away from the charge in all directions. To find , the magnitude of the field at distance from the charge, the logical Gaussian surface is a sphere centered at the charge. The electric field is normal to this surface, so the dot product of the electric field and an infinitesimal surface element involves . The flux integral is therefore reduced to , where is the magnitude of the electric field on the Gaussian surface, and is the area of the surface.

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MasteringPhysics: Assignment Print View Part A Determine the magnitude by applying Gauss's law. Part A.1 Part not displayed Express in terms of some or all of the variables/constants , , and . ANSWER: = Now consider the case that the charge has been extended along the z axis. This is generally called a line charge. The usual variable for a line charge density (charge per unit length) is , and it has units (in the SI system) of coulombs per meter. Part B
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HW #3 - Gauss' Law - MasteringPhysics Assignment Print View...

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