Gauss's law relates the electric flux (PE through a closed surface to the total charge qencl enclosed by the surface: @E = fE'dA = [12:01. You can use Gauss's law to determine the charge enclosed inside a closed surface on which the electric eld 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 zaxis. In this case, the point has been extended to a line, namely, the zaxis, 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 'r' and 6 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. Three dimensions Consider a point charge q in three-dimensional space. Symmetry requires the electric field to point directly away from the charge in all directions. To find E (r) , the magnitude of the field at distance r 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 cos(0) = 1. The flux integral is therefore reduced to J E(r) dA = E(r) A(r) , where E (r) is the magnitude of the electric field on the Gaussian surface, and A (r) is the area of the surface. (Figure 1) Part A Determine the magnitude E (r) by applying Gauss's law. Express E (r) in terms of some or all of the variables/constants q, r, and co - View Available Hint(s) AEQ ? Er) = ATEr2 Submit Previous Answers X Incorrect; Try Again; 5 attempts remainingFigure 1 013 1.5,} The figure shows a po: the center of a sphere radiant vectors E comi square part of the surf shaded, and a normal ' of the sphere in the ca \fTwo dimensions 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 By symmetry, the electric field must point radially outward from the wire at each point; that is, the field lines lie in planes perpendicular to the wire. In solving for the magnitude of the radial electric field E (r) produced by a line charge with charge density ), one should use a cylindrical Gaussian surface whose axis is the line charge. The length of the cylindrical surface L should cancel out of the expression for E (r). Apply Gauss's law to this situation to find an expression for E (r). (Figure 2) Express E (r) in terms of some or all of the variables ), r, and any needed constants. View Available Hint(s) AEd ? Er) = Submit Previous Answers X Incorrect; Try Again; 3 attempts remaining