Calculate the Magnitude of Electric Flux Through a Shaded Area on a Cube

To calculate the magnitude of the electric flux through the shaded area on a cube when a charge is placed at one of the corners, we can follow these steps:

1. Determine the Electric Field:

Electric field lines start from positive charges and end at negative charges. Since the charge is placed at a corner of the cube, it will create an electric field that points towards the center of the cube along the diagonal.

2. Calculate the Area of the Shaded Region:

Area of the shaded region is equal to the side length of the cube squared. Since the shaded region is a face of the cube, we can determine its area.

3. Use the Formula for Electric Flux:

The formula for electric flux is Φ = E * A * cos(θ), where E is the electric field, A is the area vector, and θ is the angle between them. In this case, since the electric field and the area vector are parallel, the angle is 0 degrees, and the formula simplifies to Φ = E * A.

4. Substitute Values and Calculate:

Finally, substitute the calculated electric field and area values into the formula to find the magnitude of the electric flux through the shaded area on the cube.