Suppose there are 2 cases:

A hollow cube with a point charge $+q$ at the center of one of the faces

A hollow cube with a point charge $+q$ at the midpoint of one of the edges.
In case 1, the total flux through the cube is $q/2\epsilon_0$ and in case 2 is $q/4\epsilon_0$.
How will I find the flux through the individual faces in both these cases? I am aware of the solid angle subtended by a rectangular surface formula, but the answer seems quite complex. Shouldn't it be a rather simple fraction through the faces?
Best Answer
The flux lines around a point charge are spherically symmetric, so the total flux through a surface is proportional to the solid angle subtended by the surface.
When you see questions like the one you cite they usually have some trick using symmetry. For example for the charge in the centre of the face the solid angle subtended by the whole cube is $2\pi$ and for the charge in the centre of an edge it's $\pi$. That makes it a simple fraction of the total flux $\Phi = Q/\varepsilon_0$.
However if you have a single face there is no easy way to calculate the solid angle subtended, and I'm afraid there is no alternative to grinding through the algebra.