Lei de Gauss - Exemplo resolvido

Gauss's Law Two cylinders with radii of 3 cm and 6 cm are placed in a device. These cylinders are electrically charged with a linear charge density of 5.0 × 10−6 C/m for the first and −7.0 × 10−6 C/m for the second. The cylinders are concentric, that is, the smaller cylinder is positioned inside the larger one and both share the same axis. Based on the data, the electric fields located at radii of 4 cm and 8 cm from the center of the cylinders are, in N/C, respectively: a) −4.5 × 10⁵ and 2.25 × 10⁶. b) 4.5 × 10⁵ and −2.25 × 10⁶. c) 2.25 × 10⁵ and −4.5 × 10⁶. d) 2.25 × 10⁶ and −4.5 × 10⁵. e) 2.5 × 10⁶ and −2.5 × 10⁶ Gauss's law is the law that establishes the relationship between the flux of the electric field through a closed surface and the electric charge that exists within the volume limited by this surface. Gauss's law is one of Maxwell's four equations, along with Gauss's law of magnetism, Faraday's law of induction, and Ampère-Maxwell's law. It was formulated by Carl Friedrich Gauss in 1835, but was only published after 1867.[1] Gauss was a German mathematician who made important contributions to number theory, geometry, and probability, also having contributions in astronomy and in measuring the size and shape of the Earth.[2] Electric Field Flux Figure 1: Electric field lines "piercing" a surface, showing that there is electric field flux through the surface. Since the field lines are coming out of the surface, the electric field flux is positive. The electric field flux, {\displaystyle \Phi _{E}} {\displaystyle \Phi _{E}}, is a scalar quantity and can be considered as a measure of the number of field lines that cross the surface.[2][3] It is conventionally agreed that if there are more field lines coming out of the surface than going in, the electric field flux through the surface is positive and if there are more field lines going into the surface than going out of it, the flux is negative. Furthermore, it is important to note that if the number of field lines entering the surface is equal to the number of field lines exiting the surface, then the electric field flux through the surface is zero,[2][4] as can be seen in Figure 2. To obtain the electric field flux E through a closed surface where E is non-uniform, it is necessary to divide it into infinitesimal area elements dA. A vector dA is then defined whose magnitude is dA, the direction is perpendicular to the area element, and the sense is adopted as the sense of the normal to the infinitesimal element exiting the surface. Thus, these infinitesimal elements are so small that E can be considered constant at all points of the same area element.[2] Therefore, we can define the flux of E through a surface S as follows: