When a cube is inscribed in a sphere of radius r,the length L of a side of the cube is
) If a positive point charge Q is placed at the center of the spherical surface,the ratio of the electric flux Φ_sphere at the spherical surface to the flux Φ_cube at the surface of the cube is

Question

When a cube is inscribed in a sphere of radius r,the length L of a side of the cube is

) If a positive point charge Q is placed at the center of the spherical surface,the ratio of the electric flux Φ_sphere at the spherical surface to the flux Φ_cube at the surface of the cube is

Quizplus · Accepted Answer

Since the charge is at the center, the electric field lines will be radially outward and perpendicular to the surface of the sphere and the cube. Therefore, the electric flux through both surfaces will be proportional to the areas of the surfaces. 
Since the cube is inscribed in the sphere, the diameter of the sphere is equal to the diagonal of the cube, which is L√3. Thus, the radius of the sphere is r = (L/2)√3. The surface area of the sphere is 4πr^2 and the surface area of the cube is 6L^2. 
Thus, the ratio of the flux at the surface of the sphere to the flux at the surface of the cube is

Φ_sphere / Φ_cube = (4πr^2) / (6L^2) = (4π/3) (L/2)^2 / (L^2) = (2/3) π

This ratio is independent of the radius of the sphere and the length of the side of the cube, and depends only on the geometry of the problem. Therefore, the correct answer is (C) 1.

When a Cube Is Inscribed in a Sphere of Radius