Final answer:
The electric flux through one side of a cube with a single point charge at its center is found by dividing the charge by six times the vacuum permittivity because the charge is evenly distributed across all six faces of the cube.
Step-by-step explanation:
The electric flux Φ through one side of a cube due to a single point charge q placed at its center can be deduced without the need for integration thanks to the symmetry of the cube and Gauss's Law. For a cube with a point charge at the center, the electric field is the same on each of the six faces and points radially outward from the charge. According to Gauss's Law, Φ = q/ε0, where q is the charge enclosed by the cube, and ε0 is the vacuum permittivity. Since the flux is evenly distributed among the six faces of the cube, the flux Φ through each face is q/(6ε0). For a charge of −4.9μC (−4.9 × 10−6 C), the flux through one face of the cube is:
Φ = q/(6ε0) = (−4.9 × 10−6 C)/(6ε0)