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SECTION 24.3 • Application of Gauss’s Law to Various Charge Distributions 749 Example 24.8 A Plane of Charge Find the electric field due to an infinite plane of positive Solution By symmetry, E must be perpendicular to the plane and must have the same magnitude at all points E is away from positive charges indicates that the direction E on one side of the plane must be opposite its direc- tion on the other side, as shown in Figure 24.15. A gauss- each have an area A and are equidistant from the plane. E is parallel to the curved surface—and, therefore, perpendicular to d A everywhere on the surface—condition (3) is satisfied and there is no contri- E # 2EA. Noting that the total charge inside the surface is q in # 4 A, we use Gauss’s law and find that the total flux through the gaussian surface is leading to (24.8) Because the distance from each flat end of the cylinder to the plane does not appear in Equation 24.8, we conclude 0 at any distance from the plane. That is, the field is uniform everywhere. What If? Suppose we place two infinite planes of charge parallel to each other, one positively charged and the other E # 4 2/ 0 ! E # 2E
A # q
in / 0 # 4 A / 0 What If? What if the line segment in this example were not infinitely long? Answer If the line charge in this example were of finite E is not perpen- dicular to the cylindrical surface at all points—the field It is left for you to show (see Problem 29) that the electric field inside a uniformly charged rod of finite radius Gaussian surface + + + E dA ! r (a) E (b) Figure 24.14 (Example 24.7) (a) An infinite line of charge surrounded by a cylindrical gaussian surface concentric with the line. (b) An end view shows that the electric field at the cylindrical surface is constant in magnitude and perpendicular to the surface. Figure 24.15 (Example 24.8) A cylindrical gaussian surface penetrating an infinite plane of charge. The flux is EA through each end of the gaussian surface and zero through its curved surface. E + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + A Gaussian surface E |