If a point charge $q$ is placed inside a cube (at the center), the electric flux comes out to be $q/\varepsilon_0$, which is same as that if the charge $q$ was placed at the center of a spherical shell.
The area vector for each infinitesimal area of the shell is parallel to the electric field vector, arising from the point charge, which makes the cosine of the dot product unity, which is understandable. But for the cube, the electric field vector is parallel to the area vector (of one face) at one point only, i.e., as we move away from centre of the face, the angle between area vector and electric field vector changes, i.e., they are no more parallel, still the flux remains the same?
To be precise, I guess, I am having some doubt about the angles between the electric field vector and the area vector for the cube.
Best Answer
Consider the flux through a tiny segment of a sphere. Since the electric field is parallel to the normal of the surface at all points, the flux is simply the electric field at that distance multiplied by the area of the element.
Now imagine tilting the top of the cone by an angle $\theta$ so that the corners still lie on the conical section, as seen below:
The area increases by a factor $\frac{1}{\cos\theta}$, however the electric field vector in the normal direction $E_n$ is decreased by a factor of $\cos\theta$. Therefore the flux through this surface is unchanged since flux is the product of the normal electric field component and the area.
Now imagine splitting the cube up into lots of these conical sections. Clearly the tilting of the top surfaces of these sections due to the fact it being a cube rather than a sphere does not affect the flux flowing through each area element. Therefore the total flux flowing through the cube is the same as a sphere.
Note that this was a simplified adaptation from a chapter of The Feynman Lectures on Physics which explains why the images do not quite match my explanations since I was just talking about the top surface of the conical section being tilted. Feynman explains the effect of the flux through a closed surface in a more complete way.