Animated Solution for Physics - Electrostatics: A point charge of $+12\mu \text{C}$ is at a distance $6\text{ cm}$ vertically above the centre of a square of side $12\text{ cm}$ as shown in figure. The magnitude of the electric flux through the square will be ......... $\times 10^3{\text{ N-m}}^2\text{ /C}$.

This problem is a classic example of how symmetry can turn a seemingly complex calculus problem into a simple mental exercise. Let's dive into the elegance of Gauss's Law and spatial reasoning.

We are given a square of side $12\text{ cm}$ and a point charge $q=+12\mu \text{C}$ located exactly $6\text{ cm}$ above its center. We need to find the electric flux passing through this square.

Our primary tool for finding electric flux is Gauss's Law, which states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space:

However, there is a catch. A square is an open surface. If we were to calculate the flux directly, we would have to evaluate a complex surface integral $\int \mathbf{E}\cdot d\mathbf{A}$, accounting for the varying distance and angle of the electric field at every point on the square. That sounds like a nightmare!

Instead of brute-force calculus, let's use geometry. Notice the specific numbers given in the problem: the side of the square is $12\text{ cm}$, and the height of the charge is $6\text{ cm}$. The height is exactly half the side length ($h=a/2$).

This is a massive hint! Imagine building a cube of side $12\text{ cm}$ using our given square as the bottom face. Because the charge is $6\text{ cm}$ above the center of the bottom face, it will sit perfectly at the geometric center of this newly constructed cube.

Now, we have a closed surface (the cube) enclosing our charge $q$. According to Gauss's Law, the total flux radiating outward through the entire cube is:

Because the charge is exactly at the center, the electric field lines radiate symmetrically in all directions. The cube has 6 identical square faces, and due to this perfect symmetry, the total flux is shared equally among all of them. Therefore, the flux through our single square face is simply one-sixth of the total flux:

Now, it's just a matter of plugging in the numbers. We know $q=12\times 10^{-6}\text{ C}$ and ${\epsilon }_0=8.854\times 10^{-12}{\text{ C}}^2/{\text{N-m}}^2$.

The question asks for the answer in the format of $...\times 10^3$. Let's adjust our scientific notation:

Rounding to the nearest integer as required by the format, we get $226$.

This problem beautifully illustrates how recognizing spatial symmetry can bypass tedious mathematics, a crucial skill for mastering physics!