A line charge is placed along the positive x-axis. Its linear charge density varies with distance x from the origin as = ₀ x , where ₀ is a positive constant. A hypothetical cube of side length a is positioned such that one of its corners is at the origin (0,0,0) and three of its edges lie along the positive x, y, and z axes. The total electric flux through the six faces of the cube is (Take ₀ as the permittivity of

Step-by-step solution

The portion of the line charge relevant to the cube lies along the x-axis from x = 0 to x = a . The total charge on this segment of the line is found by integrating the linear charge density: q_ segment = ₀^ a dx = ₀^ a ₀ x dx = ₀ [ x^2 2 ]₀^ a = ₀ a^2 2 This segment of the line charge lies exactly on the edge of the cube (the x-axis). The dihedral angle of the cube at this edge is 90^ , which is 1 4 of the full 360^ space surrounding the line. Therefore, the charge effectively enclosed by the cube is 1 4 of the ch