(a) Electric flux is a measure of the electric field passing through a given surface. It represents the total number of electric field lines passing through a surface and is given by the dot product of the electric field vector and the normal vector to the surface. Electric flux is defined mathematically as:
Φ = ∫E⋅dA
where Φ is the electric flux, E is the electric field vector, and dA is a differential area vector perpendicular to the surface.
Electric flux is a scalar quantity, meaning it has magnitude but no specific direction. The dot product in the formula ensures that the flux value is positive if the electric field and the normal vector have the same direction, and negative if they have opposite directions.
Now, let's use Gauss' law to obtain the expression for the electric flux through the square with a point charge above it. Gauss' law states that the electric flux through a closed surface is equal to the charge enclosed divided by the electric constant ε₀.
Φ = Q_enclosed / ε₀
In this case, the square does not form a closed surface. However, we can consider an imaginary cylindrical Gaussian surface with its curved surface enclosing the charge. The electric field is uniform and perpendicular to the Gaussian surface at every point on its curved surface. Therefore, the electric flux through the curved surface is constant, and we can express it as:
Φ = E * A
where E is the magnitude of the electric field and A is the area of the curved surface.
The charge enclosed by the Gaussian surface is the charge q, and the area of the curved surface is 2πrL, where r is the distance of the charge from the center of the square (d/2) and L is the length of the square's side (d). Therefore, the expression for the electric flux through the square is:
Φ = E * 2π(d/2)L
= πELd
(b) If the point charge is moved to a distance of 'd' from the center of the square and the side of the square is doubled, the electric flux through the square will remain the same. This is because the electric flux depends on the charge enclosed and the area of the surface, but it is independent of the distance of the charge from the surface or the size of the surface (assuming the charge remains within the surface).
Since the charge enclosed by the square remains the same, and the area of the square also doubles (assuming the side of the square is doubled), the change in the electric flux through the square is negligible. The expression for the electric flux through the square will remain as Φ = πELd.
